As of: 2026-10-25 23:49 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/topics/science/

# Science & math: everything we have

319 events (234 after June 2026), 0 models, 95 videos, 140 posts. Latest event 25 October 2026.

Importance: * minor to ***** historic. Confidence says how well the event is sourced. Each event has its own page at https://postcutoff.com/e/<id>/.

## Key moments

- 2018-12-02: AlphaFold (v1) tops the CASP13 protein-structure prediction assessment (****) https://postcutoff.com/e/2018-12-02-alphafold-1-casp13/
- 2020-02-20: Deep learning discovers halicin, a structurally new broad-spectrum antibiotic (****) https://postcutoff.com/e/2020-02-20-halicin-ai-antibiotic/
- 2020-11-30: AlphaFold 2 solves protein structure prediction at CASP14 (*****) https://postcutoff.com/e/2020-11-30-alphafold-2/
- 2022-02-16: Deep reinforcement learning controls fusion plasma in the TCV tokamak (****) https://postcutoff.com/e/2022-02-16-deepmind-tokamak-plasma-control/
- 2022-10-05: AlphaTensor discovers faster matrix multiplication algorithms, beating Strassen's 1969 record for 4×4 mod 2 (****) https://postcutoff.com/e/2022-10-05-alphatensor-matrix-multiplication/
- 2023-07-11: RFdiffusion: diffusion models design new proteins that work in the lab (****) https://postcutoff.com/e/2023-07-11-rfdiffusion-protein-design/
- 2023-11-14: GraphCast: ML weather model beats the world's best physics-based 10-day forecast on 90% of targets (****) https://postcutoff.com/e/2023-11-14-graphcast-weather/
- 2023-11-29: GNoME predicts 2.2 million new crystals, 380,000 stable, but novelty and usefulness are disputed (****) https://postcutoff.com/e/2023-11-29-gnome-millions-of-materials/
- 2023-12-14: FunSearch: an LLM finds new cap-set constructions, the first LLM discovery in open maths (****) https://postcutoff.com/e/2023-12-14-funsearch-cap-sets/
- 2024-05-08: AlphaFold 3 predicts structures and interactions of all life's molecules (****) https://postcutoff.com/e/2024-05-08-alphafold-3/
- 2024-07-25: AlphaProof and AlphaGeometry 2 reach IMO silver-medal standard (****) https://postcutoff.com/e/2024-07-25-alphaproof-imo-silver/
- 2025-02-19: Google's AI co-scientist independently reproduces an unpublished superbug discovery in 48 hours (****) https://postcutoff.com/e/2025-02-19-google-ai-co-scientist/
- 2025-02-25: AI weather forecasting goes operational: ECMWF's AIFS (Feb 2025), then NOAA's AI models (Dec 2025) (****) https://postcutoff.com/e/2025-02-25-ecmwf-aifs-operational-ai-weather/
- 2025-03-12: Sakana's AI Scientist-v2 writes the first fully AI-generated paper to pass peer review (ICLR 2025 workshop) (****) https://postcutoff.com/e/2025-03-12-sakana-ai-scientist-v2-peer-review/
- 2025-05-14: AlphaEvolve: Gemini-powered agent discovers new algorithms (****) https://postcutoff.com/e/2025-05-14-alphaevolve/
- 2025-07-21: AI systems reach gold-medal level at the International Mathematical Olympiad (*****) https://postcutoff.com/e/2025-07-21-imo-gold-ai/
- 2025-08-14: Generative AI designs new antibiotics that kill drug-resistant gonorrhoea and MRSA (****) https://postcutoff.com/e/2025-08-14-mit-generative-ai-antibiotics/
- 2025-09-10: Math Inc's Gauss agent completes the Strong Prime Number Theorem formalisation in Lean in three weeks (****) https://postcutoff.com/e/2025-09-10-math-inc-gauss-strong-pnt/
- 2025-09-12: First AI-generated complete genomes: Evo models design viable bacteriophages that kill resistant E. coli (*****) https://postcutoff.com/e/2025-09-12-ai-generated-phage-genomes/
- 2025-12-08: Genuine AI-assisted solutions to Erdős problems begin: #124 (Aristotle), #1026 (48-hour human–AI collaboration) (****) https://postcutoff.com/e/2025-12-08-ai-erdos-problems-wave-late-2025/
- 2026-01-06: Erdős problem #728 solved near-autonomously by GPT-5.2 Pro and Harmonic's Aristotle, with a Lean proof (****) https://postcutoff.com/e/2026-01-06-erdos-728-gpt-5-2-aristotle/
- 2026-02-11: DeepMind's Aletheia agent and Gemini Deep Think report autonomous Erdős solutions and new physics and CS results (****) https://postcutoff.com/e/2026-02-11-deepmind-aletheia-deep-think-science/
- 2026-02-28: Donald Knuth's 'Claude's Cycles': Claude Opus 4.6 solves an open Hamiltonian-cycle problem ('Shock! Shock!') (****) https://postcutoff.com/e/2026-02-28-knuth-claudes-cycles/
- 2026-03-01: Math Inc's Gauss formalises Viazovska's sphere-packing proofs in dimensions 8 and 24, fixing errors in the originals (****) https://postcutoff.com/e/2026-03-01-gauss-sphere-packing-formalization/
- 2026-05-03: Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets; Tao co-authors the paper (****) https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/
- 2026-05-20: OpenAI model disproves Erdős's 80-year-old unit distance conjecture (*****) https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/
- 2026-05-27: Erdős–Szemerédi sum-product conjecture shown false over the reals; a GPT-5.5 Pro agent re-disproves it in 7 of 8 runs (****) https://postcutoff.com/e/2026-05-27-sum-product-conjecture-false-over-reals/
- 2026-07-17: GPT-5.6 Sol Ultra proves the 50-year-old cycle double cover conjecture (*****) https://postcutoff.com/e/2026-07-17-cycle-double-cover-conjecture-proved/
- 2026-07-20: Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3 (*****) https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/
- 2026-07-23: AI systems score a perfect 42/42 at IMO 2026, officially graded (*****) https://postcutoff.com/e/2026-07-23-imo-2026-ai-perfect-scores/
- 2026-07-27: Neurosurgery resident uses GPT-5.6 Sol to prove Crouzeix's conjecture in a 16-hour autonomous run (****) https://postcutoff.com/e/2026-07-27-crouzeix-conjecture-proved/
- 2026-08-01: OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs (*****) https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/
- 2026-08-03: Planar Schiffer and Pompeiu conjectures disproved by two independent groups; one proof has a Lean certificate written by GPT-5.6 (****) https://postcutoff.com/e/2026-08-03-schiffer-pompeiu-conjectures-disproved/
- 2026-08-04: Alexander Perry disproves the period-index conjecture (Colliot-Thélène, 2001); a flawed ChatGPT example was the starting point (****) https://postcutoff.com/e/2026-08-04-period-index-conjecture-false/
- 2026-08-05: Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro; Tao simplifies and formalises it (****) https://postcutoff.com/e/2026-08-05-sendov-conjecture-proved/
- 2026-08-10: Claude proves more than two-thirds of Riemann zeta zeros are simple and on the critical line (up from 41.6%) (*****) https://postcutoff.com/e/2026-08-10-claude-riemann-zeta-zeros-two-thirds/
- 2026-08-13: Banach's isometric conjecture (1932) completed in the real case with key steps from ChatGPT 5.5/5.6 Pro; complex and quaternionic cases follow five days later (****) https://postcutoff.com/e/2026-08-13-banach-isometric-conjecture-solved/
- 2026-08-18: Claude autonomously designs protein binders that work in the lab against 14 of 15 targets (****) https://postcutoff.com/e/2026-08-18-claude-protein-binders-wet-lab/
- 2026-08-19: Peking University preprint claims an AI-found disproof of the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics (****) https://postcutoff.com/e/2026-08-19-yau-tian-donaldson-cscK-disproof-claim/
- 2026-08-23: Claude-assisted construction claims a complex structure on the 6-sphere, answering Hopf's 1947 problem (pending verification) (*****) https://postcutoff.com/e/2026-08-23-hopf-problem-s6-complex-structure/
- 2026-08-26: GPT-5.6 improves the Erdős–Rankin / Ford–Green–Konyagin–Maynard–Tao bound for large prime gaps (****) https://postcutoff.com/e/2026-08-26-erdos-rankin-large-prime-gaps-improved/
- 2026-08-28: Matrix Spencer conjecture proved; authors credit GPT-5.6 Sol Pro with 'the heavy-lifting' for the key lemma (****) https://postcutoff.com/e/2026-08-28-matrix-spencer-conjecture-proved/
- 2026-08-30: GPT-6 Astra lowers the bounded prime gaps record from 246 to 186 (****) https://postcutoff.com/e/2026-08-30-bounded-prime-gaps-186/
- 2026-09-03: Claude-written Lean proof claims the dying percolation conjecture θ(p_c)=0 in every dimension (*****) https://postcutoff.com/e/2026-09-03-dying-percolation-theta-pc-zero/
- 2026-09-03: GPT-6 Astra proves the Erdős–Sós conjecture (1962) with a short counting argument; mathematicians race to simplify and extend it (*****) https://postcutoff.com/e/2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra/
- 2026-09-03: Pre-release GPT-6 Astra disproves Erdős's 'first serious problem' (1931, $500) and proves the rational-exponents conjecture, all Lean-verified, in Epoch's FrontierMath Erdős runs (*****) https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/
- 2026-09-03: GPT-6 Astra autonomously disproves Smale's mean value conjecture (constant 1) in Lean; HKU mathematicians build on it to show Smale's constant exceeds 1.1018 (****) https://postcutoff.com/e/2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra/
- 2026-09-03: Google DeepMind's WeatherNext 3 learns from live satellite data: hourly 5 km global forecasts and up to 60% better precipitation skill (****) https://postcutoff.com/e/2026-09-03-weathernext-3/
- 2026-09-04: Claude produces the first complete machine-checked proof of Fermat's Last Theorem in Lean, in 11 days (*****) https://postcutoff.com/e/2026-09-04-claude-formalizes-fermats-last-theorem/
- 2026-09-04: Reed–Solomon codes list-decoded up to capacity and proximity gaps settled, with GPT-5.6 Sol and ChatGPT 5.6 Pro (Brakensiek–Chen–Putterman–Zhang–Zheng; Jeronimo) (****) https://postcutoff.com/e/2026-09-04-reed-solomon-list-decoding-capacity-ai/
- 2026-09-06: Collatz: a positive fraction of integers provably reach 1 in 10.46 ln n steps; AI-agent proof (ProofAtlas + Codex) by Lech Mazur, Lean-formalized and re-formalized by Axiom (****) https://postcutoff.com/e/2026-09-06-collatz-positive-density-log-time-mazur-codex/
- 2026-09-06: Preprint credits the 'Odin' AI research agent with a positively curved metric on S³×S³, which would disprove Hopf's Euler-characteristic sign conjecture (unverified) (****) https://postcutoff.com/e/2026-09-06-odin-ai-agent-positive-curvature-s3xs3-hopf-sign-conjecture/
- 2026-09-07: Pre-release GPT-6 Astra disproves the Köthe conjecture (1930) with a Lean-verified counterexample (****) https://postcutoff.com/e/2026-09-07-koethe-conjecture-disproved/
- 2026-09-08: OpenAI claims a Millennium Prize problem: 10,000 AI agents prove forced Navier–Stokes blow-up; priority dispute erupts (*****) https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/
- 2026-09-09: Pierce–Birkhoff conjecture (1956) disproved by a multi-agent GPT + Claude harness; counterexamples Lean-verified (****) https://postcutoff.com/e/2026-09-09-pierce-birkhoff-conjecture-disproved/
- 2026-09-10: First Phase III trial of a generative-AI-discovered drug doses first patient (Insilico's rentosertib) (****) https://postcutoff.com/e/2026-09-10-insilico-rentosertib-phase-3/
- 2026-09-14: The k-server conjecture, the 'holy grail' of online algorithms, is proved at Oxford; ChatGPT 6 Astra generalised the authors' k = 3 proof to all k (****) https://postcutoff.com/e/2026-09-14-k-server-conjecture-proved/
- 2026-09-14: Sylvester's 1879 conjecture on sums of two cubes is fully proved: Burungale and Tian settle the last case (p ≡ 8 mod 9) after Hongbo Yin's proof of the 4, 7 cases; both later papers disclose AI use (****) https://postcutoff.com/e/2026-09-14-sylvester-conjecture-proved-burungale-tian/
- 2026-09-15: Erdős–Hajnal high-girth problem (Erdős #108) disproved with ChatGPT/Codex help and a Lean proof, via the Conjectures.io bounty; experts sharpen it with GPT-6 Astra (****) https://postcutoff.com/e/2026-09-15-erdos-108-high-girth-problem-disproved-conjectures-io/
- 2026-09-16: Chvátal's 1972 conjecture proved (Chang–Liu–Liu, ChatGPT-assisted), then a GPT-6 Astra 'proof from The Book' and a Codex-built Lean formalization (****) https://postcutoff.com/e/2026-09-16-chvatal-conjecture-proved/
- 2026-09-17: ζ(5) proved irrational: Aabir Fauzan's Zenodo preprint, the first such result since Apéry's ζ(3) in 1978, is formally verified in Lean within a week, one formalization written by Claude (*****) https://postcutoff.com/e/2026-09-17-zeta-5-irrational-fauzan-lean-verified/
- 2026-09-21: Grad's 1967 conjecture on 3D plasma equilibria falls: two AI-assisted papers give three families of counterexamples (*****) https://postcutoff.com/e/2026-09-21-grad-conjecture-counterexamples/
- 2026-09-21: Courtade–Kumar 'most informative Boolean function' conjecture (2013) proved three times in two days, all with AI: Ky & Tran (ChatGPT), Google + CUHK (Gemini, Lean-verified end-to-end), Mahdavifar & Beirami (****) https://postcutoff.com/e/2026-09-21-courtade-kumar-conjecture-proved/
- 2026-09-22: Mathematician posts an unchecked ChatGPT Astra proof of the planar Mumford–Shah conjecture (1989), citing OpenAI's '100 open problems' claim (****) https://postcutoff.com/e/2026-09-22-mumford-shah-conjecture-astra-claim/
- 2026-09-22: Odlyzko–Poonen conjecture (1993) proved unconditionally: 'The proofs are due to GPT-6 Astra', which also wrote a 22,000-line Lean formalisation (****) https://postcutoff.com/e/2026-09-22-odlyzko-poonen-conjecture-proved/
- 2026-09-23: Claude agents discover a novel CRISPR-like enzyme system; Anthropic reveals its own biology wet lab (****) https://postcutoff.com/e/2026-09-23-claude-discovers-novel-enzyme-system/
- 2026-09-25: Claude (Fable 5.1 in Claude Science) computes the nine-loop six-gluon amplitude in planar N=4 super-Yang-Mills, answering a physicist's public challenge (****) https://postcutoff.com/e/2026-09-25-claude-nine-loop-amplitude-n4-sym/
- 2026-09-27: Erdős Problem #993 (1987): independent-set sequences of all forests claimed unimodal; a second proof and its Lean 4 formalization are attributed entirely to AI agents (Zhang, Li, Vallier) (****) https://postcutoff.com/e/2026-09-27-erdos-993-forest-independence-polynomials-unimodal/
- 2026-09-27: Perelman's proof of the Poincaré conjecture formalised in Lean, with AI-generated code (Chow, Qin, Liao, Khaitan) (****) https://postcutoff.com/e/2026-09-27-poincare-conjecture-lean-formalization/
- 2026-09-29: List Total Colouring Conjecture (late 1990s) disproved; counterexample found by ChatGPT 6 Astra Ultra 'with little input' (****) https://postcutoff.com/e/2026-09-29-list-total-colouring-conjecture-false-astra/
- 2026-09-30: Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue) (****) https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/
- 2026-09-30: Anonymous 'AI-generated research' account mt0-svg posts Lean-checked solutions to the Tammes problem for 15 points, Robbins' V > 2 and the Hoffman–Johnson–Junge frog-model conjecture (****) https://postcutoff.com/e/2026-09-30-mt0-svg-ai-generated-tammes-15-robbins-frog-model-lean/
- 2026-10-01: Lehmer's 1965 permutation conjecture (a research problem in Knuth's TAOCP) proved; Claude Opus 5.5 found the short hypercube proof (****) https://postcutoff.com/e/2026-10-01-lehmer-permutation-conjecture-opus-5-5/
- 2026-10-02: Kannan–Lovász–Simonovits (KLS) conjecture proved in two AI-assisted preprints: Song–Zhang's O(1) bound, then Bizeul–Klartag–Lehec ('most proofs … found by ChatGPT') (*****) https://postcutoff.com/e/2026-10-02-kls-conjecture-proved-ai-assisted/
- 2026-10-02: Klartag and Moshe prove the ε-Dvoretzky conjecture (polynomial dependence on ε); the key probabilistic idea came from a ChatGPT discussion (****) https://postcutoff.com/e/2026-10-02-epsilon-dvoretzky-conjecture-klartag-moshe/
- 2026-10-04: Linear Hadwiger conjecture proved: K_t-minor-free graphs are O(t)-colourable; proof found by GPT-6 Astra (Norin & Steiner), Lean-verified by Codex (*****) https://postcutoff.com/e/2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra/
- 2026-10-05: Claude discovers an algorithm that refutes the 3SUM and APSP hypotheses: truly subquadratic 3SUM and truly subcubic APSP (Alman & Vassilevska Williams, Lean-verified) (*****) https://postcutoff.com/e/2026-10-05-claude-refutes-3sum-apsp-hypotheses/
- 2026-10-05: Google DeepMind's AlphaProtein Novo designs new-to-nature enzymes from scratch (piperidine synthesis, DEHP degradation), with Frances Arnold's lab (****) https://postcutoff.com/e/2026-10-05-deepmind-alphaprotein-novo-enzyme-design/
- 2026-10-06: Artin–Davenport conjecture proved: every integral cubic form in 10+ variables has a nontrivial zero (Bonolis, Browning, Glas, Wang; key lemma via ChatGPT 6.0, Codex Lean auto-formalisation) (*****) https://postcutoff.com/e/2026-10-06-artin-davenport-cubic-forms-ten-variables/
- 2026-10-06: OpenAI model claims the free group factor problem: L(F₂) ≅ L(F₃), so all free group factors are isomorphic (Lean-formalized) (*****) https://postcutoff.com/e/2026-10-06-free-group-factors-isomorphism-openai/
- 2026-10-06: OpenAI release claims the rational Hodge conjecture for all CM abelian varieties, which would give the Tate conjecture for abelian varieties over finite fields (no Lean proof) (*****) https://postcutoff.com/e/2026-10-06-hodge-conjecture-cm-abelian-varieties-openai/
- 2026-10-06: OpenAI releases 722 AI-written math manuscripts (372 result families) claiming hundreds of open problems, incl. quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors (*****) https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/
- 2026-10-06: OpenAI model claims the quasi-Riemann hypothesis: no zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized) (*****) https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/
- 2026-10-06: OpenAI model claims a proof of Khot's Unique Games Conjecture, with Lean formalization (*****) https://postcutoff.com/e/2026-10-06-unique-games-conjecture-openai/
- 2026-10-06: Chen, Tserunyan & Tucker-Drob solve the Jackson–Kechris–Louveau 'finite index over treeable' problem (2002); ChatGPT 6 Astra Ultra supplied a 'crucial observation' (****) https://postcutoff.com/e/2026-10-06-finite-index-over-treeable-jkl-problem/
- 2026-10-06: Woodin's question answered: GCH below a strongly compact cardinal implies GCH everywhere (Zhixing You, 4-page proof with GPT-6 Astra) (****) https://postcutoff.com/e/2026-10-06-gch-above-strongly-compact-woodin-question/
- 2026-10-06: OpenAI model claims the irrationality exponent of π is exactly 2 (Lean-formalized), which implies the Flint–Hills series converges (****) https://postcutoff.com/e/2026-10-06-irrationality-exponent-pi-openai/
- 2026-10-06: OpenAI's math release claims the optimal approximation for metric k-median: a polynomial-time (1 + 2/e + ε) algorithm matching the known hardness, with a Lean formalization (****) https://postcutoff.com/e/2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e/
- 2026-10-06: OpenAI's maths release also claims the 3D Fourier restriction, Bochner–Riesz and local smoothing conjectures (families 077–080): the core of Stein's harmonic-analysis programme in three dimensions, manuscript only (****) https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/
- 2026-10-07: Outside researchers using AI crowdsource a tighter exponent in OpenAI's integer-multiplication result #109, from 2^-182 to about 2^-8.4 (conditional) (****) https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/
- 2026-10-07: Exact overlaps conjecture for self-similar measures on the line proved: GPT-6 Astra found the first proof (Kittle & Kogler), 47,000-line Lean formalization; OpenAI's catalogue claims it independently (****) https://postcutoff.com/e/2026-10-07-exact-overlaps-conjecture-gpt-6-astra/
- 2026-10-07: Two claimed proofs of the LeBrun–Salamon conjecture: Hu, Liu & Wan (arXiv, ChatGPT as 'auxiliary tool') and OpenAI's catalogue (family 062) (****) https://postcutoff.com/e/2026-10-07-lebrun-salamon-conjecture-proof/
- 2026-10-07: Braverman & He refute the 2004 undirected multiple-unicast (network coding) conjecture with a GPT-6-found counterexample on PG(2, 9) (****) https://postcutoff.com/e/2026-10-07-multiple-unicast-conjecture-false-gpt-6/
- 2026-10-07: Toms–Winter conjecture on C*-algebras completed by two preprints in one week; both credit AI with a key lemma (Thiel; Evington, Hua, Schafhauser, Seth, Tikuisis, White) (****) https://postcutoff.com/e/2026-10-07-toms-winter-conjecture-completed-chatgpt-astra-lemma/
- 2026-10-08: Google DeepMind's AlphaProof Nexus paper published in Science: LLM + Lean agent resolved 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures (****) https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/
- 2026-10-08: GPT-6 Astra finds a 3SAT reduction showing monotone Dualization (minimal hypergraph transversals) has no polynomial algorithm under ETH (****) https://postcutoff.com/e/2026-10-08-dualization-eth-lower-bound-gpt-6-astra/
- 2026-10-08: Engel & Mauri post a 7-page proof of the hyperkähler SYZ conjecture, two days after OpenAI's catalogue claimed it; AI used only to proofread (****) https://postcutoff.com/e/2026-10-08-hyperkahler-syz-conjecture-engel-mauri/
- 2026-10-09: Thurston's geometrization conjecture formalised in Lean: the Hamilton–Perelman proof machine-checked with GPT-6, Claude and DeepSeek agents (Chow's group) (****) https://postcutoff.com/e/2026-10-09-geometrization-conjecture-lean-formalization/
- 2026-10-09: "We need ABP": six PDE specialists publish GPT-6 Astra's proofs of the kinetic Krylov–Safonov theorem and of Pucci's 1966 conjecture, with full chats (Loher, Mooney, Mouhot, Niebel, Silvestre, Zhu) (****) https://postcutoff.com/e/2026-10-09-kinetic-krylov-safonov-pucci-conjecture-gpt-6-astra/
- 2026-10-09: Smale's sixth problem claimed for five bodies in the plane: finitely many central configurations for all positive masses, AI-found and proved in Lean (Tomar) (****) https://postcutoff.com/e/2026-10-09-smale-sixth-problem-five-bodies-lean-tomar/
- 2026-10-09: Terence Tao's Caltech lecture 'Math 2.0': AI has ended 'proof scarcity', and blind problem-solving is now 'actively harmful' to mathematics (****) https://postcutoff.com/e/2026-10-09-tao-math-2-0-caltech-lecture/
- 2026-10-10: Vaught's conjecture fails for infinitary logic: Ackerman, Freer and Knight post a Lean-verified counterexample, built over 8 months with Claude Code and Codex (****) https://postcutoff.com/e/2026-10-10-vaught-conjecture-infinitary-counterexample-lean/
- 2026-10-11: William Banks writes out OpenAI's Landau–Siegel proof with explicit constants: (1 − β) log q ≥ 0.003 for every real character, and says he verified it in full (****) https://postcutoff.com/e/2026-10-11-landau-siegel-zeros-banks-explicit-account/
- 2026-10-12: Bloch constant conjecture (Ahlfors–Grunsky, 1937) disproved: Teng Zhang shows B < 0.47186… − 1.3×10⁻⁹, with a Lean proof generated by Codex (****) https://postcutoff.com/e/2026-10-12-bloch-constant-conjecture-disproved/
- 2026-10-12: Cureton and Thorner, with AxiomProver, extend the quasi-Riemann zero-free half-plane to every Dedekind zeta function (Re s > 31/32, Lean-formalized) (****) https://postcutoff.com/e/2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover/
- 2026-10-12: Erdős Matching Conjecture (1965) proved for every fixed k and all large s, via a sharp tail bound whose k ≥ 6 cases generative AI completed (****) https://postcutoff.com/e/2026-10-12-erdos-matching-conjecture-large-matchings-ai/
- 2026-10-13: Eight extremal combinatorialists write up GPT-6 Astra's proof of the Erdős–Simonovits rational exponents conjecture (Conlon, Janzer, Liu et al.) (****) https://postcutoff.com/e/2026-10-13-rational-exponents-conjecture-proof-paper-conlon/
- 2026-10-13: Thompson's group F is nonamenable: specialists Sang-hyun Kim and Yash Lodha publish a human-written version of OpenAI's proof (****) https://postcutoff.com/e/2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition/
- 2026-10-13: Ulam's question answered: every Lie group is a group of permutations of a countable set (Monod & Thom; key construction from GPT-6 Astra) (****) https://postcutoff.com/e/2026-10-13-ulam-problem-lie-groups-countably-representable-gpt-6-astra/
- 2026-10-14: Li, Li & Liu: external-memory integer sorting beats the comparison bound, answering Aggarwal–Vitter (1988) by way of network coding, with ChatGPT/Codex and Lean (****) https://postcutoff.com/e/2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding/
- 2026-10-14: Lü, Wang & Wu claim µ(p+2) = o(π(x)), a model case of the parity problem open since at least 1989, building on an OpenAI preprint with ChatGPT's help (****) https://postcutoff.com/e/2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu/
- 2026-10-14: Bray, Dolmen, Khuri & Schmidt prove the spacetime Penrose inequality for 'maximally parameterized' data with Claude's help, and say OpenAI's Oct 6 Penrose paper appears to finish their program (****) https://postcutoff.com/e/2026-10-14-penrose-conjecture-bray-khuri-claude-openai-provenance/
- 2026-10-15: Exact asymptotic rate of Delsarte's linear programming bound for binary and spherical codes found, with key steps from Claude Opus 5.5 and GPT-5.6 Sol (****) https://postcutoff.com/e/2026-10-15-delsarte-lp-bound-exact-rate-salmon-opus/
- 2026-10-15: Hopf's Euler characteristic conjecture proved under T³ symmetry; the first proof came from ChatGPT-6-Astra 'with minimal human input' (****) https://postcutoff.com/e/2026-10-15-hopf-conjecture-t3-symmetry-chatgpt-astra/
- 2026-10-15: All 16,453,493 imaginary quadratic fields with class number ≤ 2000 determined unconditionally, using OpenAI's Siegel-zero proof made explicit in Lean with Claude Opus 5.5 (****) https://postcutoff.com/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/
- 2026-10-16: Brennan's 1978 conjecture: Smirnov, Binder, Ivrii and Prause check OpenAI's AI-generated proof and publish a much shorter proof of a stronger result (*****) https://postcutoff.com/e/2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov/
- 2026-10-16: New Gauss circle problem bound E(X) = O(X^(209/670+ε)) breaks Huxley's 5/16 threshold; authors used ChatGPT and Codex on proof strategies and checking (****) https://postcutoff.com/e/2026-10-16-gauss-circle-problem-209-670-bound/
- 2026-10-16: Lonely Runner Conjecture: Hu and Jing get the first constant-factor gain over the union bound, (1+c)/2n, after 'many rounds of discussion' with ChatGPT 6 Pro and Astra (****) https://postcutoff.com/e/2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt/
- 2026-10-16: Maximum number of Hamiltonian cycles in a tournament determined asymptotically, (cosh 1/cos 1)·(n−1)!/2ⁿ, answering Alon and proving Wormald's conjecture, with proofs 'originally found by GPT-5.6' (****) https://postcutoff.com/e/2026-10-16-max-hamiltonian-cycles-tournaments-gpt-5-6-wormald/
- 2026-10-16: OpenAI's AI-generated proof of a 2D entanglement area law fully formalized in Lean by Axiom (AxiomProver tooling + Codex), with no sorry (****) https://postcutoff.com/e/2026-10-16-openai-2d-area-law-lean-formalized-axiom/
- 2026-10-16: Smith–Toda complex V(4) exists for all primes p ≥ 11: Angeltveit extends and simplifies OpenAI's AI-written p = 1009 construction, with ChatGPT's help (****) https://postcutoff.com/e/2026-10-16-smith-toda-v4-angeltveit-improves-openai/
- 2026-10-16: Alpöge and Sussman say Claude has an (unreadable) existence argument for 4D Yang–Mills on ℝ⁴, by finishing Bałaban's 1980s program (****) https://postcutoff.com/e/2026-10-16-yang-mills-claude-balaban-program-alpoge-sussman/
- 2026-10-17: Erdős–Hajnal conjecture proved for every path P_t: Tung H. Nguyen, with ChatGPT-6 Astra proving key lemmas; a DeepMind internal model reached the same result independently, and Claude Opus 5.5 manuscripts claim P6 and all paths (*****) https://postcutoff.com/e/2026-10-17-erdos-hajnal-all-paths-nguyen-astra-deepmind/
- 2026-10-17: A non-sofic word-hyperbolic group: Sandhu and Thom build one from OpenAI's non-residually-finite hyperbolic group, starting from an AI-generated construction; Douglas–Mghirbi give an independent AI-assisted proof with Lean (****) https://postcutoff.com/e/2026-10-17-non-sofic-hyperbolic-group-sandhu-thom/
- 2026-10-19: Erdős–Szemerédi sunflower conjecture proved for every k ≥ 3; the key vector bound (Theorem 2) was found by ChatGPT 6 Astra Pro (Bucić, He, Lee) (*****) https://postcutoff.com/e/2026-10-19-erdos-szemeredi-sunflower-conjecture-gpt-6-astra-pro/
- 2026-10-19: Integer programming in standard form in N^O(d) time, improving Eisenbrand–Weismantel's N^O(d²): Zhejiang team's ideas, GPT Astra Ultra's proofs (Chen, Mao, Zhang) (****) https://postcutoff.com/e/2026-10-19-integer-programming-linear-exponent-gpt-astra-ultra/
- 2026-10-19: Kaplansky's quasitrace conjecture reported disproved by an unreleased OpenAI model; experts reproduce it and refute the monotone completeness conjecture (****) https://postcutoff.com/e/2026-10-19-quasitrace-monotone-completeness-counterexamples-openai/
- 2026-10-20: Double-cap conjecture (Witsenhausen 1974, Kalai) claimed in all dimensions, with a separate AI-built Lean proof (****) https://postcutoff.com/e/2026-10-20-double-cap-conjecture-witsenhausen-lean-ai/
- 2026-10-20: MDS conjecture settled in even characteristic: two independent proofs, one checked in Lean with Claude's help, the other written with ChatGPT (****) https://postcutoff.com/e/2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean/
- 2026-10-21: Nagata's 1959 conjecture: algebraic geometer Jenia Tevelev writes out OpenAI's 'remarkably simple' proof and calls the release 'a stunning achievement' (*****) https://postcutoff.com/e/2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition/
- 2026-10-21: Barnette's 1969 Hamiltonicity conjecture: graph theorist Zhentao Li publishes a digested, partly re-proved version of OpenAI's proof (****) https://postcutoff.com/e/2026-10-21-barnette-conjecture-openai-proof-zhentao-li-exposition/
- 2026-10-21: Hochster's 1973 small Cohen–Macaulay conjecture: an OpenAI manuscript's counterexample is surveyed, re-proved and extended to every dimension by Cristian Anghel (****) https://postcutoff.com/e/2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey/
- 2026-10-22: Eleven probabilists construct the Critical 2D KPZ process, a solution of the critical two-dimensional KPZ equation, 'through extensive interactions' with ChatGPT 5.5, 5.6 and 6 (****) https://postcutoff.com/e/2026-10-22-critical-2d-kpz-process-chatgpt/
- 2026-10-22: Barry Simon's L² conjecture (Problem 8 of his 2000 list) disproved: Becker and Yang build a counterexample largely found with ChatGPT (****) https://postcutoff.com/e/2026-10-22-simon-l2-conjecture-disproved-chatgpt/

## Timeline, oldest first

### 2018-12-02: AlphaFold (v1) tops the CASP13 protein-structure prediction assessment

****; DeepMind; Science & math; confidence medium; https://postcutoff.com/e/2018-12-02-alphafold-1-casp13/

DeepMind's first AlphaFold ranked first in the CASP13 blind assessment of protein structure prediction, an early sign that deep learning could crack the protein folding problem.

- CASP13 results announced December 2018
- Predicted inter-residue distances with a deep network, then optimized structures
- Nature paper published January 2020
- Precursor to AlphaFold 2, which essentially solved single-chain structure prediction at CASP14 (2020)

#### What happened
AlphaFold placed first overall among ~100 groups in the free-modeling category of CASP13.

#### Why it matters
Marked AI's entry into a grand challenge of biology and set up the 2020 AlphaFold 2 breakthrough.

Sources: [Improved protein structure prediction using potentials from deep learning (Nature, DOI)](https://doi.org/10.1038/s41586-019-1923-7); [Wikipedia: AlphaFold](https://en.wikipedia.org/wiki/AlphaFold)

### 2020-02-20: Deep learning discovers halicin, a structurally new broad-spectrum antibiotic

****; MIT, Broad Institute; Science & math; confidence high; https://postcutoff.com/e/2020-02-20-halicin-ai-antibiotic/

MIT's Collins and Barzilay labs (Cell, Feb 2020) trained a message-passing neural network on ~2,300 molecules. It identified halicin, a diabetes drug candidate, as a potent antibiotic that killed M. tuberculosis, carbapenem-resistant Enterobacteriaceae and pan-resistant A. baumannii, and cleared infections in mice.

- Published in Cell on 20 Feb 2020
- Screened >107 million molecules from ZINC15 in silico; of 23 top predictions tested, 8 were antibacterial
- Halicin treated C. difficile and pan-resistant A. baumannii infections in mice
- Structurally distant from known antibiotics; preclinical only

#### What happened
A graph neural network trained on growth-inhibition data screened the Drug Repurposing Hub and then 100M+ molecules, surfacing halicin and other candidates confirmed in the lab.

#### Why it matters
It launched the modern field of AI antibiotic discovery at a time of stalled antibiotic pipelines.

Sources: [A Deep Learning Approach to Antibiotic Discovery (Cell)](https://www.cell.com/cell/fulltext/S0092-8674(20)30102-1); [PubMed record](https://pubmed.ncbi.nlm.nih.gov/32084340/); [Chemistry World: AI tool screens 107 million molecules, discovers potent new antibiotics](https://www.chemistryworld.com/news/ai-tool-screens-107-million-molecules-discovers-potent-new-antibiotics/4011233.article)

### 2020-07-08: Liverpool's mobile robot chemist runs 688 experiments in 8 days and finds a 6× better photocatalyst

***; University of Liverpool; Science & math; confidence high; https://postcutoff.com/e/2020-07-08-mobile-robotic-chemist/

Andrew Cooper's group (Nature, July 2020) built a mobile robot that moved around a standard lab and ran 688 experiments over 8 days in a 10-variable space, guided by batched Bayesian optimisation. It found photocatalyst formulations about 6× more active for hydrogen production from water than the starting mixtures.

- 688 experiments, 8 days, 10-dimensional search space
- ~6× improvement in hydrogen-evolution activity
- Operated autonomously, including nights and weekends

#### What happened
A humanoid-sized mobile robot used ordinary lab instruments and an optimisation algorithm to choose and run experiments by itself.

#### Why it matters
It is a landmark for self-driving laboratories, the physical half of the "AI scientist" vision.

Sources: [A mobile robotic chemist (Nature)](https://www.nature.com/articles/s41586-020-2442-2); [C&EN: Robot runs almost 700 chemistry experiments](https://cen.acs.org/physical-chemistry/computational-chemistry/Robot-runs-almost-700-chemistry/98/i27)

### 2020-11-30: AlphaFold 2 solves protein structure prediction at CASP14

*****; DeepMind; Science & math; confidence high; https://postcutoff.com/e/2020-11-30-alphafold-2/

AlphaFold 2 achieved a median GDT score of 92.4 at CASP14, accuracy competitive with experimental methods, widely seen as solving the 50-year-old protein folding problem for single chains.

- CASP14 results announced 30 November 2020
- Median GDT of 92.4 across all targets
- Nature paper and open-source code published July 2021
- AlphaFold Protein Structure Database (with EMBL-EBI) launched July 2021; expanded to 200M+ structures in 2022
- Led to the 2024 Nobel Prize in Chemistry for Hassabis and Jumper

#### What happened
Using an attention-based architecture (Evoformer) trained on known structures, AlphaFold 2 predicted 3D protein structures from amino-acid sequence with near-experimental accuracy.

#### Why it matters
The clearest case of AI producing a major scientific breakthrough; used by millions of researchers and recognized with a Nobel Prize.

Sources: [AlphaFold: a solution to a 50-year-old grand challenge in biology (DeepMind)](https://deepmind.google/discover/blog/alphafold-a-solution-to-a-50-year-old-grand-challenge-in-biology/); [Highly accurate protein structure prediction with AlphaFold (Nature, DOI)](https://doi.org/10.1038/s41586-021-03819-2); [AlphaFold Protein Structure Database](https://alphafold.ebi.ac.uk/)

### 2021-04-29: Adam Zsolt Wagner uses reinforcement learning to find counterexamples to open graph-theory conjectures

**; Adam Zsolt Wagner; Science & math; confidence high; https://postcutoff.com/e/2021-04-29-wagner-rl-counterexamples/

Wagner's 'Constructions in combinatorics via neural networks' (arXiv 2104.14516) used a simple cross-entropy RL method to find explicit counterexamples to several published conjectures in extremal combinatorics and spectral graph theory.

- arXiv 2104.14516 (29 Apr 2021)
- Refuted several conjectures about graph eigenvalues and a Brualdi–Cao question on permanents of pattern-avoiding matrices
- Small neural network plus deep cross-entropy method; no LLM
- Wagner later joined Google DeepMind and co-authored the 2025 AlphaEvolve maths paper with Tao

#### What happened
A lone mathematician showed that off-the-shelf RL could disprove conjectures by searching for graphs that violate them.

#### Why it matters
It was the template for the 2023–2026 wave of AI counterexample finding (FunSearch, AlphaEvolve, PatternBoost, LLM counterexamples).

Sources: [Constructions in combinatorics via neural networks (arXiv 2104.14516)](https://arxiv.org/abs/2104.14516); [Reimplementation and extension (arXiv 2403.18429)](https://arxiv.org/abs/2403.18429)

### 2021-11-22: NASA's ExoMiner deep-learning model validates 301 new exoplanets from Kepler data

**; NASA Ames Research Center; Science & math; confidence high; https://postcutoff.com/e/2021-11-22-exominer-301-exoplanets/

NASA's ExoMiner neural network statistically validated 301 Kepler planet candidates as real planets in one batch, bringing the validated count to 4,569 (Astrophysical Journal, 2021).

- 301 new validated planets
- Explainable classifier mimicking the vetting steps of human experts

#### What happened
ExoMiner vetted thousands of Kepler signals and confidently validated hundreds as planets.

#### Why it matters
AI vetting has become standard for the flood of survey data from Kepler, TESS and, soon, other surveys.

Sources: [ExoMiner paper (arXiv 2111.10009)](https://arxiv.org/abs/2111.10009); [NASA JPL: new deep learning method adds 301 planets to Kepler's total count](https://www.jpl.nasa.gov/news/new-deep-learning-method-adds-301-planets-to-keplers-total-count/)

### 2021-12-01: DeepMind and mathematicians use machine learning to guide new theorems in knot theory and representation theory

***; DeepMind, University of Oxford, University of Sydney; Science & math; confidence high; https://postcutoff.com/e/2021-12-01-deepmind-knot-theory-intuition/

Davies et al. (Nature, Dec 2021) used supervised learning plus attribution to point mathematicians to hidden relationships. That led to a new theorem linking the knot signature to hyperbolic geometry, and to progress on the combinatorial invariance conjecture for Kazhdan–Lusztig polynomials.

- Nature 600:70–74 (2021)
- Knot theory: new relation between signature and the 'natural slope' (Lackenby, Juhász); follow-up in Geometry & Topology (2024)
- Representation theory: progress towards the combinatorial invariance conjecture (Williamson)
- Humans stated and proved the theorems; ML highlighted which features mattered

#### What happened
DeepMind trained models to predict one mathematical quantity from others, then used attribution to show which inputs mattered, prompting expert mathematicians to formulate and prove new results.

#### Why it matters
It was the first Nature-level demonstration of AI contributing to pure maths research, as an intuition aid rather than a prover.

Sources: [Advancing mathematics by guiding human intuition with AI (Nature)](https://www.nature.com/articles/s41586-021-04086-x); [Critical review of the paper (arXiv 2112.04324)](https://arxiv.org/abs/2112.04324)

### 2022-02-16: Deep reinforcement learning controls fusion plasma in the TCV tokamak

****; DeepMind, EPFL Swiss Plasma Center; Science & math; confidence high; https://postcutoff.com/e/2022-02-16-deepmind-tokamak-plasma-control/

DeepMind and EPFL (Nature, Feb 2022) trained a single deep-RL policy in simulation that commanded all of TCV's magnetic control coils on the real machine. It produced and held elongated, negative-triangularity and 'snowflake' plasmas, and even two separate 'droplet' plasmas at once.

- Nature 602 (Feb 2022)
- Zero-shot sim-to-real transfer: trained in a simulator, deployed directly on the tokamak
- One neural controller replaced a set of hand-designed feedback loops for 19 magnetic coils

#### What happened
A neural network learned to steer a hot plasma by adjusting magnetic coils thousands of times per second, first in simulation and then on the real reactor.

#### Why it matters
It showed that RL could replace complex hand-engineered control in fusion devices and opened the way to AI-designed plasma scenarios.

Sources: [Magnetic control of tokamak plasmas through deep reinforcement learning (Nature)](https://www.nature.com/articles/s41586-021-04301-9); [DeepMind: Accelerating fusion science through learned plasma control](https://deepmind.google/blog/accelerating-fusion-science-through-learned-plasma-control/)

### 2022-10-05: AlphaTensor discovers faster matrix multiplication algorithms, beating Strassen's 1969 record for 4×4 mod 2

****; DeepMind; Science & math; confidence high; https://postcutoff.com/e/2022-10-05-alphatensor-matrix-multiplication/

DeepMind's AlphaTensor (Nature, Oct 2022) framed matrix multiplication as a tensor-decomposition game. It found a 4×4 algorithm over GF(2) with 47 multiplications (Strassen-based: 49) and improved 5×5 to 96. Human researchers cut 5×5 further to 95 within days.

- 4×4 matrices in modular (GF(2)) arithmetic: 47 multiplications vs 49 from Strassen's 1969 method
- 5×5×5: 96 multiplications (from 98); Kauers & Moosbauer improved to 95 days later with a flip-graph method
- Found 14,236 non-equivalent 4×4 algorithms; also hardware-tuned algorithms faster on GPUs/TPUs

#### What happened
AlphaTensor, an AlphaZero descendant, searched the space of tensor decompositions and found matrix multiplication schemes using fewer scalar multiplications than any known for several sizes.

#### Why it matters
It was the first AI-found improvement to a famous algorithmic record. It prompted rapid human counter-improvements and led to AlphaEvolve's 48-multiplication complex 4×4 result in 2025.

Sources: [Discovering faster matrix multiplication algorithms with reinforcement learning (Nature)](https://www.nature.com/articles/s41586-022-05172-4); [GitHub: google-deepmind/alphatensor](https://github.com/google-deepmind/alphatensor); [Computational Complexity blog on AlphaTensor](https://blog.computationalcomplexity.org/2022/10/alpha-tensor.html)

### 2023-05-25: AI finds abaucin, a narrow-spectrum antibiotic against the superbug Acinetobacter baumannii

***; McMaster University, MIT; Science & math; confidence high; https://postcutoff.com/e/2023-05-25-abaucin-ai-antibiotic/

McMaster and MIT researchers (Nature Chemical Biology, May 2023) trained a model on ~7,500 screened molecules and found abaucin, which selectively kills A. baumannii by disrupting lipoprotein trafficking (LolE) and controlled infection in a mouse wound model.

- Nat Chem Biol 19:1342–1350 (2023)
- Narrow-spectrum: spares most other bacteria
- Mechanism: perturbs lipoprotein trafficking via LolE

#### What happened
A model trained on a modest screen predicted which compounds would inhibit A. baumannii, leading to abaucin.

#### Why it matters
It showed that AI could find narrow-spectrum antibiotics, which spare the microbiome and slow resistance.

Sources: [Deep learning-guided discovery of an antibiotic targeting Acinetobacter baumannii (Nat Chem Biol)](https://www.nature.com/articles/s41589-023-01349-8); [MIT News: Using AI, scientists find a drug that could combat drug-resistant infections](https://news.mit.edu/2023/using-ai-scientists-combat-drug-resistant-infections-0525)

### 2023-06-07: AlphaDev discovers faster small-sort routines, merged into LLVM's C++ standard library

***; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2023-06-07-alphadev-sorting/

AlphaDev (Nature, 7 Jun 2023) treated writing assembly as a game and found sort3/sort4/sort5 routines shorter than human versions; they were merged into LLVM libc++. Critics argued the gains were small tricks a compiler or GPT-4 could also find.

- Sort routines merged into LLVM libc++, used by millions of programs
- DeepMind: up to 70% faster for short sequences, ~1.7% for sequences >250k elements
- Critics: essentially a known sorting network plus one removed mov instruction; Cassio Neri published a shorter, faster sort3 (arXiv 2307.14503)

#### What happened
AlphaDev found instruction sequences for sorting 3–5 elements that saved instructions over decades-old library code. LLVM maintainers accepted them.

#### Why it matters
It was AI-discovered code shipping in core infrastructure, though experts argued about how novel the discovery really was.

Sources: [Faster sorting algorithms discovered using deep reinforcement learning (Nature)](https://www.nature.com/articles/s41586-023-06004-9); [DeepMind: AlphaDev discovers faster sorting algorithms](https://deepmind.google/blog/alphadev-discovers-faster-sorting-algorithms/); [Cassio Neri: shorter and faster than Sort3AlphaDev (arXiv 2307.14503)](https://arxiv.org/abs/2307.14503)

### 2023-07-11: RFdiffusion: diffusion models design new proteins that work in the lab

****; University of Washington Institute for Protein Design; Science & math; confidence high; https://postcutoff.com/e/2023-07-11-rfdiffusion-protein-design/

David Baker's lab (Nature, July 2023) fine-tuned RoseTTAFold as a diffusion model to generate new protein backbones for binders, symmetric assemblies and metal-binding sites. Hundreds of designs were experimentally characterised. A cryo-EM structure of a designed binder bound to influenza haemagglutinin was nearly identical to the design model.

- Nature, 11 Jul 2023; code released free and open-source in 2023
- Designs: protein binders, symmetric oligomers, enzyme active-site scaffolds, metal-binding proteins
- Successors: RFdiffusion2 (Nature Methods, Jan 2026: scaffolds for all 41 benchmark active sites vs 16 before) and RFdiffusion3 (open-sourced Dec 2025)
- Part of the work recognised by the 2024 Nobel Prize in Chemistry (Baker)

#### What happened
By adapting image-generation-style diffusion to protein structures, the Baker lab made protein design largely a matter of generating and filtering candidates on computers.

#### Why it matters
It became the workhorse of AI protein design, underlying AI antivenoms, antibodies and enzymes.

Sources: [De novo design of protein structure and function with RFdiffusion (Nature)](https://www.nature.com/articles/s41586-023-06415-8); [Baker Lab: RFdiffusion now free and open source](https://www.bakerlab.org/2023/03/30/rf-diffusion-now-free-and-open-source/); [IPD: RFdiffusion3 now available](https://www.ipd.uw.edu/2025/12/rfdiffusion3-now-available/)

### 2023-09-19: AlphaMissense classifies 89% of all 71 million possible human missense mutations

***; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2023-09-19-alphamissense/

AlphaMissense (Science, Sept 2023) scored all ~71 million possible single amino-acid substitutions in 19,233 human proteins and classified 89%: 57% likely benign and 32% likely pathogenic. Human experts had classified only 0.1%.

- 71M variants scored; 89% classified (57% likely benign, 32% likely pathogenic)
- Human experts had confidently classified only ~0.1% of missense variants
- Predictions released freely; model weights restricted

#### What happened
Built on AlphaFold-style protein modelling, AlphaMissense predicted which mutations likely disrupt protein function.

#### Why it matters
It gives clinicians a first-pass interpretation for millions of variants they otherwise could not assess.

Sources: [Accurate proteome-wide missense variant effect prediction with AlphaMissense (Science)](https://www.science.org/doi/10.1126/science.adg7492); [DeepMind: A catalogue of genetic mutations to help pinpoint the cause of diseases](https://deepmind.google/blog/a-catalogue-of-genetic-mutations-to-help-pinpoint-the-cause-of-diseases/)

### 2023-11-14: GraphCast: ML weather model beats the world's best physics-based 10-day forecast on 90% of targets

****; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2023-11-14-graphcast-weather/

GraphCast (Science, Nov 2023), a graph neural network trained on ECMWF reanalysis data, produced 10-day global forecasts in under a minute on one TPU. It beat ECMWF's HRES, the leading deterministic physics model, on 90.3% of 1,380 verification targets. It later became the basis of NOAA's operational AIGFS.

- Beat HRES on 90.3% of 1,380 targets (89.9% statistically significant)
- 0.25° resolution; a 10-day forecast in under a minute on a single TPU v4
- Basis of NOAA's operational AIGFS (Dec 2025), which uses ~99.7% less compute

#### What happened
DeepMind showed that a learned simulator could beat physics-based weather prediction on standard skill scores.

#### Why it matters
It triggered the rapid move of AI weather models into operational forecasting worldwide.

Sources: [Learning skillful medium-range global weather forecasting (Science)](https://www.science.org/doi/10.1126/science.adi2336); [DeepMind: GraphCast](https://deepmind.google/blog/graphcast-ai-model-for-faster-and-more-accurate-global-weather-forecasting/); [NOAA deploys new generation of AI-driven global weather models](https://www.noaa.gov/news-release/noaa-deploys-new-generation-of-ai-driven-global-weather-models)

### 2023-11-29: GNoME predicts 2.2 million new crystals, 380,000 stable, but novelty and usefulness are disputed

****; Google DeepMind, Lawrence Berkeley National Laboratory; Science & math; confidence high; https://postcutoff.com/e/2023-11-29-gnome-millions-of-materials/

DeepMind's GNoME (Nature, Nov 2023) used graph neural networks and active learning with DFT to predict 2.2 million new inorganic crystal structures, 380,000 of them computed to be stable. DeepMind called it '800 years' worth of knowledge'. Solid-state chemists later found 'scant evidence' of compounds that are novel, credible and useful.

- 2.2M new structures; 380k predicted stable; ~400k added to the Materials Project
- DeepMind: over 700 had already been independently synthesised by other groups
- Cheetham & Seshadri (Chem. Mater., Apr 2024): 'scant evidence for compounds that fulfill the trifecta of novelty, credibility, and utility'
- Oct 14, 2026 (C&EN, Dalmeet Singh Chawla): new outside research says the GNoME database is 'littered with duplicates of known and hypothesized crystal structures found in other databases' and contains unrealistic compounds; Google DeepMind did not comment (article partly paywalled; the study's authors and figures were not read)
- GNoME lead Ekin Doğuş Çubuk later co-founded Periodic Labs (2025)

#### What happened
DeepMind scaled ML-guided materials screening by orders of magnitude and released the predicted structures to researchers.

#### Why it matters
It is the largest AI materials-prediction effort, and a leading example of the gap between computational "discovery" and a useful new material.

Sources: [Scaling deep learning for materials discovery (Nature)](https://www.nature.com/articles/s41586-023-06735-9); [DeepMind: Millions of new materials discovered with deep learning](https://deepmind.google/blog/millions-of-new-materials-discovered-with-deep-learning/); [Cheetham & Seshadri critique (Chemistry of Materials)](https://pubs.acs.org/doi/10.1021/acs.chemmater.4c00643); [C&EN: Google DeepMind AI crystallography database includes near duplicate structures and unrealistic compounds (Oct 14, 2026; partly paywalled)](https://cen.acs.org/research-integrity/Google-DeepMind-structures-near-duplicates-questionable/104/web/2026/10)

### 2023-11-29: Berkeley's A-Lab claims 41 new materials from autonomous synthesis; after critiques Nature corrects it to 36 'inorganic' (not 'novel') materials

***; Lawrence Berkeley National Laboratory; Science & math; confidence high; https://postcutoff.com/e/2023-11-29-a-lab-autonomous-synthesis-dispute/

Published alongside GNoME, the A-Lab paper (Nature, Nov 2023) claimed a robotic lab made 41 'novel' compounds from 58 targets in 17 days. Robert Palgrave and Leslie Schoop argued that many were known compounds or ordered versions of known disordered phases, and that the diffraction analysis was flawed. In Jan 2026 Nature published a correction: the title changed from 'novel materials' to 'inorganic materials' and the headline became 36 compounds from 57 targets.

- Original claim: 41 of 58 targets made in 17 days of autonomous operation (71%)
- Palgrave: 'it's likely they didn't make any discoveries'
- Jan 2026 Author Correction: 36 compounds from 57 targets; manual re-analysis confirmed 36 of 40 reported successes
- Palgrave said the authors 'didn't really engage' with the disorder issue

#### What happened
A self-driving lab combined robotic synthesis with ML-based analysis. Independent chemists challenged whether its products were new, and the paper was eventually corrected.

#### Why it matters
It is the clearest case study of overclaiming in AI-driven science, and of how human expert scrutiny corrected it.

Sources: [Nature: A-Lab Author Correction (2026)](https://www.nature.com/articles/s41586-025-09992-y); [Chemistry World: New analysis raises doubts over autonomous lab's materials discoveries](https://www.chemistryworld.com/news/new-analysis-raises-doubts-over-autonomous-labs-materials-discoveries/4018791.article); [C&EN: Nature robot chemist paper corrected](https://cen.acs.org/research-integrity/Nature-robot-chemist-paper-corrected/104/web/2026/01)

### 2023-12-14: FunSearch: an LLM finds new cap-set constructions, the first LLM discovery in open maths

****; Google DeepMind, University of Wisconsin–Madison; Science & math; confidence high; https://postcutoff.com/e/2023-12-14-funsearch-cap-sets/

FunSearch (Nature, Dec 2023) paired a code LLM with an automated evaluator in an evolutionary loop. It found a cap set of size 512 in dimension 8 (previous best 496) and better lower bounds on the asymptotic cap-set capacity. It also found bin-packing heuristics beating first-fit and best-fit.

- Cap set in F_3^8 of size 512, beating the previous record of 496
- Improved lower bound on cap-set capacity via new admissible sets
- Outputs are programs, so humans can read how the construction works
- Co-author: mathematician Jordan Ellenberg

#### What happened
DeepMind evolved short Python programs that construct cap sets, with an LLM proposing program mutations and a scorer keeping the best.

#### Why it matters
It was the direct ancestor of AlphaEvolve and showed that LLMs' mistakes don't matter when outputs can be automatically checked.

Sources: [Mathematical discoveries from program search with large language models (Nature)](https://www.nature.com/articles/s41586-023-06924-6); [GitHub: google-deepmind/funsearch](https://github.com/google-deepmind/funsearch); [Ernest Davis: comment on FunSearch](https://cs.nyu.edu/~davise/papers/FunSearchComment.pdf)

### 2023-12-20: Explainable deep learning discovers a new structural class of antibiotics against MRSA

***; MIT, Broad Institute; Science & math; confidence high; https://postcutoff.com/e/2023-12-20-ai-new-antibiotic-structural-class/

Felix Wong, James Collins and colleagues (Nature, Dec 2023) screened ~39,000 compounds, trained graph neural networks, and used explainable substructure analysis on ~12M compounds. They found a new structural class of antibiotics active against MRSA and VRE that worked in mouse models.

- Nature, published online 20 Dec 2023
- ~39,000 compounds tested experimentally; ~12M scored computationally
- Active against MRSA and vancomycin-resistant enterococci; effective topically and systemically in mice

#### What happened
Rather than a black-box ranking, the model identified which chemical substructures drove predicted activity, leading chemists to a new antibiotic class.

#### Why it matters
New structural classes of antibiotics are rarely discovered. This one came from interpretable AI, which also showed chemists why the molecules work.

Sources: [Discovery of a structural class of antibiotics with explainable deep learning (Nature)](https://www.nature.com/articles/s41586-023-06887-8); [Broad Institute: Researchers use AI to identify new class of antibiotic candidates](https://www.broadinstitute.org/news/researchers-use-ai-identify-new-class-antibiotic-candidates)

### 2023-12-20: Coscientist: a GPT-4 agent plans and runs real chemistry experiments from plain-English prompts

***; Carnegie Mellon University; Science & math; confidence high; https://postcutoff.com/e/2023-12-20-coscientist-autonomous-chemistry/

Gabe Gomes's group (Nature, Dec 2023) built Coscientist, a GPT-4-based agent that searches documentation, writes code and drives lab automation. Across six tasks it included successfully planning and optimising palladium-catalysed cross-coupling reactions (Suzuki and Sonogashira) from a single prompt.

- GPT-4 with web search, documentation search, code execution and robotic liquid-handler control
- Successfully executed and optimised Suzuki and Sonogashira couplings
- Capability demonstration rather than a new chemical discovery

#### What happened
An LLM was connected to laboratory tools and asked in natural language to carry out reactions, which it planned, coded and ran.

#### Why it matters
It was the first peer-reviewed LLM agent operating a physical lab, a precursor of 2026's AI-run labs.

Sources: [Autonomous chemical research with large language models (Nature)](https://www.nature.com/articles/s41586-023-06792-0); [Chemistry World: first GPT-4-powered AI lab assistant](https://www.chemistryworld.com/news/first-gpt-4-powered-ai-lab-assistant-independently-directs-key-organic-reactions/4018723.article)

### 2024-01-09: Microsoft AI and PNNL screen 32 million candidates to find a solid electrolyte using ~70% less lithium

**; Microsoft, Pacific Northwest National Laboratory; Science & math; confidence high; https://postcutoff.com/e/2024-01-09-microsoft-pnnl-battery-electrolyte/

Microsoft's Azure Quantum Elements combined AI models and HPC to narrow 32 million inorganic candidates to 18 in about 80 hours. PNNL synthesised and tested the top pick, a Li–Na–Y chloride solid electrolyte reported to use about 70% less lithium, as a working prototype battery.

- 32M → 500k (stable) → 18 candidates in ~80 hours of screening
- Synthesised and built into a prototype by PNNL
- Prototype only; no commercial validation (arXiv 2401.04070)

#### What happened
A pipeline of ML property predictors filtered a huge chemical space in days, leaving a handful of candidates for chemists to make.

#### Why it matters
It is a concrete example of AI compressing the materials search funnel from years to weeks, though the result was a prototype rather than a product.

Sources: [Microsoft Azure blog: how Microsoft's AI screened over 32 million candidates to find a better battery](https://azure.microsoft.com/en-us/blog/quantum/2024/01/09/unlocking-a-new-era-for-scientific-discovery-with-ai-how-microsofts-ai-screened-over-32-million-candidates-to-find-a-better-battery/); [arXiv 2401.04070](https://arxiv.org/abs/2401.04070); [Chemistry World: Microsoft's AI system powers new battery discovery](https://www.chemistryworld.com/research/microsofts-ai-and-high-performance-computing-system-powers-new-battery-discovery/4018731.article)

### 2024-01-17: AlphaGeometry solves olympiad geometry near gold-medallist level without human demonstrations

***; Google DeepMind, New York University; Science & math; confidence high; https://postcutoff.com/e/2024-01-17-alphageometry/

AlphaGeometry (Nature, 17 Jan 2024) solved 25 of 30 IMO geometry problems from 2000–2022. The previous best system solved 10 and the average gold medallist 25.9. It combines a language model with a symbolic deduction engine and was trained on 100M synthetic proofs.

- IMO-AG-30 benchmark: 25/30 solved vs 10 for the previous state of the art (Wu's method)
- Trained entirely on 100 million synthetic theorems and proofs, no human demonstrations
- A later paper showed Wu's method plus a better deductive database rivals it (arXiv 2404.06405)
- AlphaGeometry 2 (2025) reached gold-medallist level on geometry

#### What happened
DeepMind generated synthetic geometry theorems at scale to train a language model that proposes auxiliary constructions, while a symbolic engine does the deduction.

#### Why it matters
It was a step towards the 2024 IMO silver and 2025 gold, and showed that synthetic data could replace scarce human proofs.

Sources: [Solving olympiad geometry without human demonstrations (Nature)](https://www.nature.com/articles/s41586-023-06747-5); [Nature news on AlphaGeometry](https://www.nature.com/articles/d41586-024-00145-1); [Wu's method can boost symbolic AI to rival silver medalists (arXiv 2404.06405)](https://arxiv.org/abs/2404.06405)

### 2024-02-21: AI controller predicts and avoids tearing instabilities in the DIII-D fusion reactor

***; Princeton University, Princeton Plasma Physics Laboratory, General Atomics; Science & math; confidence high; https://postcutoff.com/e/2024-02-21-ai-avoids-tokamak-tearing-instabilities/

Princeton and PPPL researchers (Nature, Feb 2024) trained an RL controller on past DIII-D data. It forecast tearing-mode instabilities up to 300 ms ahead and adjusted operating parameters in real time to avoid them during experiments while keeping high performance.

- Nature 626 (22 Feb 2024)
- Forecasts tearing instabilities up to 300 ms in advance
- Demonstrated in live DIII-D shots

#### What happened
The controller learned the precursors of tearing modes from archived experiments and steered the plasma away from them.

#### Why it matters
Instabilities that can damage reactors are a key obstacle to fusion power. Predictive AI control is a candidate solution for ITER-class devices.

Sources: [Avoiding fusion plasma tearing instability with deep reinforcement learning (Nature)](https://www.nature.com/articles/s41586-024-07024-9); [Princeton Engineering: Engineers use AI to wrangle fusion power](https://engineering.princeton.edu/news/2024/02/21/engineers-use-ai-wrangle-fusion-power-grid)

### 2024-05-08: AlphaFold 3 predicts structures and interactions of all life's molecules

****; Google DeepMind, Isomorphic Labs; Science & math; confidence high; https://postcutoff.com/e/2024-05-08-alphafold-3/

AlphaFold 3 extended structure prediction from proteins to complexes with DNA, RNA, ligands and ions, using a diffusion-based architecture, with at least 50% improvement on protein–ligand interactions over prior methods.

- Published in Nature on 8 May 2024
- Models proteins, DNA, RNA, small-molecule ligands, ions and modifications
- Diffusion module generates atomic coordinates
- Free AlphaFold Server for non-commercial research; code for academic use released November 2024

#### What happened
Google DeepMind and Isomorphic Labs released a model predicting how biomolecules fit together, aimed at drug discovery.

#### Why it matters
Moved AI structural biology from single proteins to the molecular interactions that matter for medicine.

Sources: [Accurate structure prediction of biomolecular interactions with AlphaFold 3 (Nature, DOI)](https://doi.org/10.1038/s41586-024-07487-w); [AlphaFold 3 predicts the structure and interactions of all of life's molecules (Google)](https://blog.google/technology/ai/google-deepmind-isomorphic-alphafold-3-ai-model/); [AlphaFold Server](https://alphafoldserver.com/)

### 2024-06-25: ESM3 generates esmGFP, a new fluorescent protein estimated at '500 million years of evolution' from nature

***; EvolutionaryScale; Science & math; confidence high; https://postcutoff.com/e/2024-06-25-esm3-esmgfp/

EvolutionaryScale's ESM3, a multimodal protein language model, generated esmGFP, a bright fluorescent protein only 58% identical to the closest known fluorescent protein. The authors estimate that distance equals over 500 million years of natural evolution. Published in Science (Jan 2025).

- Announced 25 Jun 2024; Science paper published online Jan 2025
- esmGFP: 58% sequence identity to the nearest known fluorescent protein
- '500 million years of evolution' is the authors' estimate

#### What happened
ESM3 was prompted with a few key residues of GFP's chromophore site and generated whole new proteins. One, esmGFP, glowed after refinement rounds.

#### Why it matters
It showed language models producing functional proteins far outside the natural sequence space.

Sources: [Simulating 500 million years of evolution with a language model (Science)](https://www.science.org/doi/10.1126/science.ads0018); [EvolutionaryScale: ESM3 release](https://www.evolutionaryscale.ai/blog/esm3-release)

### 2024-07-22: NeuralGCM: Google's hybrid physics-ML atmosphere model matches top weather forecasts and runs decades-long climate simulations

***; Google Research, ECMWF, MIT, Harvard; Science & math; confidence high; https://postcutoff.com/e/2024-07-22-neuralgcm/

In Nature (Kochkov et al., 22 July 2024) Google introduced NeuralGCM. It pairs a differentiable spectral dynamical core with neural-network physics parameterisations trained end-to-end. It was competitive with ECMWF for 1–15-day forecasts, reproduced four decades of observed temperatures in AMIP-style runs, and needed 3–5 orders of magnitude less compute than conventional models.

- Paper: 'Neural general circulation models for weather and climate', Nature 632, 1060–1066 (2024); arXiv 2311.07222
- Hybrid: physics-based dynamical core + learned column physics, trained end-to-end through the solver
- Runs at 8–40× coarser horizontal resolution than ECMWF IFS and global cloud-resolving models, giving 3–5 orders of magnitude compute savings
- Stable multi-decade climate simulations, unlike pure-ML weather emulators at the time

#### What happened
Unlike GraphCast-style end-to-end emulators, NeuralGCM kept a numerical dynamical core and learned only the unresolved physics. That made it stable enough for climate-length runs.

#### Why it matters
It showed that ML can reach climate modelling, not only weather forecasting, and made differentiable hybrid GCMs a serious research direction.

Sources: [Nature: Neural general circulation models for weather and climate](https://www.nature.com/articles/s41586-024-07744-y); [arXiv 2311.07222](https://arxiv.org/abs/2311.07222); [Google Research: NeuralGCM harnesses AI to better simulate long-range global precipitation](https://research.google/blog/neuralgcm-harnesses-ai-to-better-simulate-long-range-global-precipitation/)

### 2024-07-25: AlphaProof and AlphaGeometry 2 reach IMO silver-medal standard

****; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2024-07-25-alphaproof-imo-silver/

Google DeepMind's AlphaProof (RL + Lean formal proofs) and AlphaGeometry 2 solved 4 of 6 problems at the 2024 International Mathematical Olympiad, scoring 28/42 — silver-medal level, one point short of gold.

- Announced 25 July 2024
- Score: 28/42 (gold cutoff was 29)
- AlphaProof solved two algebra problems and one number theory problem, including the hardest problem
- AlphaGeometry 2 solved the geometry problem
- Some problems took up to three days of compute (humans get 9 hours)
- Full AlphaProof method published in Nature on 12 Nov 2025 (RL on millions of auto-formalised problems plus test-time RL)

#### What happened
DeepMind's systems, operating on problems manually translated into the Lean formal language, were graded by IMO medalists.

#### Why it matters
First AI to reach medal level at the IMO; a year later, natural-language LLMs reached gold.

Sources: [AI achieves silver-medal standard solving International Mathematical Olympiad problems (Google DeepMind)](https://deepmind.google/discover/blog/ai-solves-imo-problems-at-silver-medal-level/); [AlphaGeometry: Solving olympiad geometry without human demonstrations (Nature, DOI)](https://doi.org/10.1038/s41586-023-06747-5); [Olympiad-level formal mathematical reasoning with reinforcement learning (AlphaProof, Nature 2025)](https://www.nature.com/articles/s41586-025-09833-y)

### 2024-09-05: AlphaProteo designs high-affinity protein binders, including the first AI-designed VEGF-A binder

***; Google DeepMind; Science & math; confidence medium; https://postcutoff.com/e/2024-09-05-alphaproteo/

DeepMind's AlphaProteo generated protein binders for 7 targets with 9–88% experimental success rates (88% for BHRF1) and 3–300× better affinities than prior methods. It produced the first successful AI-designed binder for VEGF-A.

- Experimental binding success 9–88% across 7 targets
- Affinities 3–300× better than the best previous methods on several targets
- Technical report, not peer-reviewed at announcement

#### What happened
DeepMind released a binder-design system and reported wet-lab results from partner labs.

#### Why it matters
High one-shot success rates cut the months of screening normally needed to find a binder.

Sources: [DeepMind: AlphaProteo generates novel proteins for biology and health research](https://deepmind.google/blog/alphaproteo-generates-novel-proteins-for-biology-and-health-research/); [MobiHealthNews: Google DeepMind unveils AlphaProteo](https://www.mobihealthnews.com/news/google-deepmind-unveils-alphaproteo-ai-drug-design)

### 2024-11-20: AlphaQubit: neural decoder sets accuracy record for quantum error correction on Google's Sycamore

***; Google DeepMind, Google Quantum AI; Science & math; confidence high; https://postcutoff.com/e/2024-11-20-alphaqubit-quantum-error-correction/

AlphaQubit (Nature, Nov 2024), a recurrent-transformer decoder for the surface code, made 6% fewer errors than tensor-network decoding and 30% fewer than correlated matching on real Sycamore data at code distances 3 and 5. It is not yet fast enough for real-time use.

- Pre-trained on simulated data, fine-tuned on Sycamore experimental data
- Distance 3 (17 qubits) and distance 5 (49 qubits)
- Caveat: too slow for real-time decoding on superconducting hardware at the time

#### What happened
DeepMind trained a neural network to infer which errors occurred in a quantum processor from noisy stabiliser measurements.

#### Why it matters
Better decoding lowers the overhead of fault-tolerant quantum computing.

Sources: [Learning high-accuracy error decoding for quantum processors (Nature)](https://www.nature.com/articles/s41586-024-08148-8); [Google: AlphaQubit](https://blog.google/innovation-and-ai/models-and-research/google-deepmind/alphaqubit-quantum-error-correction/)

### 2024-12-04: GenCast: diffusion-based ensemble forecast beats ECMWF's ENS on 97% of targets

***; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2024-12-04-gencast-ensemble-weather/

GenCast (Nature, Dec 2024) is a diffusion model producing probabilistic 15-day ensemble forecasts. It beat ECMWF's ENS, the leading operational ensemble, on 97.2% of 1,320 targets and on 99.8% at lead times beyond 36 hours, generating a 15-day ensemble member in about 8 minutes on one TPU.

- 97.2% of 1,320 targets better than ENS; 99.8% beyond 36 h
- Better prediction of extreme weather, tropical-cyclone tracks and wind-power output
- Code and weights released for research

#### What happened
DeepMind applied image-style diffusion to the atmosphere, sampling many plausible futures rather than one.

#### Why it matters
Ensembles drive decisions about extreme-weather risk. AI now leads here too, feeding into the WeatherNext models used by forecasters.

Sources: [Probabilistic weather forecasting with machine learning (Nature)](https://www.nature.com/articles/s41586-024-08252-9); [DeepMind: GenCast](https://deepmind.google/blog/gencast-predicts-weather-and-the-risks-of-extreme-conditions-with-sota-accuracy/)

### 2024-12-20: OpenAI o3 scores 25% on FrontierMath research-level maths benchmark, amid funding disclosure controversy

***; OpenAI, Epoch AI; Science & math; confidence high; https://postcutoff.com/e/2024-12-20-frontiermath-o3-25-percent/

Epoch AI's FrontierMath (Nov 2024) contains unpublished research-level problems on which models scored under 2%. On 20 Dec 2024 OpenAI claimed 25.2% for o3. It then emerged that OpenAI had funded the benchmark and had access to most problems. Released o3 scored lower in independent tests.

- FrontierMath paper v1: 7 Nov 2024; prior models <2%
- o3 claimed 25.2% (aggressive test-time compute setting) on 20 Dec 2024
- OpenAI's funding and data access disclosed only in paper v5 (20 Dec 2024); Epoch said it should have been more transparent
- Later records: Gemini 3 Pro 38% (Tiers 1-3) and 19% (Tier 4) in Nov 2025; GPT-5.2 Pro 31% on Tier 4 in Jan 2026

#### What happened
OpenAI previewed o3 with a headline FrontierMath score an order of magnitude above prior models. The benchmark's independence was then questioned when OpenAI's funding and access came to light.

#### Why it matters
It was the first sign that research-level maths was yielding to reasoning models, and an early lesson in benchmark governance and conflicts of interest.

Sources: [Epoch AI: OpenAI and FrontierMath](https://epoch.ai/latest/openai-and-frontiermath); [The Decoder: OpenAI quietly funded independent math benchmark](https://the-decoder.com/openai-quietly-funded-independent-math-benchmark-before-setting-record-with-o3/); [TechRepublic: independent FrontierMath score for o3](https://www.techrepublic.com/article/news-openai-generative-ai-models-frontiermath-score/)

### 2025-01-15: AI-designed proteins neutralise deadly snake-venom toxins and protect mice

***; University of Washington Institute for Protein Design, Technical University of Denmark; Science & math; confidence high; https://postcutoff.com/e/2025-01-15-ai-designed-antivenom/

Baker lab and DTU researchers (Nature, Jan 2025) used RFdiffusion to design small proteins that bind and neutralise cobra three-finger toxins. Depending on dose, toxin and design, 80–100% of mice survived otherwise lethal doses.

- Designed binders against short- and long-chain three-finger toxins
- 80–100% survival in mice given lethal doses
- Small, stable proteins could be cheaper to make than antibody-based antivenoms

#### What happened
The team generated binders computationally, tested a small number in the lab, and confirmed protection in animal models.

#### Why it matters
Snakebite kills tens of thousands of people a year, mainly in poor regions. Cheap, designed antitoxins show AI protein design aimed at neglected diseases.

Sources: [De novo designed proteins neutralize lethal snake venom toxins (Nature)](https://www.nature.com/articles/s41586-024-08393-x); [Baker Lab: Neutralizing deadly snake toxins](https://www.bakerlab.org/2025/01/15/neutralizing-deadly-snake-toxins/); [DTU: AI-designed proteins neutralise snake toxins](https://www.dtu.dk/english/newsarchive/2025/01/ai-designed-proteins-neutralise-snake-toxins)

### 2025-01-16: Microsoft's MatterGen generates materials to order; flagship result later challenged as a known compound

***; Microsoft Research; Science & math; confidence medium; https://postcutoff.com/e/2025-01-16-mattergen/

MatterGen (Nature, Jan 2025) is a diffusion model that generates stable inorganic materials with target properties. In the flagship test, TaCr2O6 was generated for a 200 GPa bulk modulus and measured at 169 GPa after synthesis. A 2026 critique in Materials Horizons argues the synthesised disordered phase matches a compound reported in 1972 that was in MatterGen's training data.

- Target bulk modulus 200 GPa; measured 169 GPa (<20% error)
- Critique (Materials Horizons, 2026): synthesised Ta1/3Cr2/3O2 is equivalent to Ta1/2Cr1/2O2 reported in 1972 (seen via secondary summary)
- Released open-source with MatterSim

#### What happened
Microsoft moved from screening to generating materials directly, and validated one design in the lab.

#### Why it matters
Generative materials design is promising, but as with GNoME and A-Lab, "new material" claims need crystallographic scrutiny.

Sources: [A generative model for inorganic materials design (Nature)](https://www.nature.com/articles/s41586-025-08628-5); [Microsoft Research: MatterGen](https://www.microsoft.com/en-us/research/blog/mattergen-a-new-paradigm-of-materials-design-with-generative-ai/); [whataifound.org: MatterGen finding and critique](https://whataifound.org/finding/2025-01-16-mattergen)

### 2025-02-19: Google's AI co-scientist independently reproduces an unpublished superbug discovery in 48 hours

****; Google, Google DeepMind, Imperial College London, Stanford University; Science & math; confidence high; https://postcutoff.com/e/2025-02-19-google-ai-co-scientist/

Google's Gemini 2.0–based multi-agent 'AI co-scientist' (announced 19 Feb 2025) generated hypotheses that were validated in the lab. It proposed AML drug-repurposing candidates, and liver-fibrosis drugs active in human organoids. Its top-ranked hypothesis for how cf-PICI genetic elements spread between bacteria matched Imperial College's unpublished, experimentally confirmed finding. The system was published in Nature on 19 May 2026.

- Agents for generation, reflection, ranking (tournament), evolution and meta-review on Gemini 2.0
- cf-PICI: 5 ranked hypotheses in 48 hours; the top one (hijacking tails from diverse phages) matched José Penadés's unpublished result; both papers later in Cell (Sep 2025)
- Liver fibrosis: 2 of the co-scientist's recommended epigenetic drugs were anti-fibrotic in human hepatic organoids (vorinostat reduced TGFβ-induced chromatin changes by 91%)
- AML: repurposing candidates inhibited tumour viability in cell lines
- Caveats: evaluation not blind or pre-registered; Google staff co-authors; 'decade-long mystery solved in 2 days' is press framing

#### What happened
Google built a multi-agent system that debates and ranks research hypotheses. Partner labs tested its suggestions, and one matched an unpublished result the humans had spent years on.

#### Why it matters
It was the most-cited early example of an LLM system generating a correct, non-obvious scientific hypothesis. At Google I/O 2026 it became part of the "Gemini for Science" product.

Sources: [Co-Scientist paper (Nature, 2026)](https://www.nature.com/articles/s41586-026-10644-y); [Cell: AI co-scientist and the cf-PICI mechanism](https://www.cell.com/cell/fulltext/S0092-8674(25)00973-0); [bioRxiv: cf-PICI hypothesis generated by AI co-scientist](https://www.biorxiv.org/content/10.1101/2025.02.19.639094v1.full); [Advanced Science: AI-assisted liver fibrosis drug repurposing](https://advanced.onlinelibrary.wiley.com/doi/full/10.1002/advs.202508751); [HPCwire: Google unveils AI scientist](https://www.hpcwire.com/2025/02/26/google-unveils-ai-scientist-that-could-transform-research/)

### 2025-02-19: Evo 2: a 40B-parameter genome language model trained on DNA from all domains of life

***; Arc Institute, Stanford University, NVIDIA; Science & math; confidence high; https://postcutoff.com/e/2025-02-19-evo-2-genome-model/

Arc Institute, Stanford and NVIDIA released Evo 2 (7B and 40B parameters) in Feb 2025, trained on genomes across bacteria, archaea and eukaryotes. It predicts variant effects and generates genome-scale sequences. Published in Nature on 4 Mar 2026, and used to design the first AI-generated viable phage genomes.

- Open weights, 7B and 40B parameters, 1M-base context
- Nature publication 4 Mar 2026 (DOI 10.1038/s41586-026-10176-5)
- 88k+ GitHub downloads and 8M+ API requests in its first year, per Arc

#### What happened
Arc released one of the largest fully open biology models, trained on trillions of DNA bases.

#### Why it matters
It made genome-scale generative design possible, most notably whole viable bacteriophages.

Sources: [Arc Institute: Evo 2 one year later](https://arcinstitute.org/news/evo-2-one-year-later); [Wikipedia: Evo (AI)](https://en.wikipedia.org/wiki/Evo_(AI))

### 2025-02-25: AI weather forecasting goes operational: ECMWF's AIFS (Feb 2025), then NOAA's AI models (Dec 2025)

****; ECMWF, NOAA; Science & math; confidence high; https://postcutoff.com/e/2025-02-25-ecmwf-aifs-operational-ai-weather/

On 25 Feb 2025 the European Centre for Medium-Range Weather Forecasts made its machine-learned AIFS Single model operational alongside its physics model. It was up to 20% better on tropical-cyclone tracks and used about 1,000× less energy per forecast. The AIFS ensemble followed on 1 Jul 2025. On 17 Dec 2025 NOAA deployed AIGFS (GraphCast-based), AIGEFS and the hybrid HGEFS operationally.

- AIFS Single: operational 25 Feb 2025, ~28 km grid, ~1,000× less energy, up to 20% better cyclone tracks
- AIFS ENS operational 1 Jul 2025; both upgraded to v2 on 12 May 2026
- NOAA (17 Dec 2025): AIGFS uses 99.7% less compute; AIGEFS uses 9% of the physics ensemble's compute and gains 18–24 h of skill; HGEFS is billed as the first operational hybrid AI/physics ensemble
- ECMWF Director-General Florence Rabier: 'This milestone will transform weather science and predictions.'

#### What happened
Within about 15 months of GraphCast's publication, the world's leading forecast centres began issuing official forecasts from machine-learned models.

#### Why it matters
It is one of the fastest transitions of AI research into critical public infrastructure.

Sources: [ECMWF: AI forecasts become operational](https://www.ecmwf.int/en/about/media-centre/news/2025/ecmwfs-ai-forecasts-become-operational); [NOAA deploys new generation of AI-driven global weather models](https://www.noaa.gov/news-release/noaa-deploys-new-generation-of-ai-driven-global-weather-models); [CACM: AI weather forecasting goes operational](https://cacm.acm.org/news/ai-weather-forecasting-goes-operational/)

### 2025-03-12: Sakana's AI Scientist-v2 writes the first fully AI-generated paper to pass peer review (ICLR 2025 workshop)

****; Sakana AI, University of British Columbia, University of Oxford; Science & math; confidence high; https://postcutoff.com/e/2025-03-12-sakana-ai-scientist-v2-peer-review/

On 12 Mar 2025 Sakana AI reported that a paper generated end-to-end by The AI Scientist-v2 (idea, code, experiments, analysis, writing) scored 6, 7, 6 at an ICLR 2025 workshop, above the acceptance threshold; it was withdrawn by prior agreement. The system and its limits were later published in Nature (26 Mar 2026).

- Workshop: ICLR 2025 'I Can't Believe It's Not Better' (ICBINB); reviewer scores 6, 7, 6 (avg 6.33), higher than ~55% of human-written submissions
- Reviewers knew some submissions might be AI-generated but not which; the paper was withdrawn after review as agreed with organisers
- Workshop acceptance, not a main-conference paper; the result was a negative result on compositional regularisation
- Nature paper (2026): automated reviewer reached 69% balanced accuracy; paper quality rises with the underlying model
- Admitted weaknesses: naive ideas, weak rigour, hallucinated citations

#### What happened
Sakana AI, with UBC and Oxford collaborators, submitted three papers written entirely by The AI Scientist-v2 to an ICLR 2025 workshop with the organisers' consent. One received scores of 6, 7 and 6 — above the acceptance bar — and was withdrawn before publication, as planned, because norms for AI-authored papers did not exist. The first version of the system had been released in August 2024; a peer-reviewed description appeared in Nature on 26 March 2026.

#### Why it matters
It was the first demonstration that a fully automated pipeline could clear human peer review, even at a workshop with a higher acceptance rate than main tracks. It set off debate about AI-generated papers flooding venues, which later led to arXiv and conference policy changes.

Sources: [Sakana AI: The AI Scientist generates its first peer-reviewed scientific publication](https://sakana.ai/ai-scientist-first-publication/); [Sakana AI: The AI Scientist published in Nature](https://sakana.ai/ai-scientist-nature/); [Nature news on the AI Scientist paper](https://www.nature.com/articles/d41586-026-00899-w); [The AI Scientist (v1) paper, arXiv 2408.06292](https://arxiv.org/abs/2408.06292)

### 2025-05-14: AlphaEvolve: Gemini-powered agent discovers new algorithms

****; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2025-05-14-alphaevolve/

Google DeepMind's AlphaEvolve combined Gemini models with evolutionary search and automated evaluation to discover new algorithms, including a way to multiply 4×4 complex matrices with 48 scalar multiplications, improving on Strassen's 1969 algorithm.

- Announced 14 May 2025
- 4×4 complex-valued matrix multiplication with 48 scalar multiplications
- Matched state of the art on ~75% and improved on ~20% of 50+ open math problems tested, per DeepMind
- A scheduling heuristic recovers on average 0.7% of Google's worldwide compute resources
- Kissing number in 11 dimensions: lower bound raised from 592 to 593
- The 48-multiplication result is for complex-valued, non-commutative 4×4 multiplication; a June 2025 human follow-up gave a 48-multiplication scheme with rational coefficients (arXiv 2506.13242)
- Nov 2025: Georgiev, Gómez-Serrano, Tao and Wagner applied AlphaEvolve to 67 problems (see related entry)

#### What happened
DeepMind described an agent that iteratively writes and evaluates code, already deployed across Google's data centers, chip design and AI training.

#### Why it matters
A concrete example of LLM-based systems making novel discoveries and improving the infrastructure that trains them — an early form of recursive improvement.

Sources: [AlphaEvolve: A Gemini-powered coding agent for designing advanced algorithms (Google DeepMind)](https://deepmind.google/discover/blog/alphaevolve-a-gemini-powered-coding-agent-for-designing-advanced-algorithms/); [AlphaEvolve: A coding agent for scientific and algorithmic discovery (arXiv)](https://arxiv.org/abs/2506.13131); [Independent verification of the 48-multiplication algorithm (GitHub)](https://github.com/PhialsBasement/AlphaEvolve-MatrixMul-Verification); [Human follow-up: 48 multiplications with rational coefficients (arXiv 2506.13242)](https://arxiv.org/abs/2506.13242)

### 2025-05-20: FutureHouse's Robin multi-agent system proposes ripasudil as a new treatment candidate for dry AMD

***; FutureHouse; Science & math; confidence high; https://postcutoff.com/e/2025-05-20-futurehouse-robin-ripasudil/

FutureHouse's Robin generated the hypotheses, analyses and figures that identified ripasudil, a glaucoma drug, as a candidate for dry age-related macular degeneration. Ripasudil increased phagocytosis in retinal pigment epithelium cells and upregulated ABCA1 about 3×. Humans ran the bench work; the project took 2.5 months. Published in Nature on 19 May 2026.

- Robin proposed enhancing RPE phagocytosis as a mechanism, then ripasudil (ROCK inhibitor) as the drug
- ABCA1 upregulated ~3× (RNA-seq follow-up proposed by Robin)
- Caveat: Robin's analysis agent reported a 7.5× phagocytosis effect; human re-analysis of the same data gave 1.75×
- No clinical data; in vitro only

#### What happened
Robin chained literature-search and data-analysis agents to go from disease to mechanism to drug candidate, with human lab work in between.

#### Why it matters
It was an early end-to-end AI-driven discovery loop in biology. The overstated effect size is a reminder that AI analyses need human re-checking.

Sources: [Robin paper (Nature, 2026)](https://www.nature.com/articles/s41586-026-10652-y); [FutureHouse: Demonstrating end-to-end scientific discovery with Robin](https://www.futurehouse.org/research-announcements/demonstrating-end-to-end-scientific-discovery-with-robin-a-multi-agent-system)

### 2025-05-21: Microsoft's Aurora foundation model beats operational forecasts for air quality, waves, cyclones and weather

***; Microsoft Research; Science & math; confidence high; https://postcutoff.com/e/2025-05-21-microsoft-aurora-earth-model/

Aurora (Nature, May 2025) is an Earth-system foundation model pre-trained on over a million hours of geophysical data. After fine-tuning it beat operational systems at air-quality, ocean-wave, tropical-cyclone-track and high-resolution weather forecasting, at far lower computational cost.

- Pre-trained on >1M hours of diverse atmospheric data
- Outperformed operational forecasts in 4 domains after fine-tuning
- Microsoft cites ~5,000× lower compute cost than numerical models (company figure)

#### What happened
Microsoft showed that the pre-train-then-fine-tune recipe of LLMs also works for the whole Earth system.

#### Why it matters
One model can be adapted cheaply to new environmental prediction tasks, including air pollution and ocean waves.

Sources: [A foundation model for the Earth system (Nature)](https://www.nature.com/articles/s41586-025-09005-y); [Microsoft Source: Aurora goes beyond weather forecasting](https://news.microsoft.com/source/features/ai/microsofts-aurora-ai-foundation-model-goes-beyond-weather-forecasting/)

### 2025-05-27: Intology's 'Zochi' AI system gets a paper into the ACL 2025 main conference

***; Intology; Science & math; confidence medium; https://postcutoff.com/e/2025-05-27-intology-zochi-acl-2025/

In May 2025 Intology said its autonomous research agent Zochi produced 'Tempest', a paper on multi-turn LLM jailbreaking via tree search, that was accepted to the main conference of ACL 2025 (acceptance rate ~20%) — claimed as the first AI-generated paper to pass peer review at an A* main venue.

- Paper: 'Tempest: Automatic Multi-Turn Jailbreaking of LLMs with Tree Search'
- Meta-review score 4/5; Intology claims it ranked in the top 8.2% of submissions
- Human role per Intology: manuscript preparation only (figures, citation formatting, minor fixes)
- Autonomy claims are self-reported; exact announcement day not verified (month precision)

#### What happened
Intology's agent Zochi generated a method (Tempest) for automatically jailbreaking language models over multiple conversation turns using tree search, ran the experiments and drafted the paper, which passed ACL 2025 main-track peer review.

#### Why it matters
It moved AI-authored research from workshop level (Sakana, March 2025) to a selective main track within months, though the degree of autonomy could not be independently audited.

Sources: [Intology: Zochi's paper accepted to ACL 2025](https://www.intology.ai/blog/zochi-acl); [ACL 2025 main conference papers](https://2025.aclweb.org/program/main_papers/); [LessWrong discussion: Zochi publishes a paper](https://www.lesswrong.com/posts/LtsgfGsXpiLTSGpaW/zochi-publishes-a-paper)

### 2025-06-25: AlphaGenome predicts how DNA variants affect thousands of gene-regulation signals from 1 Mb of sequence

***; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2025-06-25-alphagenome/

DeepMind's AlphaGenome reads up to 1 million DNA bases and predicts 5,930 human (1,128 mouse) genomic signals, including expression, chromatin accessibility and splicing, at base-pair resolution. It covers the 98% of the genome that does not code for proteins. Published in Nature on 28 Jan 2026.

- Input: up to 1 Mb of DNA; outputs 5,930 human tracks
- State of the art on most variant-effect benchmarks at announcement
- Nature paper 28 Jan 2026 (vol 649); API for non-commercial research

#### What happened
DeepMind extended from protein structure to how DNA sequence controls gene activity, releasing a model and API.

#### Why it matters
Most disease-linked variants are non-coding. AlphaGenome gives researchers a way to predict what they do.

Sources: [DeepMind: AlphaGenome — AI for better understanding the genome](https://deepmind.google/blog/alphagenome-ai-for-better-understanding-the-genome/); [Nature vol 649 issue 8099 (AlphaGenome paper)](https://www.nature.com/nature/volumes/649/issues/8099); [Science Media Centre: expert reaction to AlphaGenome](https://www.sciencemediacentre.org/expert-reaction-to-paper-on-google-deepminds-alphagenome/)

### 2025-07-21: AI systems reach gold-medal level at the International Mathematical Olympiad

*****; Google DeepMind, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2025-07-21-imo-gold-ai/

At IMO 2025, an advanced Gemini Deep Think model (officially graded) and an experimental OpenAI reasoning model (graded by former medalists) each solved 5 of 6 problems for 35/42 points — gold-medal standard — working end-to-end in natural language within the 4.5-hour time limits.

- OpenAI announced its result on 19 July 2025; Google DeepMind on 21 July 2025
- Both scored 35/42, solving 5 of 6 problems
- Google DeepMind's result was officially certified by IMO coordinators
- Natural-language proofs, no formal translation, within competition time limits
- Formal provers: Harmonic's Aristotle produced Lean-verified solutions to 5 of 6 problems (gold-equivalent; arXiv 2510.01346); ByteDance Seed-Prover got an IMO-certified 30 points in-contest and later completed P1–P5
- One year earlier, AlphaProof reached silver with formal Lean proofs and days of compute

#### What happened
Two general-purpose LLM reasoning systems achieved gold-medal scores at the world's top high-school math competition.

#### Why it matters
A long-standing AI grand challenge fell years earlier than many forecasters expected, showcasing the power of RL-trained reasoning.

Sources: [Advanced version of Gemini with Deep Think officially achieves gold-medal standard at the IMO (Google DeepMind)](https://deepmind.google/blog/advanced-version-of-gemini-with-deep-think-officially-achieves-gold-medal-standard-at-the-international-mathematical-olympiad/); [OpenAI announcement on X](https://x.com/OpenAI/status/1946594928945148246); [OpenAI Model Earns Gold-Medal Score at International Math Olympiad (Scientific American)](https://www.scientificamerican.com/article/openai-model-earns-gold-medal-score-at-international-math-olympiad-and/); [Harmonic Aristotle IMO 2025 paper (arXiv 2510.01346)](https://arxiv.org/abs/2510.01346); [ByteDance Seed-Prover IMO 2025 result](https://seed.bytedance.com/en/blog/bytedance-seed-prover-achieves-silver-medal-score-in-imo-2025)

### 2025-07-29: Stanford's 'Virtual Lab' of AI agents designs SARS-CoV-2 nanobodies validated in the lab

***; Stanford University, Chan Zuckerberg Biohub; Science & math; confidence high; https://postcutoff.com/e/2025-07-29-virtual-lab-ai-agents-nanobodies/

James Zou's group (Nature, 2025) had an LLM 'principal investigator' agent run a team of AI scientist agents. The team built a pipeline combining ESM, AlphaFold-Multimer and Rosetta and designed 92 nanobodies. Two showed improved binding to recent SARS-CoV-2 variants (JN.1 or KP.3) while keeping binding to the ancestral spike.

- Agents: PI agent plus specialist agents (immunology, computational biology, ML) and a critic
- 92 nanobodies designed; 2 with improved binding to JN.1 or KP.3
- Human role: high-level feedback and all wet-lab work; preprint Nov 2024, Nature 2025

#### What happened
Instead of a single model, a simulated research group of LLM agents held "meetings", chose tools and designed an experiment that humans ran.

#### Why it matters
It was a peer-reviewed demonstration of multi-agent AI doing interdisciplinary research design with real lab outcomes.

Sources: [The Virtual Lab of AI agents designs new SARS-CoV-2 nanobodies (Nature)](https://www.nature.com/articles/s41586-025-09442-9); [GitHub: zou-group/virtual-lab](https://github.com/zou-group/virtual-lab)

### 2025-07-30: Interpretable neural network discovers new non-reciprocal force laws in dusty plasma

***; Emory University; Science & math; confidence high; https://postcutoff.com/e/2025-07-30-ai-discovers-dusty-plasma-physics/

Emory physicists (PNAS, July 2025) trained a physics-structured neural network on 3D particle trajectories from dusty-plasma experiments. It learned the non-reciprocal forces between particles with over 99% accuracy and overturned standard assumptions: particle charge is not simply proportional to radius, and the distance dependence of the forces is not universal. The work won the 2026 PNAS Cozzarelli Prize.

- PNAS vol 122 issue 31 (2025); ScienceDaily repost Apr 2026 ('AI just discovered new physics in the fourth state of matter')
- >99% accuracy in describing non-reciprocal interparticle forces
- Corrects long-held assumptions in dusty-plasma theory
- Justin Burton: 'We showed that we can use AI to discover new physics. Our AI method is not a black box.'

#### What happened
Instead of fitting a pre-assumed force law, the team built physical structure into a neural network and let it learn the interactions from data. The learned laws contradicted textbook assumptions.

#### Why it matters
It is a clean example of AI discovering new physical laws that humans can interpret, rather than just making predictions.

Sources: [ScienceDaily: AI just discovered new physics in the fourth state of matter](https://www.sciencedaily.com/releases/2026/04/260422044635.htm); [Emory News: AI and dusty plasma](https://news.emory.edu/features/2025/07/esc_ai_dusty_plasma_30-07-2025/index.html); [Emory: scientists receive Cozzarelli Prize](https://news.emory.edu/stories/2026/05/emory-scientists-receive-cozzarelli-prize-discovery-new-physics-dusty-plasma); [arXiv 2310.05273 (preprint)](https://arxiv.org/abs/2310.05273)

### 2025-08-14: Generative AI designs new antibiotics that kill drug-resistant gonorrhoea and MRSA

****; MIT; Science & math; confidence high; https://postcutoff.com/e/2025-08-14-mit-generative-ai-antibiotics/

MIT's Collins lab (Cell, Aug 2025) used generative models to design more than 36 million candidate compounds from scratch. Lead NG1 kills multidrug-resistant Neisseria gonorrhoeae and DN1 kills MRSA, clearing skin infections in mice. Both act on bacterial membranes by novel mechanisms and are structurally unlike any known antibiotic.

- >36 million compounds generated (fragment-based and unconstrained generation)
- NG1: active against multidrug-resistant N. gonorrhoeae; DN1: cleared MRSA skin infections in mice
- Novel membrane-targeting mechanisms; preclinical only

#### What happened
The team moved from screening existing libraries to generating new molecules, filtering tens of millions of designs down to a few synthesised leads.

#### Why it matters
It showed generative AI exploring chemical space beyond existing compound libraries for one of medicine's most urgent needs.

Sources: [MIT News: Using generative AI, researchers design compounds that can kill drug-resistant bacteria](https://news.mit.edu/2025/using-generative-ai-researchers-design-compounds-kill-drug-resistant-bacteria-0814); [Euronews: MIT scientists use AI to develop new antibiotics for gonorrhoea and MRSA](https://www.euronews.com/health/2025/08/15/mit-scientists-use-ai-to-develop-new-antibiotics-for-stubborn-gonorrhoea-and-mrsa)

### 2025-08-20: GPT-5 Pro proves an improved convex-optimisation bound, which humans had already surpassed

**; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2025-08-20-gpt-5-pro-convex-optimization-proof/

OpenAI's Sébastien Bubeck reported that GPT-5 Pro, in about 17 minutes, proved that gradient descent on L-smooth convex functions yields a convex sequence of function values for step sizes up to 1.5/L. The paper's v1 had proved it for 1/L. However, the authors' own v2 had already proved the tight 1.75/L bound.

- Problem: for which step sizes η is the optimisation curve of gradient descent convex? v1 proved η ≤ 1/L and gave a counterexample above 1.75/L
- GPT-5 Pro proved η ≤ 1.5/L by a different argument; Bubeck checked it
- The human authors' updated version had already closed the gap at 1.75/L
- Bubeck: 'Claim: gpt-5-pro can prove new interesting mathematics.'

#### What happened
Bubeck gave GPT-5 Pro the open question from v1 of a paper; the model produced a valid proof of an intermediate bound.

#### Why it matters
It was one of the first widely discussed cases of an LLM producing correct new research-level mathematics. The fact that humans had already done better also foreshadowed later disputes over novelty.

Sources: [Sébastien Bubeck on X](https://x.com/SebastienBubeck/status/1958198661139009862); [whataifound.org: GPT-5 convex bound](https://whataifound.org/finding/2025-08-gpt5-convex-bound); [What does GPT-5's new math claim actually mean?](https://allthings.how/what-does-gpt-5s-new-math-claim-actually-mean/)

### 2025-09-04: DeepMind's Deep Loop Shaping cuts LIGO control noise 30–100×

***; Google DeepMind, Caltech, Gran Sasso Science Institute; Science & math; confidence high; https://postcutoff.com/e/2025-09-04-deepmind-ligo-deep-loop-shaping/

In Science (Sept 2025), DeepMind, LIGO/Caltech and GSSI reported an RL control method trained with frequency-domain rewards. Tested on hardware at LIGO Livingston, it reduced control noise in the 10–30 Hz band by more than 30×, and up to 100× in sub-bands, beating the design goal.

- >30× noise reduction in the 10–30 Hz observation band (up to 100× in sub-bands)
- Demonstrated on LIGO Livingston hardware
- Could let LIGO detect more and heavier black-hole mergers and intermediate-mass black holes

#### What happened
An RL controller learned to stabilise LIGO's mirrors while injecting far less noise into the frequencies where gravitational waves are measured.

#### Why it matters
It extends the reach of one of physics' most sensitive instruments without new hardware.

Sources: [Improving cosmological reach of a gravitational wave observatory using Deep Loop Shaping (Science)](https://www.science.org/doi/10.1126/science.adw1291); [Caltech: Artificial intelligence helps boost LIGO](https://www.caltech.edu/about/news/artificial-intelligence-helps-boost-ligo)

### 2025-09-10: Math Inc's Gauss agent completes the Strong Prime Number Theorem formalisation in Lean in three weeks

****; Math Inc; Science & math; confidence high; https://postcutoff.com/e/2025-09-10-math-inc-gauss-strong-pnt/

Math Inc (Christian Szegedy) announced that its autoformalization agent Gauss completed Terence Tao and Alex Kontorovich's Strong Prime Number Theorem project in Lean in about 3 weeks, producing ~25,000 lines of Lean and over 1,000 theorems and definitions. Human experts had worked on the project for 18+ months.

- ~25,000 lines of Lean, 1,000+ theorems and definitions, code public on GitHub
- Human project began in 2024 and had stalled on complex-analysis prerequisites
- Announcement day approximate (10–11 Sep 2025)

#### What happened
Gauss read the human blueprint of the Strong PNT project and wrote the missing Lean formalisations, including a large amount of complex analysis.

#### Why it matters
Autoformalization at this scale points to a future where new proofs, including AI-generated ones, are routinely machine-checked. That matters as AI floods mathematics with claimed proofs.

Sources: [Math Inc: Gauss](https://www.math.inc/gauss); [GitHub: math-inc/strongpnt](https://github.com/math-inc/strongpnt); [Math Inc announcement on X](https://x.com/mathematics_inc/status/1966194751847461309)

### 2025-09-12: First AI-generated complete genomes: Evo models design viable bacteriophages that kill resistant E. coli

*****; Arc Institute, Stanford University; Science & math; confidence high; https://postcutoff.com/e/2025-09-12-ai-generated-phage-genomes/

Brian Hie's lab used the Evo 1 and Evo 2 genome language models to generate whole ΦX174-like bacteriophage genomes. Of ~285–300 synthesised designs, 16 were viable. Some rapidly overcame ΦX174-resistant E. coli, and one used an evolutionarily distant DNA-packaging protein. Preprint 12 Sep 2025; published in Science on 6 Aug 2026.

- Generated full ~5.4 kb ΦX174-family genomes; ~285–300 synthesised, 16 viable
- AI phage cocktails overcame ΦX174-resistant E. coli strains
- Cryo-EM showed one phage using a packaging protein from a distant lineage
- Raised biosecurity discussion about generative design of self-replicating agents
- Biosecurity debate (The Cryptonomist, Oct 16, 2026, summarizing): Johns Hopkins Center for Health Security specialists wrote that 'The ability to compose viral genomes using generative AI now exists; the governance to safely steer it does not' and urged that work on viruses infecting eukaryotes not be pursued; Jordi García Ojalvo (Pompeu Fabra) judged this experiment lower-risk because each genome needs individual lab testing and the viable yield was low (16 of about 300 designs built). The Johns Hopkins piece itself was not read

#### What happened
The team prompted genome language models to write complete phage genomes, synthesised hundreds, and found 16 that infected and killed bacteria, including strains resistant to the natural phage.

#### Why it matters
It is a milestone toward AI-designed life forms and phage therapies against resistant bacteria, and a biosecurity flashpoint.

Sources: [The Cryptonomist: AI generated viral genomes yield 16 viable phages, prompting safety concerns (Oct 16)](https://en.cryptonomist.ch/2026/10/16/ai-generated-viral-genomes/); [bioRxiv: generative design of novel bacteriophages with genome language models](https://www.biorxiv.org/content/10.1101/2025.09.12.675911v1); [Arc Institute: first AI-designed synthetic phage](https://arcinstitute.org/news/hie-king-first-synthetic-phage); [Stanford News: Evo 2 AI tool designs E. coli-killing bacteriophages (Science, Aug 2026)](https://news.stanford.edu/stories/2026/08/evo-2-ai-tool-e-coli-killer-bacteriophages); [C&EN: AI program designs new bacteriophages](https://cen.acs.org/biological-chemistry/genomics/ai-program-designs-new-bacteriophages/104/web/2026/08)

### 2025-09-17: DeepMind and mathematicians use neural networks to find new unstable singularities in fluid equations

***; Google DeepMind, New York University, Stanford University, Brown University; Science & math; confidence high; https://postcutoff.com/e/2025-09-17-deepmind-unstable-singularities-fluids/

A DeepMind-led team (with Tristan Buckmaster and Javier Gómez-Serrano) used physics-informed neural networks and high-precision optimisation to find new families of unstable self-similar blow-up solutions for the incompressible porous media and Boussinesq equations (3D Euler with boundary), accurate to near machine precision. This was a numerical discovery, not a proof.

- arXiv 2509.14185 (Sep 2025)
- Multiple new unstable self-similar blow-up profiles; empirical formula relating blow-up rate to order of instability
- Accuracy near double-precision round-off, enough to support future computer-assisted proofs
- Does not resolve the Navier–Stokes Millennium Problem

#### What happened
Unstable singularities are thought to be what any Navier–Stokes blow-up would look like, but they are almost impossible to find numerically. The team's neural-network method found whole families of them.

#### Why it matters
It was the groundwork of the AI-plus-computer-assisted-proof approach to fluid blow-up that culminated in the disputed 2026 Navier–Stokes claims.

Sources: [Discovery of unstable singularities (arXiv 2509.14185)](https://arxiv.org/abs/2509.14185); [Physics World: neural networks discover unstable singularities in fluid systems](https://physicsworld.com/a/neural-networks-discover-unstable-singularities-in-fluid-systems/)

### 2025-09-22: AlphaEvolve finds gadgets that prove new NP-hardness of approximation bounds for MAX-k-CUT

**; Google Research, Google DeepMind; Science & math; confidence medium; https://postcutoff.com/e/2025-09-22-alphaevolve-hardness-of-approximation/

Google researchers used AlphaEvolve to discover gadget reductions proving it is NP-hard to approximate MAX-4-CUT within 0.987 and MAX-3-CUT within 0.9649. They also built near-extremal Ramanujan graphs of up to 163 nodes for average-case hardness results; checking the gadgets was sped up ~10,000×.

- arXiv 2509.18057 'Reinforced Generation of Combinatorial Structures'
- MAX-4-CUT inapproximability 0.987; MAX-3-CUT 0.9649
- Correctness of the final theorems checked by standard (non-AI) verification

#### What happened
AlphaEvolve searched for finite combinatorial gadgets whose properties imply hardness theorems. Standard verification then turned the found objects into proofs.

#### Why it matters
AI-found objects became ingredients of rigorous complexity-theory theorems, not just numeric improvements.

Sources: [Reinforced Generation of Combinatorial Structures (arXiv 2509.18057)](https://arxiv.org/abs/2509.18057); [Google Research: AI as a research partner — advancing theoretical CS with AlphaEvolve](https://research.google/blog/ai-as-a-research-partner-advancing-theoretical-computer-science-with-alphaevolve/)

### 2025-09-27: Scott Aaronson credits GPT-5 with a key step in a quantum complexity proof

***; UT Austin, CWI, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2025-09-27-aaronson-gpt-5-qma-proof/

In 'Limits to black-box amplification in QMA' (Aaronson and Witteveen, arXiv 2509.21131), GPT-5-Thinking suggested the key function Tr[(I−E(θ))^−1] used in the proof. Aaronson called it the first paper of his where a key technical step came from AI.

- Result: black-box amplification cannot push QMA completeness error below doubly exponential or soundness error below exponential
- Aaronson: 'Within a half hour, it had suggested to look at the function…'
- Aaronson: 'if a student had given it to me, I would've called it clever'
- Blog post 'The QMA Singularity', 27 Sep 2025

#### What happened
Stuck on a technical step, Aaronson asked GPT-5 for help. Within about half an hour it proposed analysing a resolvent-trace function, which worked.

#### Why it matters
It was a credible, first-person account from a top theorist of an LLM contributing a genuine idea to a published result.

Sources: [Scott Aaronson: The QMA Singularity](https://scottaaronson.blog/?p=9183); [Limits to black-box amplification in QMA (arXiv 2509.21131)](https://arxiv.org/abs/2509.21131); [The Quantum Insider: GPT-5 serves as research assistant](https://thequantuminsider.com/2025/09/29/gpt-5-serves-as-research-assistant-in-proving-one-of-quantum-computing-theorys-trickiest-theorems/)

### 2025-10-15: Google's C2S-Scale 27B model generates a new cancer-immunotherapy hypothesis confirmed in living cells

***; Google Research, Google DeepMind, Yale University; Science & math; confidence high; https://postcutoff.com/e/2025-10-15-c2s-scale-gemma-cancer-hypothesis/

C2S-Scale 27B, a Gemma-based single-cell model, simulated over 4,000 drugs in two immune contexts. It predicted that the CK2 inhibitor silmitasertib boosts tumour antigen presentation only with low-dose interferon present. In living cells the combination raised MHC-I antigen presentation by ~50%. The link had not been reported before.

- Virtual screen of >4,000 drugs in 'immune-context-positive' vs '-neutral' settings
- Silmitasertib (CX-4945) + low-dose interferon: ~50% increase in antigen presentation in vitro
- In vitro only; no animal or clinical data; preprint

#### What happened
Researchers asked the model which drugs would amplify immune signals only in an immune-active context. Its top novel prediction held up in lab tests.

#### Why it matters
It is evidence that scaling biological foundation models can yield testable, novel hypotheses, though only in vitro so far.

Sources: [Google: How a Gemma model helped discover a new potential cancer therapy pathway](https://blog.google/technology/ai/google-gemma-ai-cancer-therapy-discovery/); [DDW: Google AI model reveals new way to improve immunotherapy](https://www.ddw-online.com/google-ai-model-reveals-new-way-to-improve-immunotherapy-38114-202510/)

### 2025-10-16: Google DeepMind partners with Commonwealth Fusion Systems to optimise and control the SPARC tokamak with AI

***; Google DeepMind, Commonwealth Fusion Systems; Science & math; confidence high; https://postcutoff.com/e/2025-10-16-deepmind-cfs-torax-fusion/

DeepMind announced a research partnership with Commonwealth Fusion Systems (CFS) for CFS's SPARC tokamak, which aims to be the first magnetic-confinement device to produce net fusion energy. The work uses DeepMind's open-source JAX plasma simulator TORAX, RL and evolutionary search to find high-output operating scenarios, and RL controllers for real-time tasks such as spreading exhaust heat on the reactor wall. Google is also an investor in CFS.

- TORAX: open-source, differentiable plasma transport simulator written in JAX; CFS: it 'saved us countless hours'
- Three strands: fast simulation (TORAX), searching operating scenarios with RL/evolutionary algorithms, and RL real-time control (e.g. heat-load distribution)
- Builds on DeepMind's 2022 RL tokamak magnetic-control work with EPFL's Swiss Plasma Center (TCV)
- Google has invested directly in CFS

#### What happened
DeepMind and CFS said they would use AI to plan and run SPARC's plasma campaigns before the machine reaches full power, with TORAX as the shared simulation layer.

#### Why it matters
It moves AI plasma control from academic demos (TCV, DIII-D) into the commissioning plan of a privately built machine that aims for net energy.

Sources: [Google DeepMind: Bringing AI to the next generation of fusion energy](https://deepmind.google/blog/bringing-ai-to-the-next-generation-of-fusion-energy/); [TORAX on GitHub](https://github.com/google-deepmind/torax)

### 2025-10-17: OpenAI researchers claim GPT-5 'solved' 10 Erdős problems; the solutions were already in the literature

***; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2025-10-17-gpt-5-erdos-problems-controversy/

In mid-October 2025 OpenAI's Kevin Weil tweeted that GPT-5 'found solutions to 10 (!) previously unsolved Erdős problems'. Thomas Bloom, who runs erdosproblems.com, called this 'a dramatic misrepresentation': GPT-5 had found existing papers solving problems listed as open only because he did not know of them. The tweets were deleted.

- Claim (deleted tweet by Kevin Weil): 'GPT-5 found solutions to 10 (!) previously unsolved Erdős problems and made progress on 11 others'
- Bloom: GPT-5 'found references, which solved these problems, that I personally was unaware of'
- Demis Hassabis: 'This is embarrassing.' Yann LeCun also mocked the claim
- What was real: GPT-5 was an effective literature-search tool, and several problems' statuses were updated

#### What happened
OpenAI researchers publicised GPT-5 "solutions" to Erdős problems. The site's maintainer explained that "open" on his site meant only that he did not know of a solution, and that GPT-5 had surfaced old papers.

#### Why it matters
The episode set the standard of scepticism for later AI maths claims, and led to Tao's public wiki tracking exactly what AI contributed to each Erdős problem. It also showed the real, less glamorous value of AI literature search.

Sources: [TechCrunch: OpenAI's 'embarrassing' math](https://techcrunch.com/2025/10/19/openais-embarrassing-math/); [The Decoder: OpenAI researcher announced a GPT-5 math breakthrough that never happened](https://the-decoder.com/leading-openai-researcher-announced-a-gpt-5-math-breakthrough-that-never-happened/); [Terence Tao's wiki: AI contributions to Erdős problems](https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems)

### 2025-10-22: Agents4Science 2025: first conference where AI must be first author and reviewer

***; Stanford University, Together AI; Science & math; confidence high; https://postcutoff.com/e/2025-10-22-agents4science-conference/

Agents4Science 2025 (22 Oct 2025, virtual) required AI systems as first authors and used GPT-5, Gemini 2.5 and Claude Sonnet 4 as reviewers: 315 submissions, 253 reviewed, 48 accepted, making AI-authored science an explicit experiment.

- 315 submissions; 62 desk-rejected; 253 reviewed by three LLM reviewers (GPT-5, Gemini 2.5, Claude Sonnet 4)
- Top 79 also got human expert review; 48 papers accepted
- Organised by James Zou's group at Stanford with Together AI
- Secondary reports say only a handful of accepted papers were fully AI-generated (unverified figure)

#### What happened
Stanford researchers ran a conference in which AI agents had to be listed as first authors and LLMs did first-round reviewing, to study openly what AI-driven research looks like.

#### Why it matters
It made AI authorship an explicit, measurable experiment rather than a hidden practice, and produced data on the strengths and failure modes of AI reviewers.

Sources: [Agents4Science analysis paper (arXiv 2511.15534)](https://arxiv.org/abs/2511.15534); [Agents4Science accepted papers](https://agents4science.stanford.edu/accepted-papers.html); [Nature news on the AI-authored conference](https://www.nature.com/articles/d41586-025-03363-3); [Science News: a science conference tests AI agents](https://www.sciencenews.org/article/science-conference-test-ai-agents)

### 2025-10-24: Genentech's GNEprop screens 1.4 billion virtual compounds and finds 82 new antibacterial hits

**; Genentech, NVIDIA, Mila; Science & math; confidence high; https://postcutoff.com/e/2025-10-24-genentech-gneprop-antibacterial-screening/

In Nature Biotechnology (24 Oct 2025), Genentech researchers with NVIDIA and Mila described GNEprop, a graph neural network trained on a ~2-million-compound phenotypic screen against sensitized E. coli. Used to screen more than 1.4 billion synthetically accessible molecules virtually, it found 82 compounds with confirmed antibacterial activity. The hit rate was about 90 times higher than the original high-throughput screen, and several scaffolds were new.

- Training data: ~2 million small molecules screened experimentally against a sensitized E. coli strain
- Virtual screen of >1.4 billion synthetically accessible compounds; 82 confirmed actives; ~90-fold higher hit rate than HTS
- GNEprop includes explainability (active motifs) and out-of-distribution detection for structural novelty vs known antibiotics
- Authors include Gabriele Scalia, Steven T. Rutherford and Tommaso Biancalani (Genentech BRAID / Infectious Diseases / Computational Chemistry)
- Preprint first posted on bioRxiv in Sept 2024; Nature Biotechnology ran an accompanying commentary

#### What happened
Genentech combined a huge wet-lab screen with a graph neural network, then used the model to search a 1.4-billion-molecule virtual library. The model's picks were far more likely to kill bacteria than randomly screened compounds, and several had new scaffolds.

#### Why it matters
Industrial-scale evidence for ML-guided antibiotic discovery, following MIT's halicin and abaucin work. It is a hit-finding result; no candidate has entered the clinic.

Sources: [Nature Biotechnology: Deep-learning-based virtual screening of antibacterial compounds](https://www.nature.com/articles/s41587-025-02814-6); [Nature Biotechnology commentary: Deep learning speeds the search for new antibiotic scaffolds](https://www.nature.com/articles/s41587-025-02806-6); [bioRxiv preprint (Sept 2024)](https://www.biorxiv.org/content/10.1101/2024.09.11.612340v1); [STAT (sponsored): How AI is supercharging antibiotic discovery](https://www.statnews.com/sponsor/2026/01/12/how-ai-is-supercharging-antibiotic-discovery/)

### 2025-11-05: Baker lab designs antibodies from scratch with atomic accuracy using RFdiffusion

***; University of Washington Institute for Protein Design; Science & math; confidence high; https://postcutoff.com/e/2025-11-05-ai-designed-antibodies-rfdiffusion/

In Nature (Nov 2025) the Baker lab reported de novo design of VHH nanobodies, scFvs and full antibodies against chosen epitopes. Cryo-EM confirmed atomically accurate binding poses and CDR loops for influenza haemagglutinin and C. difficile toxin B. Chai Discovery's Chai-2 separately reported ~16% hit rates for zero-shot antibody design.

- Targets included influenza HA and C. difficile toxin TcdB; cryo-EM matched designs at atomic level
- Chai-2 (bioRxiv, Jul 2025): ~16% de novo antibody hit rate; binders for ~50% of 52 targets with ≤20 designs each (preprint)

#### What happened
After years of designing small binders, AI protein design reached antibodies, the most important class of biologic drugs.

#### Why it matters
Computational antibody design could replace months of animal immunisation and library screening in drug discovery.

Sources: [Atomically accurate de novo design of antibodies with RFdiffusion (Nature)](https://www.nature.com/articles/s41586-025-09721-5); [GeekWire: Nobel winner's lab notches AI-designed antibodies that hit their targets](https://www.geekwire.com/2025/nobel-winners-lab-notches-another-breakthrough-ai-designed-antibodies-that-hit-their-targets/); [Chai-2 zero-shot antibody design (bioRxiv)](https://www.biorxiv.org/content/10.1101/2025.07.05.663018v1)

### 2025-11-05: Tao, Gómez-Serrano, Georgiev and Wagner test AlphaEvolve on 67 maths problems

***; Google DeepMind, UCLA, Brown University; Science & math; confidence high; https://postcutoff.com/e/2025-11-05-alphaevolve-tao-67-problems/

In 'Mathematical exploration and discovery at scale' (arXiv 2511.02864), Bogdan Georgiev, Javier Gómez-Serrano, Terence Tao and Adam Zsolt Wagner ran AlphaEvolve on 67 problems in analysis, combinatorics, geometry and number theory. It rediscovered the best known constructions in most cases and improved several. Some runs were chained with Deep Think and AlphaProof to produce proofs.

- 67 problems; a public repository with per-problem notebooks
- Rediscovered state-of-the-art constructions in most cases and improved on several
- Pipeline: AlphaEvolve (constructions) → Deep Think (informal proof) → AlphaProof (formal proof) in some cases
- Tao blog post, 5 Nov 2025

#### What happened
Leading mathematicians stress-tested AlphaEvolve on a large, varied problem set and published both the successes and the failures.

#### Why it matters
Coming from Tao, it gave the maths community a credible, balanced picture of what AI search could do, just before the 2026 surge.

Sources: [Mathematical exploration and discovery at scale (arXiv 2511.02864)](https://arxiv.org/abs/2511.02864); [Terence Tao: Mathematical exploration and discovery at scale](https://terrytao.wordpress.com/2025/11/05/mathematical-exploration-and-discovery-at-scale/); [GitHub: alphaevolve_repository_of_problems](https://github.com/google-deepmind/alphaevolve_repository_of_problems)

### 2025-11-05: Edison Scientific's Kosmos AI scientist claims six months of research per run

***; Edison Scientific, FutureHouse; Science & math; confidence medium; https://postcutoff.com/e/2025-11-05-kosmos-ai-scientist/

In early November 2025 FutureHouse spin-out Edison Scientific launched Kosmos, an autonomous AI scientist that reads ~1,500 papers and runs ~42,000 lines of analysis code per 12-hour run; beta users estimated one run equals ~6 months of their work, and 79.4% of its statements were judged accurate. It reported 7 discoveries, 3 reproducing unpublished findings.

- Typical run: 12 hours, ~1,500 papers read, ~42,000 lines of code executed (structured 'world model' shared across agents)
- 79.4% of conclusions judged accurate by independent scientists
- 7 discoveries across metabolomics, materials, neuroscience, genetics: 3 reproduced unpublished/preprint findings, 4 presented as novel
- '6 months of work in one day' is a beta-user estimate, not an independent measurement

#### What happened
Kosmos runs many parallel literature-search and data-analysis agents coordinated through a shared structured world model, producing reports in which every statement is traced to code or a paper.

#### Why it matters
It is an early commercial "AI scientist" whose headline value is reproducing months-long analyses overnight. The strongest evidence is that it independently reached conclusions matching unpublished human work.

Sources: [Edison Scientific: Announcing Kosmos](https://edisonscientific.com/news/announcing-kosmos); [Kosmos: An AI Scientist for Autonomous Discovery (arXiv 2511.02824)](https://arxiv.org/abs/2511.02824); [Alzforum: Introducing Kosmos, AI scientist makes discoveries overnight](https://www.alzforum.org/news/research-news/introducing-kosmos-ai-scientist-makes-discoveries-overnight)

### 2025-11-20: OpenAI publishes 'Early science acceleration experiments with GPT-5', including four new math results

***; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2025-11-20-openai-gpt-5-science-acceleration/

On 20 Nov 2025 OpenAI and academic co-authors, including Timothy Gowers, released case studies of GPT-5 contributing to research in maths, physics, astronomy, computer science, biology and materials science. The paper includes four new mathematical results checked by the human authors. It frames GPT-5 as an expert-guided collaborator, not an autonomous discoverer.

- arXiv 2511.16072; authors include Sébastien Bubeck, Timothy Gowers, Alex Lupsasca, Mehtaab Sawhney, Mark Sellke, Derya Unutmaz, Kevin Weil
- Four new maths results verified by humans, including an Erdős-problem result by Sawhney and Sellke with GPT-5
- Physics: GPT-5 Pro re-derived Lupsasca's hidden SL(2,R) symmetries of the Kerr black-hole wave equation, a rediscovery of a known result that needed a warm-up prompt
- Biology: from an unpublished chart, GPT-5 Pro proposed a mechanism (IL-2 interference) for how brief 2-deoxyglucose exposure pushes CD4+ T cells toward a Th17-like state, and correctly predicted a held-out experiment in Derya Unutmaz's lab
- Collaborators came from Vanderbilt, UC Berkeley, Columbia, Oxford, Cambridge, LLNL and the Jackson Laboratory

#### What happened
A month after an embarrassing overclaim about Erdős problems, OpenAI published a more careful, multi-author record of where GPT-5 had actually helped working scientists, and where it failed.

#### Why it matters
It marked a shift from benchmark claims to documented research contributions, and set the template for OpenAI's 2026 "OpenAI for Science" results.

Sources: [Early science acceleration experiments with GPT-5 (arXiv 2511.16072)](https://arxiv.org/abs/2511.16072); [OpenAI: Accelerating science with GPT-5](https://openai.com/index/accelerating-science-gpt-5/); [Alex Lupsasca on GPT-5 Pro and black-hole symmetries (OpenAI Academy)](https://academy.openai.com/public/blogs/alex-lupsasca-gpt-5-pro-black-hole-physics-hidden-symmetries); [OpenAI: GPT-5 and an immunology mystery](https://openai.com/index/gpt-5-immunology-mystery/)

### 2025-12-01: AI searches 100 million Hubble images in 2.5 days, finding ~1,400 anomalies including 800+ never described

**; European Space Agency; Science & math; confidence high; https://postcutoff.com/e/2025-12-01-ai-finds-1400-anomalies-hubble-archive/

ESA researchers (Astronomy & Astrophysics, Dec 2025) used AnomalyMatch to scan 99.6 million Hubble Legacy Archive cutouts in about 2.5 days. They found ~1,400 anomalous objects, over 800 previously undescribed, including 86 new candidate gravitational lenses, jellyfish and ring galaxies, and objects that defy classification.

- 99.6M image cutouts in ~2.5 days
- ~1,400 anomalies; >800 not previously described; 86 candidate gravitational lenses
- Humans inspected all flagged images

#### What happened
A semi-supervised anomaly detector swept the full Hubble archive, and experts reviewed the top-ranked oddities.

#### Why it matters
It shows the AI-first search pattern that surveys such as Rubin/LSST and Euclid will depend on.

Sources: [ESA/Hubble: heic2603](https://esahubble.org/news/heic2603/); [ESA: 1,400 quirky objects found in Hubble's archive](https://www.esa.int/Science_Exploration/Space_Science/1400_quirky_objects_found_in_Hubble_s_archive); [arXiv 2505.03508](https://arxiv.org/abs/2505.03508)

### 2025-12-01: Physics Letters B paper built on a GPT-5 idea draws criticism that it tests the wrong thing

**; Michigan State University, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2025-12-01-hsu-gpt-5-physics-paper-dispute/

Physicist Steve Hsu published a Physics Letters B paper whose main idea, applying the Tomonaga–Schwinger formalism to test state-dependent (nonlinear) quantum mechanics, came from GPT-5. He called it the 'first research article in theoretical physics in which the main idea came from an AI'. Jonathan Oppenheim argued the criterion detects nonlocality rather than nonlinearity, and Peter Woit called it 'Theoretical Physics Slop'.

- Claim (Hsu): 'first research article in theoretical physics in which the main idea came from an AI'
- Rebuttal: Oppenheim, arXiv 2512.07809
- Peer-reviewed publication did not prevent a substantive correctness dispute

#### What happened
A physicist credited GPT-5 with the core idea of a peer-reviewed paper, and other physicists argued the idea was flawed.

#### Why it matters
It is a cautionary example: AI-originated ideas can pass peer review and still be wrong or misframed.

Sources: [The Decoder: Physicist Steve Hsu publishes research built around a core idea generated by GPT-5](https://the-decoder.com/physicist-steve-hsu-publishes-research-built-around-a-core-idea-generated-by-gpt-5/); [Oppenheim rebuttal (arXiv 2512.07809)](https://arxiv.org/abs/2512.07809); [Peter Woit: Theoretical Physics Slop](https://www.math.columbia.edu/~woit/wordpress/?p=15362)

### 2025-12-06: AxiomProver produces machine-checked Lean proofs for all 12 Putnam 2025 problems

***; Axiom Math; Science & math; confidence medium; https://postcutoff.com/e/2025-12-06-axiomprover-putnam-2025/

Axiom Math's autonomous Lean 4 prover solved 8 of 12 problems of the 6 Dec 2025 Putnam competition within exam time and the remaining 4 in the following days, all as machine-checked Lean proofs published on GitHub.

- Putnam 2025 held 6 Dec 2025; 8/12 solved within the exam window, 12/12 after extra time
- Proofs are formal Lean 4 and publicly released
- Axiom says no human scored 12/12, but the 12/12 includes solutions found after the deadline
- Not an official entry; self-reported timing

#### What happened
Axiom Math ran its prover on the 2025 Putnam problems, producing formal Lean 4 proofs that any Lean installation can check.

#### Why it matters
Formal verification removes grading disputes like those around informal IMO proofs, and showed formal provers catching up with informal LLMs on hard competition maths.

Sources: [GitHub: AxiomMath/putnam2025 (Lean proofs)](https://github.com/AxiomMath/putnam2025); [Axiom Math: From seeing why to checking everything](https://axiommath.ai/research/from-seeing-why-to-checking-everything/)

### 2025-12-08: Genuine AI-assisted solutions to Erdős problems begin: #124 (Aristotle), #1026 (48-hour human–AI collaboration)

****; Harmonic, Google DeepMind, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2025-12-08-ai-erdos-problems-wave-late-2025/

In Nov–Dec 2025 AI tools produced the first genuinely new (if modest) solutions to Erdős problems. Harmonic's Aristotle proved a version of #124 in Lean autonomously (29 Nov). Erdős #1026 (posed 1975) was fully solved within ~48 hours by humans combining Aristotle, AlphaEvolve, GPT and deep-research tools (7–9 Dec). Terence Tao warned these were 'long-tail' problems.

- #124 (from a 1995 paper): Aristotle proved it autonomously in Lean from the formal statement; Bloom noted it was the easier of two variants, and Tao's wiki lists it as partial
- #1026: Aristotle proved the key case c(k²)=1/k in Lean (7 Dec); full answer c(k²+2a+1) = k/(k²+a) assembled by 8–9 Dec
- Tao on #1026: 'It was only through the combined efforts of all the contributors and their tools that all these key inputs were able to be assembled within 48 hours.'
- #367: partial result by Alexeev, van Doorn and Tao with Aristotle and Gemini Deep Think (Nov 2025)
- #707 ($1000 problem): Alexeev & Mixon disproved it with ChatGPT-assisted Lean checks, then found Marshall Hall Jr. had a counterexample in 1947
- Tao: such results 'do not meet the hyped up goal of AI autonomously solving major mathematical open problems'

#### What happened
After the October 2025 fiasco, a distributed community of mathematicians and amateurs began systematically attacking the ~1,100 Erdős problems with AI tools, with results logged on Tao's wiki. The first genuinely new results arrived within weeks.

#### Why it matters
Erdős problems became the first large, public, verifiable scoreboard for AI in research mathematics, and set the stage for 2026's much larger results.

Sources: [Terence Tao's wiki: AI contributions to Erdős problems](https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems); [Terence Tao: The story of Erdős problem #1026](https://terrytao.wordpress.com/2025/12/08/the-story-of-erdos-problem-126/); [erdosproblems.com forum: problem #124](https://www.erdosproblems.com/forum/thread/124); [Xena Project: formalization of Erdős problems](https://xenaproject.wordpress.com/2025/12/05/formalization-of-erdos-problems/); [Alexeev & Mixon on Erdős #707 (arXiv 2510.19804)](https://arxiv.org/abs/2510.19804)

### 2026-01-06: Erdős problem #728 solved near-autonomously by GPT-5.2 Pro and Harmonic's Aristotle, with a Lean proof

****; OpenAI, Harmonic; Science & math; confidence high; https://postcutoff.com/e/2026-01-06-erdos-728-gpt-5-2-aristotle/

On 4–6 Jan 2026 amateur Kevin Barreto relayed an informal argument from GPT-5.2 Pro to Harmonic's Aristotle, which formalised it in Lean. It was widely accepted as the first Erdős problem solved essentially autonomously by AI with no prior solution in the literature. Terence Tao said the win 'says more about speed than difficulty'.

- Jan 4: first run solved an ambiguous reading of the problem; Jan 5: GPT-5.2 Pro upgraded the argument to the intended statement; Jan 6: Aristotle formalised it
- Tao: 'a near-autonomous solution that has not been reproduced in existing literature'
- Human role: prompting and relaying only; Barreto clarified no mathematical hint was given
- Write-up: arXiv 2601.07421

#### What happened
An amateur used two commercial AI systems in tandem: one to find a proof and one to formally verify it. After fixing a misreading of the problem statement, the pipeline produced a Lean-checked solution.

#### Why it matters
It showed the combination of informal LLM reasoning with formal verification as a practical, trustworthy workflow for research maths that non-experts could run.

Sources: [Resolution of Erdős Problem #728: a writeup of Aristotle's Lean proof (arXiv 2601.07421)](https://arxiv.org/abs/2601.07421); [Terence Tao's wiki: AI contributions to Erdős problems](https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems); [The Decoder: Tao says GPT-5.2 Pro cracked an Erdős problem but warns the win says more about speed than difficulty](https://the-decoder.com/terence-tao-says-gpt-5-2-pro-cracked-an-erdos-problem-but-warns-the-win-says-more-about-speed-than-difficulty/)

### 2026-02-05: GPT-5 autonomously runs 36,000 experiments in Ginkgo's cloud lab, cutting protein-synthesis cost 40%

***; OpenAI, Ginkgo Bioworks; Science & math; confidence medium; https://postcutoff.com/e/2026-02-05-gpt-5-ginkgo-autonomous-lab/

OpenAI and Ginkgo Bioworks reported that GPT-5, in a closed loop with Ginkgo's automated cloud lab, tested over 36,000 cell-free protein synthesis reaction compositions on 580 plates over six rounds. It cut the cost of producing sfGFP by 40% ($422/g vs $698/g), with reagent cost 57% lower, reaching a new state of the art within three rounds.

- 6 closed-loop rounds; 36,000+ compositions; 580 plates
- Cost $422/g vs $698/g of sfGFP (−40%); reagent cost −57%
- The optimised mix is now sold commercially by Ginkgo
- bioRxiv preprint (Feb 2026); not yet peer-reviewed

#### What happened
GPT-5 designed each round of experiments, Ginkgo's robots ran them, and the results fed back to the model.

#### Why it matters
It is a concrete, economically meaningful result from an LLM driving a physical lab end-to-end.

Sources: [OpenAI: GPT-5 lowers protein synthesis cost](https://openai.com/index/gpt-5-lowers-protein-synthesis-cost/); [bioRxiv preprint](https://www.biorxiv.org/content/10.64898/2026.02.05.703998v1); [R&D World: GPT-5 autonomously ran 36,000 protein-synthesis experiments](https://www.rdworldonline.com/openais-gpt-5-autonomously-ran-36000-protein-synthesis-experiments-in-ginkgo-bioworks-cloud-lab/)

### 2026-02-10: Isomorphic Labs unveils IsoDDE drug-discovery engine, hailed as 'an AlphaFold 4' — but proprietary

***; Isomorphic Labs, Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2026-02-10-isomorphic-isodde/

On 10 Feb 2026 DeepMind spin-off Isomorphic Labs released a 27-page technical report on IsoDDE, a proprietary drug-discovery engine that outperforms AlphaFold 3-era tools and Boltz-2 on protein–ligand binding, affinity and antibody-structure prediction; outside scientists called it "on the scale of an AlphaFold 4" but lamented the lack of details.

- Announced 2026-02-10 via a 27-page technical report; model not released
- Beats Boltz-2 and physics-based methods at binding-affinity prediction; state of the art on antibody–target interactions; generalises to molecules unlike its training data
- Mohammed AlQuraishi: 'a major advance, on the scale of an AlphaFold4... The problem is that we know nothing of the details.'

#### What happened
Isomorphic Labs, led by Demis Hassabis, described IsoDDE, a successor-class system to AlphaFold 3 aimed at drug discovery, in a technical report without releasing code or weights.

#### Why it matters
Shows AI structural biology continuing to advance fast, but also a shift from DeepMind's open-science AlphaFold tradition toward proprietary commercial models.

Sources: [Nature: 'An AlphaFold 4' — scientists marvel at DeepMind drug spin-off's exclusive new AI](https://www.nature.com/articles/d41586-026-00365-7); [Scientific American (reprint of Nature news)](https://www.scientificamerican.com/article/an-alphafold-4-scientists-marvel-at-deepmind-drug-spin-offs-exclusive-new-ai/)

### 2026-02-11: DeepMind's Aletheia agent and Gemini Deep Think report autonomous Erdős solutions and new physics and CS results

****; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2026-02-11-deepmind-aletheia-deep-think-science/

Google DeepMind described Aletheia, a Gemini Deep Think–based maths research agent. It autonomously solved Erdős problems #652, #654 and #1040 and resolved #1051, which led to a peer-reviewed generalisation. A semi-autonomous sweep of 700 open Erdős problems resolved 4 and found existing literature solutions for several more. With 18 external researchers, Deep Think also produced a cosmic-string gravitational-radiation result and refuted a decade-old online-optimisation conjecture.

- Aletheia paper: arXiv 2602.10177; up to 90% on IMO-ProofBench Advanced
- Autonomous: Erdős #652, #654, #1040; #1051 resolved and generalised
- 700-problem sweep: 4 open questions resolved; several 'open' problems found already solved in the literature
- Physics: a new Gegenbauer-polynomial solution removing singularities in cosmic-string gravitational radiation calculations
- One paper (eigenweights) classed by DeepMind as essentially autonomous and publishable

#### What happened
DeepMind packaged Gemini Deep Think into an agent that generates, checks and revises proofs, and released a batch of results across maths, CS and physics.

#### Why it matters
Google's answer to OpenAI's maths push showed that several labs could now produce publishable research-level results, although many were on problems nobody had seriously attacked.

Sources: [Google DeepMind: Accelerating mathematical and scientific discovery with Gemini Deep Think](https://deepmind.google/blog/accelerating-mathematical-and-scientific-discovery-with-gemini-deep-think/); [Aletheia paper (arXiv 2602.10177)](https://arxiv.org/abs/2602.10177); [InfoQ: DeepMind Aletheia agentic math](https://www.infoq.com/news/2026/04/deepmind-aletheia-agentic-math/)

### 2026-02-13: GPT-5.2 conjectures, and an OpenAI model proves, that 'single-minus' gluon tree amplitudes are nonzero

***; OpenAI, Institute for Advanced Study, Harvard University, University of Cambridge, Vanderbilt University; Science & math; confidence medium; https://postcutoff.com/e/2026-02-13-gpt-5-2-single-minus-gluon-amplitudes/

A preprint by Guevara, Lupsasca, Skinner, Strominger and OpenAI's Kevin Weil showed that tree-level single-minus gluon amplitudes, long assumed to vanish, are nonzero in a 'half-collinear' region of (2,2)-signature kinematics. GPT-5.2 Pro conjectured the general formula from the n=3–6 cases, and an internal OpenAI model produced a proof in about 12 hours, which the humans checked. A graviton extension followed on 4 Mar 2026.

- GPT-5.2 Pro guessed the closed-form all-n formula from small cases; an internal model proved it in ~12 hours
- Follow-up (4 Mar 2026): extension to gravitons, with the paper drafted by GPT-5.2 Pro
- Critique (Hugging Face blog): the physics framing was human work; the result applies only in non-physical (2,2) signature on a measure-zero kinematic slice; the loophole may have been noted by Witten in 2003

#### What happened
Leading amplitude theorists used OpenAI models to guess and prove a general formula in a corner of gauge-theory kinematics.

#### Why it matters
It is a showcase of LLMs as conjecture engines in theoretical physics, though critics dispute its physical significance and the AI's share of the insight.

Sources: [OpenAI: New result in theoretical physics](https://openai.com/index/new-result-theoretical-physics/); [OpenAI: Extending single-minus amplitudes to gravitons](https://openai.com/index/extending-single-minus-amplitudes-to-gravitons/); [Hugging Face blog: critical look at GPT and single-minus gluons](https://huggingface.co/blog/dlouapre/gpt-single-minus-gluons); [The Quantum Insider: AI spots what physicists missed in gluon scattering](https://thequantuminsider.com/2026/02/13/ai-scientist-spots-what-physicists-missed-in-gluon-scattering/)

### 2026-02-14: 'First Proof' challenge: AI solves about half of 10 unpublished research problems set by mathematicians

***; Google DeepMind, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-02-14-first-proof-challenge/

Eleven mathematicians released 10 unpublished research-level problems on 5 Feb 2026 and answers on 14 Feb. DeepMind's Aletheia got 6/10 by majority expert assessment. OpenAI got at least 5 likely correct and retracted one claimed solution. Scientific American called the results 'mixed'.

- 10 problems from the authors' own unpublished research; answers revealed 14 Feb 2026
- Aletheia: problems 2, 5, 7, 8, 9, 10 judged correct by majority (experts split on #8)
- OpenAI: problems 4, 5, 6, 9, 10 likely correct; retracted claim on #2

#### What happened
Mathematicians created a contamination-proof test using problems from their own unpublished work, and AI labs submitted solutions within a week.

#### Why it matters
It gave a cleaner measure than olympiads of whether AI can do research maths: at the time, about half the time.

Sources: [First Proof challenge](https://1stproof.org/); [OpenAI: First Proof submissions](https://openai.com/index/first-proof-submissions/); [Scientific American: First Proof is AI's toughest math test yet — the results are mixed](https://www.scientificamerican.com/article/first-proof-is-ais-toughest-math-test-yet-the-results-are-mixed/)

### 2026-02-28: Donald Knuth's 'Claude's Cycles': Claude Opus 4.6 solves an open Hamiltonian-cycle problem ('Shock! Shock!')

****; Anthropic, Stanford University; Science & math; confidence high; https://postcutoff.com/e/2026-02-28-knuth-claudes-cycles/

Donald Knuth published a note opening 'Shock! Shock!' describing how Claude Opus 4.6 found, in about an hour of guided exploration, a general construction decomposing the arcs of a 3D torus digraph on m³ vertices into three Hamiltonian cycles for all odd m. Knuth had worked on the problem for weeks for a future TAOCP volume. He then proved Claude's construction correct.

- Note dated 28 Feb 2026, revised 4 Mar 2026
- Graph: vertices (i,j,k) mod m, arcs increment one coordinate; goal: split all arcs into 3 directed Hamiltonian cycles
- Claude found the odd-m construction in 31 guided explorations over about an hour; the even case remains largely open
- Knuth wrote the proof; the result was later formalised in Lean (kim-em/KnuthClaudeLean)

#### What happened
Knuth posed a Hamiltonian-cycle decomposition problem he planned for TAOCP. Working through it with Claude Opus 4.6 over dozens of explorations, a collaborator got a pattern that works for every odd m. Knuth then proved it and wrote up the story.

#### Why it matters
Coming from one of computing's most respected and AI-sceptical elders, the note became a cultural marker that frontier LLMs could contribute original mathematical constructions.

Sources: [Donald Knuth: Claude's Cycles (PDF)](https://www-cs-faculty.stanford.edu/~knuth/papers/claude-cycles.pdf); [GitHub: kim-em/KnuthClaudeLean (Lean formalisation)](https://github.com/kim-em/KnuthClaudeLean); [Adafruit blog: Don Knuth wrote a paper thanking Claude](https://blog.adafruit.com/2026/03/03/don-knuth-wrote-a-paper-thanking-claude-for-solving-an-open-math-problem/)

### 2026-03-01: Math Inc's Gauss formalises Viazovska's sphere-packing proofs in dimensions 8 and 24, fixing errors in the originals

****; Math Inc; Science & math; confidence high; https://postcutoff.com/e/2026-03-01-gauss-sphere-packing-formalization/

Math Inc's Gauss agent completed the Lean formalisation of Maryna Viazovska's Fields-Medal proofs of optimal sphere packing in dimensions 8 (5 days) and 24 (~2 weeks), about 180,000 lines. Along the way it found and fixed a sign error and an incomplete step in the published proofs.

- Dimension 8: 5 days, code grew from ~20k to ~60k lines; dimension 24: ~2 weeks
- Final code ~180k lines (some sources say ~200k)
- Found a sign error in Proposition 7 (dim 8) and an incomplete step in Appendix A (dim 24)
- Write-up arXiv 2604.23468; exact announcement day not verified

#### What happened
Gauss took over a partial human Lean project on sphere packing and finished both dimensions, reporting the errors it found in the literature.

#### Why it matters
AI autoformalization reached Fields-Medal-level proofs, strengthening the case that formal verification can keep up with the flood of AI-generated mathematics.

Sources: [Formalizing sphere packing in dimensions 8 and 24 (arXiv 2604.23468)](https://arxiv.org/abs/2604.23468); [GitHub: math-inc/Sphere-Packing-Lean](https://github.com/math-inc/Sphere-Packing-Lean)

### 2026-03-10: AlphaEvolve improves lower bounds for nine classical Ramsey numbers

***; Google; Science & math; confidence high; https://postcutoff.com/e/2026-03-10-alphaevolve-ramsey-lower-bounds/

Google researchers used AlphaEvolve to construct graphs improving the lower bounds of nine small Ramsey numbers, including R(3,13) ≥ 61, R(4,16) ≥ 174 and R(4,19) ≥ 219 (arXiv 2603.09172).

- R(3,13): 60→61; R(3,18): 99→100
- R(4,13): 138→139; R(4,14): 147→148; R(4,15): 158→159
- R(4,16): 170→174; R(4,18): 205→209; R(4,19): 213→219; R(4,20): 234→237
- Authors: Nagda, Raghavan, Thakurta

#### What happened
AlphaEvolve evolved programs that build large graphs with no big cliques or independent sets, beating the previously best known constructions.

#### Why it matters
Small Ramsey numbers are among the most-studied computational problems in combinatorics; AI-found improvements across nine at once showed the reach of evolutionary LLM search.

Sources: [Ramsey lower bounds via AlphaEvolve (arXiv 2603.09172)](https://arxiv.org/abs/2603.09172); [Wikipedia: Ramsey's theorem (background)](https://en.wikipedia.org/wiki/Ramsey%27s_theorem)

### 2026-03-10: Google, Imperial College and the NHS: AI second reader finds 25% of interval breast cancers missed by screening (two Nature Cancer papers)

***; Google, Imperial College London, NHS; Science & math; confidence medium; https://postcutoff.com/e/2026-03-10-google-ai-mammography-nature-cancer-nhs/

On March 10, 2026 Google Research, Imperial College London and the UK's NHS published two studies in Nature Cancer on an experimental AI system for breast-cancer screening. On past mammograms of about 125,000 women, the AI flagged 25% of the "interval cancers" that double reading by two specialists had missed, and found more cancers overall with fewer false positives at first screens. A second study, on more than 50,000 women, estimated that using the AI as the second reader could cut reading workload by about 40% (Google's figure; trade press reports 32.1% for the retrospective analysis). Google restated the results in its Oct 12, 2026 health research roundup.

- Two papers in Nature Cancer: s43018-026-01127-0 (accuracy) and s43018-026-01128-z (workload and arbitration); the papers themselves were not readable by our fetcher (login redirect), so numbers come from Google's posts and trade press
- Accuracy study: mammograms of 125,000 women; the AI detected 25% of interval cancers (cancers diagnosed between screening rounds after a normal result) that had been missed; more invasive cancers and more cancers overall than expert radiologists; fewer false positives at first-time screens (Google, Mar 10)
- ICT&health (Mar 11): about 125,000 women aged 50–70 from five NHS screening programmes, screened 2015–2016; 115,973 scans in the final analysis; detection rose from 7.54 to 9.33 cancers per 1,000 women with AI as second reader; for first-time screeners recalls fell 39.3% and detection rose 8.8%
- Workload: Google says an AI second reader was estimated to reduce screening workload by 40% (study of over 50,000 women); ICT&health reports a 32.1% fall in scan-reading workload in the retrospective study. The two figures are not reconciled here
- Real-time feasibility study: over 9,000 cases (ICT&health: 9,266) across 12 NHS screening sites in London; AI output did not influence patient care; AI took 17.7 minutes per assessment on average vs 2.08 days for the first human reader (ICT&health)
- Arbitration: in a simulated review, specialist arbitration panels sometimes overruled cancers that the AI had correctly flagged (Google)
- Context: NHS screening uses double reading; each specialist reads about 5,000 scans a year (Google). The UK has a 29% shortfall of clinical radiologists (ICT&health)
- Oct 12, 2026: Google's roundup 'Applying technology to understand and treat disease' (Yossi Matias, Pushmeet Kohli) repeats: 125,000 women, 25% of missed interval cancers, workload reduction 'up to 40%'

#### What happened
The NHS breast screening programme has every mammogram read by two specialists. Google's experimental system was tested as a
replacement for the second reader. On historical scans of about 125,000 women it found a quarter of the interval cancers, the ones
that appeared between screening rounds after a normal result. A companion paper modelled the workflow and estimated a large cut in
reading work. A small real-time run at 12 London sites showed the system can return results in minutes, without affecting care.

Google's Oct 12, 2026 roundup of its health research (90 peer-reviewed papers in five years, by its count) lists these studies next to
AlphaFold drug repurposing for Chagas disease, AlphaGenome Atlas, AlphaProtein Novo, DeepSomatic and a flu forecast ranked first by
the CDC for the 2025–26 season. Google.org says it gave over $60 million in 2026 to groups applying such models to disease.

#### Why it matters
Interval cancers are the main failure of screening, and the UK lacks radiologists. The result is mostly retrospective: whether an AI
reader improves outcomes in live screening is being tested in prospective trials (ICT&health mentions the EDITH trial across 30
sites). The arbitration finding is a warning that humans may overrule correct AI calls.

#### Unverified
The Nature Cancer papers were not read directly; titles, author lists and exact publication dates of each paper are not confirmed.
The 40% and 32.1% workload figures come from different sources.

Sources: [Google: How AI can improve breast cancer detection in the UK (Mar 10, 2026)](https://blog.google/innovation-and-ai/technology/health/google-ai-breast-cancer-detection/); [Nature Cancer paper 1 (s43018-026-01127-0)](https://www.nature.com/articles/s43018-026-01127-0); [Nature Cancer paper 2 (s43018-026-01128-z)](https://www.nature.com/articles/s43018-026-01128-z); [ICT&health: AI matches radiologists in breast cancer screening study (Mar 11, 2026)](https://www.icthealth.org/news/ai-matches-radiologists-in-breast-cancer-screening-study); [Google: Applying technology to understand and treat disease (Oct 12, 2026)](https://blog.google/innovation-and-ai/technology/health/ai-health-research/)

### 2026-03-25: RAVEN machine-learning pipeline validates 118 new planets in TESS data

**; University of Warwick; Science & math; confidence medium; https://postcutoff.com/e/2026-03-25-raven-tess-118-new-planets/

Warwick's RAVEN pipeline analysed 2.2 million stars observed by TESS and validated 118 new planets and over 2,000 vetted candidates (nearly 1,000 of them new), including ultra-short-period planets and planets in the 'Neptunian desert' (MNRAS, 2026).

- 2.2M stars from TESS's first four years; 118 newly validated planets; >2,000 vetted candidates, nearly 1,000 new
- ~9–10% of Sun-like stars host a close-in (<16-day) planet, with uncertainties up to 10× smaller than Kepler's; Neptunian-desert planets occur around ~0.08% of Sun-like stars
- Paper arXiv 2603.22597; Warwick press release Mar 2026 (day approximate); MNRAS

#### What happened
An ML vetting pipeline processed millions of TESS light curves and validated over a hundred planets.

#### Why it matters
It continues AI's role as the main filter for exoplanet surveys.

Sources: [Warwick: AI approach uncovers dozens of hidden planets in TESS data](https://warwick.ac.uk/news/pressreleases/ai-approach-uncovers-dozens-of-hidden-planets/); [RAVEN TESS paper (arXiv 2603.22597)](https://arxiv.org/abs/2603.22597); [ScienceDaily: RAVEN validates 118 new planets](https://www.sciencedaily.com/releases/2026/05/260502233926.htm)

### 2026-03-30: Erdős Problem #650 solved exactly, f(m) = min(m, ⌈2√m⌉), with a strategy proposed by ChatGPT and a Lean proof by Aristotle; first Justin Sun Prize awards in October

**; OpenAI, Harmonic; Science & math; confidence high; https://postcutoff.com/e/2026-03-30-erdos-650-exact-value-chatgpt-aristotle/

On 30 March 2026 Wouter van Doorn, Yanyang Li and Quanyu Tang posted arXiv 2603.28636, which determines exactly the function in Erdős Problem #650 (matching integers to distinct multiples in an interval): f(m) = min(m, ⌈2√m⌉). The abstract says the proof strategy "was first proposed by ChatGPT" and the argument was "made fully rigorous and formally verified in Lean by Aristotle"; the exposition is human-written. On 5 Oct 2026 the three were named the first recipients of the crypto-funded Justin Sun Prize for this and related Erdős work (sponsored press release; no amount stated).

- Paper: 'Optimal bounds for an Erdős problem on matching integers to distinct multiples', arXiv 2603.28636 (math.CO, math.NT), submitted 30 March 2026, 8 pages; Wouter van Doorn, Yanyang Li, Quanyu Tang
- Problem (paper): f(m) is the largest integer such that for every set A of m positive integers and every open interval I of length 2a_m there are at least f(m) disjoint pairs (a, b) with a ∈ A dividing b ∈ I. Result: f(m) = min(m, ⌈2√m⌉) for all m
- History (erdosproblems.com/650): Erdős and Surányi (1959) proved f(m) ≥ √m; Erdős and Selfridge gave f(m) ≤ 2⌈√m⌉; GPT 5.4 Pro, prompted by He, Li and Tang, proved f(m) ≤ ⌈2√m⌉, and GPT 5.4 Pro with Aristotle gave a matching lower bound; the site lists the problem as solved by [VLT26]
- AI disclosure (abstract): 'The proof was obtained through an AI-assisted workflow: the proof strategy was first proposed by ChatGPT, and the detailed argument was subsequently made fully rigorous and formally verified in Lean by Aristotle. The exposition and final proofs presented here are entirely human-written.'
- The problem's formal statement is in Google DeepMind's Formal Conjectures repository (ErdosProblems/650.lean)
- Justin Sun Prize (announced Geneva, Oct 5, 2026; carried as TRON-commissioned sponsored content by The Block and Cointelegraph): van Doorn, Tang and Li are the program's 'first confirmed awards', for six Erdős problems: #650; #369, #457 and #469 (van Doorn, Lean-checkable proofs); #1044 (Tang); #1196 (Tang and Li within a wider team). Paid in USDT on TRON or USDC on Ethereum; no amounts or judges named
- Prize site (hejustinsun.com/prize): motto 'Quod probatur, solvitur'; accepts complete Lean proofs, solutions without Lean, or previously catalogued work; no distinction between human and AI contributors

#### What happened
Erdős Problem #650 asks, for a set of m integers, how many of them can always be paired with distinct multiples lying in a short
interval. Erdős and Surányi had shown at least √m in 1959. In early 2026 GPT 5.4 Pro, prompted by researchers, improved the upper
bound to ⌈2√m⌉ and, with Harmonic's Aristotle, found a matching lower bound (as recorded on erdosproblems.com). On 30 March 2026
Wouter van Doorn, Yanyang Li and Quanyu Tang posted the exact answer, f(m) = min(m, ⌈2√m⌉), stating in the abstract that ChatGPT
proposed the proof strategy and Aristotle made it rigorous and verified it in Lean, while the final text is human-written.

The result returned to the news in October 2026, when the Office of Justin Sun announced in Geneva (Oct 5) that the three were the
first recipients of the Justin Sun Prize, a crypto-paid prize for Erdős problems that accepts Lean proofs and does not distinguish
human from AI contributors. The announcement circulated as TRON-commissioned sponsored content (The Block, Cointelegraph) and names
no amounts or judges.

#### Why it matters
A small, clean example of the 2026 workflow for open problems: a chat model proposes the idea, a formal prover closes the gaps in
Lean, and humans simplify and write it up. The prize shows new money (here from a crypto entrepreneur) attaching itself to
AI-assisted Erdős results, which the erdosproblems.com maintainers had begun to ration by October (see the related entry on the site
freezing proof claims).

Sources: [arXiv 2603.28636: Optimal bounds for an Erdős problem on matching integers to distinct multiples](https://arxiv.org/abs/2603.28636); [Erdős Problem #650 (erdosproblems.com)](https://www.erdosproblems.com/650); [Formal Conjectures: ErdosProblems/650.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/650.lean); [The Block (sponsored): Justin Sun Prize's New Round Honors Human–AI Collaboration on Erdős Problems](https://www.theblock.co/sponsored/2026-10-05-justin-sun-prizes-new-round-honors-human-ai-collaboration-on-erdos-problems-417743); [Justin Sun Prize](https://hejustinsun.com/prize)

### 2026-05-01: GPT-5.5 Pro-assisted construction lowers the smallest known Borsuk counterexample dimension from 64 to 63

***; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-05-01-borsuk-conjecture-dimension-63/

In May 2026 Max Grinsztajn, assisted by OpenAI's GPT-5.5 Pro, built a 321-point set in R^63 that cannot be split into 64 parts of smaller diameter, so Borsuk's conjecture fails in dimension 63 (b(63) ≥ 65). The previous smallest known failing dimension, 64, had stood since 2013. A second, independent AI-generated version (GPT-5.6 Sol) was posted to arXiv in August and withdrawn because the result already existed.

- Construction: 320-point Jenrich–Brouwer core from the G2(4) strongly regular graph in a codimension-2 subspace of R^63, plus one projected and rescaled point
- Result: 321 points, any subset of smaller diameter has at most 5 points, so at least 65 parts are needed (b(63) ≥ 65)
- Open range for Borsuk's conjecture moves from 4 ≤ n ≤ 63 to 4 ≤ n ≤ 62
- Repository README: 'The construction and proof were obtained with assistance from GPT-5.5 Pro'; exact verification script plus Sage-checkable certificates (no Lean proof)
- Recorded in Tao's optimization-constants table (constant 28a) as [Gri2026]
- arXiv 2608.12561 (Yibo Ji, 12 Aug 2026): same 321-point set 'generated entirely by ChatGPT using GPT 5.6 Sol'; withdrawn 14 Aug 2026 because the construction had already been published
- Yibo Ji's arXiv withdrawal comment (2608.12561v2): 'Just found out that this counterexample had already been posted elsewhere: https://github.com/maaxgrin/borsuk-63-counterexample and https://nickk124.github.io/borsuk/'

#### What happened
Borsuk asked in 1933 whether every bounded set in R^n can be split into n+1 pieces of smaller diameter. Kahn and Kalai showed in 1993 that the answer is no in high dimensions, and later work pushed the smallest known failing dimension down to 64 (Jenrich, 2013, from Bondarenko's construction). In May 2026 Max Grinsztajn, working with GPT-5.5 Pro, added one carefully projected point to the 320-point Jenrich–Brouwer set and got a 63-dimensional counterexample. His repository ships an exact verification script and certificates.

The exact day is not known. Wikipedia dates the result to May 2026. In August 2026 Yibo Ji posted the same kind of 321-point construction to arXiv, saying it was "generated entirely by ChatGPT using GPT 5.6 Sol". He withdrew it two days later because the result was already published. Secondary sources also mention an independent find by "Konz", which we have not verified.

#### Why it matters
It is a clean, checkable improvement to a well-known geometry record. Two separate human+model pairs reached it within a few months, which suggests these gaps are now within easy reach of frontier models.

Sources: [GitHub: maaxgrin/borsuk-63-counterexample (paper PDF + verifier)](https://github.com/maaxgrin/borsuk-63-counterexample); [Tao et al. optimization constants: constant 28a (Borsuk)](https://teorth.github.io/optimizationproblems/constants/28a.html); [arXiv 2608.12561: An AI Generated Counterexample to Borsuk Problem in Dimension 63 (withdrawn)](https://arxiv.org/abs/2608.12561); [Wikipedia: Borsuk's conjecture](https://en.wikipedia.org/wiki/Borsuk%27s_conjecture); [Wikipedia: List of mathematical discoveries by artificial intelligence](https://en.wikipedia.org/wiki/List_of_mathematical_discoveries_by_artificial_intelligence)

### 2026-05-03: Amateur with GPT-5.4 Pro 'vibe-maths' a 60-year-old Erdős conjecture on primitive sets; Tao co-authors the paper

****; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-05-03-erdos-1196-primitive-sets/

23-year-old amateur Liam Price gave GPT-5.4 Pro a single prompt. In about 80 minutes it sketched a proof of Erdős problem #1196, the 1966 Erdős–Sárközy–Szemerédi conjectures on primitive sets and divisibility chains, using Markov chains with von Mangoldt weights. Professionals including Terence Tao and Jared Lichtman turned it into a paper (arXiv 2605.00301) that also gives a short new proof of the Erdős primitive set conjecture.

- Problem open since 1966 (~60 years)
- Proof sketch by GPT-5.4 Pro in ~80 minutes from one prompt by Liam Price; escalated by Kevin Barreto
- Paper authors include Tao, Alexeev, Barreto, Lichtman, Price and others
- Lichtman (who proved the Erdős primitive set conjecture in 2022) said the argument looked like it came 'from The Book'
- erdosproblems.com lists #1196 as PROVED; formalisation reported underway
- AI disclosure (abstract, arXiv 2605.00301): 'We introduce a new method for bounding Erdős sums of primitive sets, suggested from output of GPT-5.4 Pro, based on Markov chains with von Mangoldt weights.'

#### What happened
An amateur prompted a public model, which found a proof strategy via random divisibility chains. The problem's leading experts confirmed and extended it within days.

#### Why it matters
It was the first AI solution to a well-known, decades-old Erdős conjecture that specialists had actively worked on, not just an obscure entry. It came weeks before the unit-distance disproof.

Sources: [Primitive sets and von Mangoldt chains (arXiv 2605.00301)](https://arxiv.org/abs/2605.00301); [Terence Tao: Primitive sets and von Mangoldt chains — Erdős problem #1196 and beyond](https://terrytao.wordpress.com/2026/05/03/primitive-sets-and-von-mangoldt-chains-erdos-problem-1196-and-beyond/); [Scientific American: Amateur armed with ChatGPT vibe-maths a 60-year-old problem](https://www.scientificamerican.com/article/amateur-armed-with-chatgpt-vibe-maths-a-60-year-old-problem/)

### 2026-05-09: Google DeepMind's 'AI co-mathematician' sets FrontierMath Tier 4 record and helps resolve a Kourovka Notebook problem

***; Google DeepMind, University of Oxford; Science & math; confidence high; https://postcutoff.com/e/2026-05-09-deepmind-ai-co-mathematician/

DeepMind's agentic 'AI co-mathematician' on Gemini 3.1 Pro scored 48% (23/48) on FrontierMath Tier 4, versus 19% for Gemini 3.1 Pro alone and 39.6% for GPT-5.5 Pro. It helped Oxford's Marc Lackenby resolve Kourovka Notebook Problem 21.10 in group theory; a reviewer agent caught a flaw that Lackenby then fixed.

- arXiv 2605.06651
- FrontierMath Tier 4: 48% vs Gemini 3.1 Pro 19%, GPT-5.5 Pro 39.6%, Claude Opus 4.7 22.9%
- Earlier record: GPT-5.2 Pro 31% (15/48) in Jan 2026, per Epoch AI
- Semon Rezchikov: 'I would rank, aesthetically, its general style of proofs as the best one of any models'

#### What happened
DeepMind wrapped Gemini in a team of generator, reviewer and literature agents designed to work alongside a mathematician.

#### Why it matters
FrontierMath Tier 4, built to resist AI for years, was nearly half solved 18 months after launch. The Lackenby collaboration showed the agent catching its own errors.

Sources: [AI co-mathematician (arXiv 2605.06651)](https://arxiv.org/abs/2605.06651); [Epoch AI: new record on FrontierMath Tier 4 (Jan 2026)](https://epochai.substack.com/p/new-record-on-frontiermath-tier-4); [The Rundown: Google DeepMind's powerful AI co-mathematician](https://www.therundown.ai/p/google-deepmind-powerful-ai-co-mathematician)

### 2026-05-12: GPT-5.5 Pro finds counterexample disproving McKean's 1966 conjecture and the Gaussian completely monotone conjecture

***; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-05-12-gaussian-completely-monotone-disproved/

Gu and Sellke (arXiv 2605.11656) presented an explicit probability measure, found by GPT-5.5 Pro, for which the 5th time-derivative of entropy along the heat flow is positive. This disproves the Gaussian completely monotone conjecture, McKean's 1966 Gaussian-optimality conjecture (1-D) and Toscani's 2015 entropy power conjecture.

- arXiv 2605.11656 (12 May 2026)
- Counterexample found by GPT-5.5 Pro; proof written by the human authors
- Follow-ups: a hexagonal multidimensional counterexample (arXiv 2605.18081) and log-concave families (2608.30275)
- AI disclosure (Gu–Sellke abstract, arXiv 2605.11656): 'The explicit counterexample was found by GPT-5.5 Pro.'
- AI disclosure (Zou–Fan–Gao–Wang abstract, arXiv 2605.18081, which disproves the Cheng–Geng log-convexity conjecture in dimension ≥ 2): 'The explicit two-dimensional counterexample was found by GPT-5.5 Pro.'

#### What happened
OpenAI researcher Mark Sellke and a co-author used GPT-5.5 Pro to search for a distribution violating a 60-year-old monotonicity conjecture, and it produced one.

#### Why it matters
It is part of 2026's striking pattern of AI counterexamples: models proved especially good at finding objects that break long-believed conjectures. Suvrit Sra documented 15+ such counterexamples in "GPT, the Counterexample Machine".

Sources: [Gu & Sellke: counterexample to the GCM conjecture (arXiv 2605.11656)](https://arxiv.org/abs/2605.11656); [Follow-up: multidimensional counterexample (arXiv 2605.18081)](https://arxiv.org/abs/2605.18081); [Suvrit Sra: GPT, the Counterexample Machine (arXiv 2608.29595)](https://arxiv.org/abs/2608.29595)

### 2026-05-20: OpenAI model disproves Erdős's 80-year-old unit distance conjecture

*****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-05-20-ai-disproves-erdos-unit-distance-conjecture/

On 2026-05-20 OpenAI announced that an internal model found a counterexample to Erdős's 1946 unit-distance conjecture using algebraic number theory — widely described as the first historically significant proof produced by an AI; Timothy Gowers said he would recommend it to the Annals of Mathematics 'without any hesitation'. A wave of AI-assisted Erdős-problem solutions followed through summer 2026.

- Counterexample: a grid construction where g(N) exceeds a fixed multiple of N^(1+ε), ε ≈ 6.24×10^-38 (Physics World)
- Method: algebraic number theory (Golod–Shafarevich class field towers, building on Ellenberg–Venkatesh and Hajir–Maire–Ramakrishna)
- Same-day human exposition and verification (arXiv 2605.20695) by Alon, Bloom, Gowers, Litt, Sawin, Shankar, Tsimerman, V. Wang and Matchett Wood
- Will Sawin made the exponent explicit (1.014, later 1.0318) and showed this method cannot exceed about 1.2143; Kevin Buzzard reports it was later formalised in Lean
- Gowers: 'quite an important moment in the history of mathematics'; Jozsef Solymosi: 'I was most surprised by the depth of the solution'
- Timothy Gowers: would recommend Annals of Mathematics publication 'without any hesitation'
- Erdős #728 (Jan 4 2026) solved by amateurs Barreto & Price with GPT-5.2 Pro, formally verified with Aristotle
- Erdős #1196 (May 2026): paper co-authored by Barreto, Price, Terence Tao, Jared Duker Lichtman and others
- Aug 1 2026: OpenAI said unreleased model 'Astra' made 10 further advances incl. three more Erdős problems
- erdosproblems.com status at Quanta's Aug 2026 article: 565 solved, 652 open

#### What happened
The unit distance problem asks how many pairs of points among N points in the plane can be exactly distance 1 apart; Erdős conjectured an upper bound of N^(1+o(1)). OpenAI's model constructed a counterexample.
Nine leading mathematicians commented on the result. Meanwhile amateurs using GPT-5.x and teams with Terence Tao resolved other Erdős problems, and Google DeepMind reported solving 9 of 353 open problems at a few hundred dollars each.

#### Why it matters
This is the moment AI crossed from solving competition problems to settling a famous open research conjecture, reshaping debate about AI's role in mathematics. (Confidence medium: primary OpenAI post not fetched; details from reputable press.)

Sources: [OpenAI: model disproves discrete geometry conjecture](https://openai.com/index/model-disproves-discrete-geometry-conjecture/); [Human exposition of the counterexample (arXiv 2605.20695)](https://arxiv.org/abs/2605.20695); [Gil Kalai: Amazing — Erdős unit distance problem was disproved by AI](https://gilkalai.wordpress.com/2026/05/21/amazing-erdos-unit-distance-problem-was-disproved-it-was-achieved-by-ai/); [Quanta: Why the legendary Erdős problems are falling to AI](https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/); [Scientific American: AI just solved an 80-year-old Erdős problem](https://www.scientificamerican.com/article/ai-just-solved-an-80-year-old-erdos-problem-and-mathematicians-are-amazed/); [Physics World: AI-led solutions of Erdős problems spark debate](https://physicsworld.com/a/ai-led-solutions-of-erdos-problems-spark-debate-over-the-future-of-mathematics/); [MAA: AI solves an 80 year-old Erdős problem](https://maa.org/math-values/ai-solves-an-80-year-old-erdos-problem/); [Slate: Did A.I. really solve a math problem mathematicians couldn't?](https://slate.com/technology/2026/06/math-chatgpt-erdos-problem-solved-open-ai.html)

### 2026-05-27: Erdős–Szemerédi sum-product conjecture shown false over the reals; a GPT-5.5 Pro agent re-disproves it in 7 of 8 runs

****; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-05-27-sum-product-conjecture-false-over-reals/

Inspired by the AI disproof of the unit-distance conjecture, Bloom, Sawin, Schildkraut and Zhelezov proved on 27 May 2026 that the Erdős–Szemerédi sum-product conjecture is false over the real numbers. They built sets A with |A+A| and |AA| ≤ |A|^(2−c). A July 2026 paper (arXiv 2607.20525) showed a GPT-5.5 Pro agent autonomously generated correct disproofs in 7 of 8 independent trials, some with new constructions.

- Human paper: arXiv 2605.28781 (27 May 2026), 'inspired' by OpenAI's unit-distance disproof, which used related algebraic-number-theory ideas
- AI replication: GPT-5.5 Pro agent, three-stage prompting pipeline, correct disproofs in 7/8 runs; some avoid units by using L^p-type regions of algebraic integers
- The 1983 conjecture (max(|A+A|,|AA|) ≥ |A|^(2−ε)) remains open over the integers
- AI disclosure (Huang, arXiv 2607.20525, abstract): 'we present a simple agent built on GPT-5.5 Pro. Using a problem-agnostic, three-stage prompting pipeline -- proof-plan proposal, proof construction, and review -- the agent autonomously generated correct proofs that the sum-product conjecture is false over R in 7 of 8 independent trials; in the remaining trial, it identified an unresolved gap in its argument.'

#### What happened
The number-theoretic idea behind the AI's unit-distance counterexample prompted human experts to attack a second famous Erdős conjecture, which fell within a week. A later study showed the AI could have done it alone.

#### Why it matters
It shows AI ideas spreading into human research and then being reproduced autonomously: a feedback loop between AI and human mathematicians.

Sources: [The sum-product conjecture is false for real numbers (arXiv 2605.28781)](https://arxiv.org/abs/2605.28781); [GPT-5.5 Pro agent disproofs of the sum-product conjecture over R (arXiv 2607.20525)](https://arxiv.org/abs/2607.20525)

### 2026-06-05: Computationally designed broad coronavirus vaccine is safe and immunogenic in first human trial

***; University of Cambridge, DIOSynVax; Science & math; confidence medium; https://postcutoff.com/e/2026-06-05-ai-designed-coronavirus-vaccine-trial/

A Phase 1 trial in 39 volunteers found that a vaccine antigen designed entirely by computer (Cambridge / DIOSynVax, Jonathan Heeney) was safe and raised immune responses against SARS-CoV-2, SARS and bat coronaviruses. It was reported as the first time a vaccine whose active ingredient was created entirely through computer simulations was tested in people.

- Phase 1, 39 volunteers; Journal of Infection (2026)
- Broad responses against SARS-CoV-2, SARS-CoV-1 and bat sarbecoviruses
- Design used computational structure-based antigen design and ML; exact AI contribution less specific than headlines suggest

#### What happened
An antigen engineered in silico to present conserved coronavirus epitopes completed a first-in-human trial.

#### Why it matters
It is an early human validation of computer-designed vaccine antigens, relevant to pandemic preparedness.

Sources: [ScienceDaily: computer-designed coronavirus vaccine tested in people](https://www.sciencedaily.com/releases/2026/06/260605023357.htm); [DIOSynVax](https://www.diosynvax.com/)

### 2026-06-29: Machine-learning screen predicts two new kagome superconductors, confirmed in the lab

**; Aalto University, Rice University; Science & math; confidence high; https://postcutoff.com/e/2026-06-29-ml-screened-kagome-superconductors/

Päivi Törmä's group at Aalto combined ML pre-screening with quantum-geometry calculations to predict superconductivity in YRu3B2 and LuRu3B2. Rice University synthesised both and confirmed superconductivity at 0.81 K and 0.95 K (Physical Review Research).

- Tc: 0.81 K (YRu3B2), 0.95 K (LuRu3B2), far from room temperature
- Törmä: 'This approach will greatly speed up superconductor discovery.'
- Press headlines about a 'race to room-temperature superconductors' overstate the result

#### What happened
A theory-plus-ML pipeline picked candidates, and experimental partners confirmed them.

#### Why it matters
It is a modest but clean prediction-then-confirmation loop in superconductor research, a field full of hype.

Sources: [ScienceDaily: Aalto/Rice ML-screened kagome superconductors (Jul 2026)](https://www.sciencedaily.com/releases/2026/07/260701205006.htm); [Futura Sciences: AI unveils two materials](https://www.futura-sciences.com/en/shock-in-science-ai-unveils-two-materials-that-could-change-everything_39019/)

### 2026-07-01: AI-assisted counterexample answers Grothendieck's question on finite flat group schemes, merged into Mathlib

***; OpenAI, Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-07-01-grothendieck-group-scheme-counterexample/

Akhil Mathew, using OpenAI's and Anthropic's models, found a finite locally free group scheme of order 4 over a non-reduced finite ring with 2⁹ elements that is not killed by 4 (it is killed by 8). This answers Grothendieck's question negatively. The Lean proof was merged into Mathlib on 3 Aug 2026.

- Known positive cases: commutative (Deligne), reduced base (Grothendieck); pure characteristic-p case still open
- Found by studying deformations of α₂×α₂
- Kevin Buzzard attributes discovery to OpenAI's Sol and autoformalisation to Claude Fable
- Mathlib PR #41748 (Counterexamples/GrothendieckPower.lean), merged 3 Aug 2026

#### What happened
In the same weeks as the Jacobian counterexample, Mathew used frontier models to find and formalise a counterexample to a question from the foundations of algebraic geometry.

#### Why it matters
Kevin Buzzard said this mattered more to him than the Erdős results because it lies in "an area of mathematics that I personally find more interesting". AI was now reaching core modern algebraic geometry.

Sources: [Benjamin Antieau: Akhil Mathew and AI](https://antieau.github.io/2026/08/10/akhil-mathew-ai.html); [Xena Project: Human mathematicians are being out-counterexampled](https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/)

### 2026-07-14: Berkeley's Phillip Kerger uses GPT-5.6 Sol and a 10-page prompt to close a 30-year gap in derivative-free convex optimization

***; UC Berkeley, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-07-14-gpt-5-6-zeroth-order-convex-lower-bound/

In July 2026 UC Berkeley IEOR teaching professor Phillip Kerger posted a proof that deterministic optimization of a convex Lipschitz function over a d-dimensional ball with only exact function values needs Ω(d²/log(d+1)) queries. This nearly matches Protasov's 1996 ~d² algorithm and closes a gap that had stood since 1996 (the previous lower bound was ~d). Kerger says GPT-5.6 Sol produced the proof in one ~2.5-hour session from a 10-page prompt modelled on OpenAI's cycle-double-cover method. He verified the core lower bound in Lean. It was the first widely noticed outside replication of that "long prompt" method.

- Paper: arXiv 2607.13335, 'Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values' (Phillip Kerger, submitted July 14, 2026, 36 pages); it also gives Õ(d²·2ⁿ) results for mixed-integer problems
- Result: Ω(d²/log(d+1)) deterministic exact-value query lower bound vs O(d² log² d) upper bound from Protasov's 1996 method; the previously applicable lower bound was Ω(d)
- AI role (Kerger's Medium post, as reported by temperature2 and AI Weekly): a ~10-page prompt with background, techniques and direction; the model produced a complete proof in ~2.5 hours without intervention. Kerger had worked on the gap for about a year. AI Weekly names the model 'GPT-5.6 Sol Pro'
- Lean 4/mathlib companion repo (PhillipKerger/zero-order-bounds-lean-verification, created July 14) fully verifies the paper's deterministic d^(-1/2)-accuracy lower bound, including Brunn–Minkowski and Urysohn machinery; the upper-bound and transfer results are outside the formalization
- The arXiv abstract itself does not mention AI. The AI-assistance claim rests on the author's own blog post
- Attention: r/math post 'After OpenAI's CDC proof announcement, GPT-5.6 used a prompt to close a 30-year gap in convex optimization' reached the Hacker News front page (601 points, July 18)
- Follow-up: Zhang, Zhang, Qi and Lin (arXiv 2607.16558, July 18) proved near-optimal lower bounds for randomized algorithms in the same exact-value setting (no AI mentioned)

#### What happened
A few days after OpenAI's cycle double cover proof, Kerger applied the same kind of long, structured prompt to a problem he had worked
on for a year. The question is how many function evaluations a deterministic method needs to minimize a convex function when it sees
only exact values and no gradients. Protasov's 1996 method needs about d² evaluations, but the best lower bound was only about d. The
model's argument establishes a near-quadratic lower bound. Kerger then had the core result machine-checked in Lean and published the
paper, the Lean code and (reportedly) the prompt and chat logs.

#### Why it matters
Most of the summer's AI-math headlines came from the labs themselves. This was an outside, single-author result with a formal proof,
on a well-defined classical gap, and it suggested the recipe transfers to other areas of mathematics. Caveat: the Medium post with
the step-by-step AI account could not be read directly here, because of a Cloudflare block. Details of the AI's role come from
secondary summaries of it.

Sources: [arXiv 2607.13335: Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization](https://arxiv.org/abs/2607.13335); [GitHub: PhillipKerger/zero-order-bounds-lean-verification (Lean 4 companion)](https://github.com/PhillipKerger/zero-order-bounds-lean-verification); [Phillip Kerger (Medium): An AI-assisted breakthrough in convex optimization](https://medium.com/@kerger.p/an-ai-assisted-breakthrough-in-convex-optimization-an-optimization-problem-dating-back-30-years-a-db5c631119de); [temperature2: GPT-5.6 closes a second 30-year math gap without OpenAI](https://temperature2.com/p/2026-07-19-gpt-5-6-second-math-proof-outside-openai/); [AI Weekly: GPT-5.6 Sol Pro closes 30-year gap in zeroth-order convex optimization, Lean-verified](https://aiweekly.co/node/6956); [Hacker News discussion (601 points)](https://news.ycombinator.com/item?id=48957779); [arXiv 2607.16558: Near-optimal lower bounds for randomized algorithms (follow-up)](https://arxiv.org/abs/2607.16558)

### 2026-07-17: GPT-5.6 Sol Ultra proves the 50-year-old cycle double cover conjecture

*****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-07-17-cycle-double-cover-conjecture-proved/

In mid-July 2026 OpenAI released a preprint crediting GPT-5.6 Sol Ultra, 'in less than an hour', with a proof of the cycle double cover conjecture (Szekeres 1973, Seymour 1979): every bridgeless graph has a collection of cycles covering each edge exactly twice. Independent expositions by graph theorists Sang-il Oum and Jim Geelen followed.

- OpenAI released the proof as a PDF (cdn.openai.com); arXiv 2607.15399 is Jim Geelen's 3-page notes 'OpenAI's proof of the Cycle Double Cover Theorem' ('my attempt to clarify, to myself, the OpenAI proof … This write-up offers no new insights'); Oum's exposition is arXiv 2607.16356 (17 Jul 2026)
- Proof attributed entirely to GPT-5.6 Sol Ultra; the write-up was done with Codex
- Independent checks and expositions by Sang-il Oum and Jim Geelen; a public Lean formalisation is reported but not verified here

#### What happened
OpenAI published a proof of the cycle double cover conjecture that it attributed wholly to its top model. Leading graph theorists independently re-expounded and checked the argument within days.

#### Why it matters
Along with the Jacobian counterexample the same week, it marked the point where famous named conjectures, not just Erdős-list problems, began falling to AI.

Sources: [OpenAI: cycle double cover proof (PDF)](https://cdn.openai.com/pdf/04d1d1e4-bc75-476a-97cf-49055cd98d31/cdc_proof.pdf); [Jim Geelen: OpenAI's proof of the Cycle Double Cover Theorem (arXiv 2607.15399)](https://arxiv.org/abs/2607.15399); [Sang-il Oum: exposition of the proof (arXiv 2607.16356)](https://arxiv.org/abs/2607.16356); [AI Weekly: OpenAI attributes cycle double cover proof to GPT-5.6 Sol Ultra](https://aiweekly.co/alerts/openai-attributes-cycle-double-cover-proof-to-gpt-56-sol-ultra)

### 2026-07-20: Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3

*****; Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/

Anthropic mathematician Levent Alpöge posted an explicit polynomial map F: C³→C³ with constant Jacobian determinant −2 that is not injective, found with Claude Fable 5. This refutes Keller's 1939 Jacobian conjecture in every dimension n≥3; the two-variable case remains open. Within days mathematicians produced infinite families, a geometric explanation and counterexamples in all dimensions above 2.

- Announced on X on 19–20 July 2026 ('hello there the jacobian conjecture is false thanx'); no paper at first
- Explicit map with constant Jacobian −2 sending three points to one; checkable by hand or computer algebra
- Akhil Mathew suggested the problem; Claude Fable 5 found the map
- Follow-ups: infinite family (Gallagher, 20 Jul); 'tangent-sweep' explanation (Speyer, 23 Jul); Tao's 'digestion' (21 Jul); Shuhong Gao, arXiv 2608.00222, including a degree-4 3-D example
- The Fields Medallists' September letter criticised announcing it by tweet
- 17 Oct 2026: Michiel de Bondt (arXiv 2610.24889) applied economical stabilization to a tweaked Alpöge map: counterexamples to HJC(15), the Vanishing Conjecture VC(28) and HVC(30), low-degree ones to JC(12,3), HJC(18,3), VC(34,4), HVC(36,4), and a cubic-linear (Drużkowski) counterexample in dimension 90, all over the (Gaussian) integers. Disclosure: 'This manuscript was written by Google Chrome AI, under supervision of the author … With the exception of section 5, all ideas in the paper are the author's own. The AI-model came with the 7-term formula that we use for xi xj xk and did not mention the 4-term formula.' Unrefereed preprint

#### What happened
Alpöge announced the counterexample in a one-line tweet with the explicit map. Because anyone can verify it by expanding a determinant, it was confirmed within hours, and a burst of human follow-up work explained and generalised it.

#### Why it matters
The Jacobian conjecture is one of the most famous open problems in algebra. Its refutation by an AI-found formula is among the most shocking AI results in mathematics so far, and fed the debate about how such results should be announced.

Sources: [Terence Tao: A digestion of the Jacobian conjecture counterexample](https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/); [Shuhong Gao: counterexamples in all dimensions >2 (arXiv 2608.00222)](https://arxiv.org/abs/2608.00222); [Xena Project: Human mathematicians are being out-counterexampled](https://xenaproject.wordpress.com/2026/07/20/human-mathematicians-are-being-outcounterexampled/); [de Bondt: Alpöge's counterexample to the JC, Economic Stabilization, and Zhao's Vanishing Conjecture (arXiv 2610.24889)](https://arxiv.org/abs/2610.24889); [ScienceDaily: Claude Fable 5 AI finds a tiny formula that topples an 87-year-old math conjecture](https://www.sciencedaily.com/releases/2026/08/260804034634.htm)

### 2026-07-22: Dmitry Rybin posts a GPT-5.6 Pro counterexample to the ~30-year-old Dinitz–Garg–Goemans (Goemans cost) conjecture on unsplittable flows

***; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-07-22-dinitz-garg-goemans-conjecture-disproved-gpt-5-6-pro/

On 22 July 2026 Dmitry Rybin posted on X that the Dinitz–Garg–Goemans conjecture, open for about 30 years, is false, with a small graph found in a chat with GPT-5.6 Pro: a fractional flow of cost 58 for which every unsplittable flow with capacity violation at most 15 costs at least 60. He shared the full chat. The post had ~4.9M views. No paper or peer review has followed as far as we found.

- Rybin on X (22 July 2026, ~4.88M views by Oct 19, fxtwitter): 'Dinitz-Garg-Goemans conjecture is false. This graph theory problem was open for ~30 years. The graph below has fractional flow cost 58. Any unsplittable flow (with capacity violation <=15) has cost at least 60.' The post links the ChatGPT conversation with GPT-5.6 Pro
- What is refuted (DataCamp): the cost version, Goemans' conjecture that a cheap splittable flow can be turned into an unsplittable one keeping both congestion and cost bounded; the Dinitz–Garg–Goemans congestion theorem itself is not in question
- Construction (as reported by 36Kr and DataCamp): one source, three terminals with demands 15, 10 and 15; fractional cost 58 vs ≥ 60 for any unsplittable routing within the congestion budget. DataCamp could not reproduce the exact graph from a primary source
- Checks: DataCamp says several people checked the arithmetic and found it consistent and one researcher placed the instance in an infinite three-parameter family; QbitAI/36Kr says the deliverables included proof certificates, an exhaustive enumeration program and LaTeX source. No named expert statement found
- Seen by us only on 19 Oct 2026, through Fireship's video 'Summer math fell to the machines' (Aug 19); Chinese tech press (36Kr, 24 July) covered it the same week together with Shouqiao Wang's six Erdős problems

#### What happened
Rybin asked GPT-5.6 Pro for a complete counterexample to the conjecture and posted the resulting graph and the conversation. The claim spread
widely on X (one of the most viewed AI-maths posts of the summer) and in Chinese tech media. We read the X post (via fxtwitter), DataCamp and
36Kr; the open-since year is DataCamp's "roughly 1999", which fits Rybin's "~30 years". The construction details are secondary; verify against
the chat before relying on them.

#### Why it matters
It is one of several AI-found counterexamples posted in July 2026 (see the Jacobian, Hessian and Rota-flats entries of the same week) and was
missed by this dataset for three months because it lived only on X and in a chat log, with no arXiv paper.

Sources: [Dmitry Rybin on X: Dinitz-Garg-Goemans conjecture is false (22 July 2026)](https://x.com/DmitryRybin1/status/2079904005652893709); [ChatGPT conversation with GPT-5.6 Pro (shared by Rybin)](https://chatgpt.com/share/6a60b2eb-0b64-83ee-9c76-7931ca1de063); [DataCamp: GPT-5.6 and the Dinitz-Garg-Goemans conjecture](https://www.datacamp.com/blog/gpt-5-6-dinitz-garg-goemans-conjecture); [36Kr: GPT-5.6 disproves 30-year graph theory conjecture; Peking University alumnus solves six Erdős problems in five days (24 July)](https://eu.36kr.com/en/p/3908956256605577)

### 2026-07-22: Columbia PhD student Shouqiao Wang says he solved six open Erdős problems in five days with GPT-5.6 Sol in Codex, and publishes his workflow

***; OpenAI, Columbia University; Science & math; confidence medium; https://postcutoff.com/e/2026-07-22-shouqiao-wang-six-erdos-problems-gpt-5-6-sol-codex/

On 22 July 2026 Shouqiao Wang, a Columbia PhD candidate, posted "I solved 6 open Erdős problems in 5 days, using @OpenAI GPT-5.6 Sol", with a thread giving his Codex workflow and prompts. Coverage says he tried about 13 problems (46% success), one run lasting 32 hours. The post had ~1.6M views. We have not identified the problem numbers or any independent verification.

- Wang on X (22 July 2026, ~1.59M views by Oct 19, fxtwitter): 'I solved 6 open Erdős problems in 5 days, using @OpenAI GPT-5.6 Sol. I have a math background, but the Codex workflow I used does not require deep mathematical knowledge. Here's exactly how I approached it, including my prompts'
- Per 36Kr (24 July): about 13 problems attempted, 46% success rate; one problem ran continuously for 32 hours; process of attempt, diagnosis, draft proof and adversarial audit; one solved problem was a topic Terence Tao had previously studied (no Tao statement quoted)
- Method (per coverage of his thread): pick problems mathematicians already discuss, define in advance what counts as a complete solution, list weaker results and trap cases up front, then run multi-hour searches for counterexamples before drafting proofs
- Not established: which Erdős problem numbers, whether erdosproblems.com marked them solved, and any expert review; the thread replies were not read (fxtwitter returns only the first post)

#### What happened
Wang posted his result and a step-by-step thread on how he drove GPT-5.6 Sol through Codex on Erdős problems. The thread drew wide attention
for its claim that the workflow needs little mathematical expertise. We read only the first post (fxtwitter) and 36Kr's summary.

#### Why it matters
It is an early public recipe for turning a coding agent into a problem-solving loop on open maths problems, published months before the
Erdős-problem wave of September–October 2026. Its claims still need checking against erdosproblems.com.

Sources: [Shouqiao Wang on X: I solved 6 open Erdős problems in 5 days (22 July 2026)](https://x.com/Qiaoqiao2001/status/2080003441821163958); [36Kr: GPT-5.6 disproves 30-year graph theory conjecture; Peking University alumnus solves six Erdős problems in five days (24 July)](https://eu.36kr.com/en/p/3908956256605577)

### 2026-07-23: AI systems score a perfect 42/42 at IMO 2026, officially graded

*****; Huawei, Xiaohongshu (RedNote); Science & math; confidence high; https://postcutoff.com/e/2026-07-23-imo-2026-ai-perfect-scores/

For the first time AI achieved full marks at the International Mathematical Olympiad: at IMO 2026 in Shanghai, Huawei's 'Celia' and Xiaohongshu/RedNote's 'dots-note-3.0' each scored 42/42, with solutions graded by IMO organisers after the human contest; only 7 of 666 human contestants got perfect scores. Other labs (OpenAI, Anthropic, Moonshot, Axiom) also claimed 42/42.

- Perfect 42/42 (all six problems) for Huawei 'Celia' and RedNote 'dots-note-3.0' under the IMO's formal AI evaluation process
- Process: AI received problems only after human contestants finished; strict time limit; no human intervention; graded by IMO organisers
- Humans: 7 of 666 contestants achieved full marks (IMO held in Shanghai)
- Per commentator Deedy Das (quoted by TechXplore), OpenAI, Anthropic, Axiom Math and Moonshot's Kimi K3 also reached 42/42 (not all officially graded)
- Context: 2024 best AI = silver (4/6 problems over 2-3 days); 2025 = gold-level 35/42 (Google DeepMind, OpenAI)
- No AI was an official medal-eligible contestant

#### What happened
At IMO 2026 (Shanghai), several AI systems solved all six problems. The two officially graded perfect scores came from Chinese companies not usually considered frontier labs:
Huawei (Celia) and Xiaohongshu/RedNote (dots-note-3.0, its first IMO entry). Multiple US labs and Moonshot also reported perfect solutions. Commentator Deedy Das: "The frontier of AI has officially moved well past IMO math."

#### Why it matters
Olympiad math is now saturated as an AI benchmark just one year after the first gold-level results; attention shifts to research-level math (FrontierMath Tier 4, Erdős problems).

Sources: [TechXplore: AI catches up with humans to score 100% at top math contest](https://techxplore.com/news/2026-07-ai-humans-score-math-contest.html); [SCMP: RedNote's AI model first to achieve flawless score at maths Olympiad](https://www.scmp.com/tech/article/3361482/worlds-first-ai-model-earn-perfect-score-maths-olympiad-comes-chinas-rednote); [Taipei Times: AI models score 100 percent at top math competition](https://www.taipeitimes.com/News/world/archives/2026/07/24/2003861308); [Malay Mail: Huawei, Xiaohongshu AI storm Olympiad](https://www.malaymail.com/news/tech-gadgets/2026/07/23/huawei-xiaohongshu-ai-storm-olympiad-join-maths-elite-with-perfect-100pc-score/228720); [France 24 / AFP: AI catches up with humans to score 100% at top maths contest](https://www.france24.com/en/live-news/20260723-ai-catches-up-with-humans-to-score-100-at-top-maths-contest); [Deedy Das on X: self-run IMO 2026 results for frontier models](https://x.com/deedydas/status/2079409461874332066); [NVIDIA AI on X: Nemotron 3 Ultra graded 30/42 by IMO team](https://x.com/NVIDIAAI/status/2079642933058244704)

### 2026-07-24: Hessian conjecture refuted in five variables, derived from Claude-found Jacobian counterexample

***; Independent researchers; Science & math; confidence high; https://postcutoff.com/e/2026-07-24-hessian-conjecture-counterexample/

Five days after Levent Alpöge's Claude Fable 5-assisted counterexample to the Jacobian conjecture, Guowu Meng and Liang Yang used "Schur descent" on it to build a five-variable counterexample to the related Hessian conjecture. The Hessian conjecture now holds for n≤3, fails for n≥5, and is open only for n=4.

- arXiv 2607.22198, submitted 2026-07-24 (revised 07-27)
- Explicit polynomial in 5 variables, degree 14, constant Hessian determinant 128, with non-injective gradient
- Derived from Alpöge's Jacobian counterexample; the paper itself does not report AI use

#### What happened
Guowu Meng and Liang Yang turned Alpöge's three-variable Jacobian counterexample into a five-variable counterexample to the Hessian conjecture.

#### Why it matters
It shows how AI-found results feed quickly into human follow-up work. It also leaves one clean open case, n=4.

Sources: [arXiv 2607.22198: A five-variable counterexample to the Hessian conjecture](https://arxiv.org/abs/2607.22198); [Terence Tao: A digestion of the Jacobian conjecture counterexample](https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/)

### 2026-07-24: Rota's 1970 unimodality conjecture for matroid flats disproved, with ChatGPT 5.6 Pro suggesting a key construction

***; OpenAI, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-07-24-rota-flats-unimodality-disproved/

In July–August 2026 three arXiv papers overturned classic matroid conjectures with AI help. Matt Larson (2607.02208, Jul 2) disproved Mason's log-concavity conjecture for flat counts and White's exchange conjecture, using ChatGPT 5.5 Pro (and Claude Opus 4.8) to search for counterexamples. Divoux, Lowen and Wang (2607.22515, Jul 24) then disproved Rota's 1970 conjecture that the numbers of flats by rank form a unimodal sequence; ChatGPT 5.6 Pro supplied a q-lift example they generalized. A follow-up (2608.07342, Aug 7) showed the sequence can have arbitrarily many peaks.

- Rota's conjecture (1970): the Whitney numbers of the second kind (number of flats of each rank) of every matroid form a unimodal sequence
- arXiv 2607.02208 (Matt Larson, Jul 2, 2026): counterexamples to Mason's conjecture (log-concavity of flat counts; e.g. W_74^2 < W_73·W_75 for a graphic matroid of a generalized theta graph with three 26-edge paths and one edge) and to White's conjecture on symmetric exchanges (a rank-9 binary matroid on 18 elements)
- Larson's AI use: 'I prompted ChatGPT 5.5 Pro to look for a counterexample around 20 times'; the paper also credits Claude Opus 4.8
- arXiv 2607.22515 (Alexander Divoux, Chayim Lowen, Shouda Wang, Jul 24, 2026; 8 pages): 'We give counterexamples to Rota's 1970 conjecture'; built from Larson's matroid via Whittle's q-lift
- AI role in 2607.22515: 'We used ChatGPT 5.6 Pro to search for a counterexample to this weaker property. Based on a suggestion of the authors, ChatGPT 5.6 Pro returned a non-convex example built using a q-lift'; the paper was 'written entirely by the authors'
- arXiv 2608.07342 (Divoux, Larson, Lowen, Wang, Aug 7, 2026): flat counts can have arbitrarily many peaks
- Listed in Table 3 of the AI4Math survey arXiv 2608.24961 as a conjecture disproved with ChatGPT 5.6 Pro

#### What happened
Larson first used ChatGPT 5.5 Pro, prompted around 20 times with hints (for example, look at matroids realizable over F5 and at failures at high
indices), to find a graphic matroid that breaks Mason's log-concavity conjecture. Divoux, Lowen and Wang then asked ChatGPT 5.6 Pro to look for an
example of a weaker property; it returned a q-lift construction, which the authors turned into counterexamples to Rota's unimodality conjecture.
All four then showed the flat-count sequence can have many peaks.

#### Why it matters
Rota's conjecture was one of the best-known open problems about matroids. The chain is a clear example of the 2026 pattern: AI systems search for
counterexamples, and humans generalize and write up the results, with the AI's role stated precisely in the papers.

Sources: [arXiv 2607.22515: Matroid flat counts are not unimodal](https://arxiv.org/abs/2607.22515); [arXiv 2607.02208: Counterexamples to two conjectures about matroids (Larson)](https://arxiv.org/abs/2607.02208); [arXiv 2608.07342: Matroid flat counts can have many peaks](https://arxiv.org/abs/2608.07342); [arXiv 2608.24961: The Gold Rush in AI4Math (Table 3)](https://arxiv.org/abs/2608.24961)

### 2026-07-24: Terence Tao's ICM 2026 public lecture 'Mathematics in the age of AI' calls a crisis in the foundations of mathematical values

***; International Congress of Mathematicians, UCLA; Science & math; confidence high; https://postcutoff.com/e/2026-07-24-tao-icm-mathematics-in-the-age-of-ai/

On 24 Jul 2026, at the International Congress of Mathematicians in Philadelphia, Terence Tao gave the public lecture "Mathematics in the age of AI". He argued that mathematics is entering a "crisis in the foundations of mathematical values and practices", comparable to the 1900–1930 foundations crisis. Setting aside the capability debate, he asked what the community's goals should be if strong AI capability arrives. An essay version is arXiv 2608.16753.

- Venue: ICM 2026 public lecture, Pennsylvania Convention Center, Philadelphia, 24 Jul 2026 (7:15 pm)
- Frames an 'AI Capability Conjecture' (weak vs strong forms) and conditions on it being true, then asks the orthogonal 'Goals and Values Question'
- Uses problem-solving as a case study: from 'solve as many unsolved problems as possible' to results that are verified, clearly communicated, digested and incorporated into the definitive theory
- Recommendation reported by press: results that cannot be shown correct and properly attributed, or explained by their authors, should not be published; disclose tool use
- Slide footnote: 'All em-dashes in these slides were human-generated.'
- Essay: arXiv 2608.16753 (17 Aug 2026, 12 pages, submitted to the ICM 2026 Proceedings)
- Tao also published an AI-collated summary of his AI views and an AI-conducted 'hard hitting' interview of himself

#### What happened
Tao's public lecture at the quadrennial ICM compared the present moment to the early-20th-century crisis in foundations. That crisis ended with a rigorous, standardized framework. Tao said the community now needs to codify its *values* in the same way. He deliberately did not argue about which AI capabilities are real. He treated a "reasonably strong" capability conjecture as a working hypothesis and asked what mathematicians actually want. Press described the lecture as more foreboding than his earlier comments.

#### Why it matters
It was the most prominent framing of AI-and-mathematics at the field's main quadrennial event. It came just before the wave of AI results (Astra's ten advances, Navier–Stokes) and the community statements that followed (Fields Medallists' letter, Palomar, SAIR).

Sources: [Tao: slides 'Mathematics in the age of AI' (PDF)](https://teorth.github.io/tao-web/slides/age-of-ai-icm-2026.pdf); [arXiv 2608.16753: Mathematics in the age of AI (essay)](https://arxiv.org/abs/2608.16753); [Tao on Mathstodon: slides uploaded, AI-made summary and interview](https://mathstodon.xyz/@tao/116977934921819775); [Terence Tao on AI in mathematics (and beyond), AI-collated summary](https://teorth.github.io/tao-web/ai-views.html); [Tao: AI 'interview' on his AI views](https://teorth.github.io/tao-web/ai-views-interview.html); [Scientific American: If AI can do math, what's the point of mathematicians?](https://www.scientificamerican.com/article/mathematicians-confront-the-ai-apocalypse/); [Simons Foundation: Watch: Terence Tao on AI and why we do math](https://www.simonsfoundation.org/2026/08/13/fields-medalist-terence-tao-on-artificial-intelligence-and-why-we-do-math/); [YouTube recording (uploaded by Alvaro Lozano-Robledo)](https://www.youtube.com/watch?v=sxAe4HJceFQ)

### 2026-07-25: Sharp thin-shell inequality: Var(|X|²) ≤ 8n for isotropic log-concave vectors, with a proof 'found by GPT-5.6 Pro' (Chen & Klartag)

***; ETH Zurich, Weizmann Institute of Science, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-07-25-sharp-thin-shell-inequality-gpt-5-6-pro/

On 25 July 2026 Yuansi Chen and Boaz Klartag posted arXiv 2607.23307, which determines the optimal constant in the thin-shell theorem: for every isotropic log-concave random vector X in R^n, Var(|X|²) ≤ 8n, with equality for independent centered exponential coordinates. The abstract says "The proof was found by GPT-5.6 Pro in response to prompts supplied by the first-named author"; both authors verified, revised and rewrote the argument. Unrefereed preprint by two leading specialists; the chat log is attached to the arXiv submission.

- Paper: 'Digesting the proof of the sharp thin-shell inequality', Yuansi Chen and Boaz Klartag, arXiv 2607.23307 [math.MG], 25 July 2026, 23 pages; arXiv comment: 'Statement of AI use included. Chat log is in the ancillary files as a pdf'
- Theorem 1.1: for any isotropic, log-concave random vector X in R^n, E(|X|² − n)² ≤ 8n. 'The constant 8 is optimal: equality is attained when X_1, …, X_n are independent, identically distributed, standard, centered exponential random variables'
- Among isotropic random vectors uniform on convex bodies in R^n, Var(|X|²) is maximized by the uniform distribution on a regular simplex; the paper also gives a sharp bound on the Hilbert–Schmidt norm of the third-moment tensor
- Background: the thin-shell (variance) conjecture of Anttila–Ball–Perissinaki and Bobkov–Koldobsky asked for a universal constant; it 'was recently proven by Klartag and Lehec' (arXiv 2507.15495, 2025) after bounds by Fleury, Guédon–Milman, Lee–Vempala, Jambulapati–Lee–Vempala and Guan. The new paper gives the exact constant by a different method
- Method: a weighted Riemannian manifold associated with log-concave moment measures and the Monge–Ampère equation, studied by Klartag in 2014; 'The main improvement … comes from a concise yet effective analysis of the 3rd-derivatives tensor of the potential'. Earlier arguments used concentration on the sphere or Eldan's stochastic localization
- AI disclosure (abstract): 'The proof was found by GPT-5.6 Pro in response to prompts supplied by the first-named author, following general discussions between the two authors concerning log-concave moment measures. The prompts referred to the paper “Logarithmically-concave moment measures I” and suggested bootstrapping a bound on the second trace moment.'
- AI use statement: 'GPT-5.6 Pro produced the initial proof in response to prompts from the first-named author. Both authors verified and revised the argument, and rewrote it to make it more accessible. GPT-5.6 was then used to polish the writing.' Section 3 contains 'the 3rd-moment computations by GPT-5.6 Pro'
- Acknowledgement: Chen 'thanks Ronen Eldan for providing access to GPT Pro and for encouraging its use'
- Reception: Gil Kalai's 22 Sept 2026 essay on AI and mathematics uses the paper as his example of AI simplifying mathematics. No refereed version or independent written check found as of 13 Oct 2026

#### What happened
A log-concave random vector in high dimension, once normalized to mean zero and identity covariance, has most of its mass in a thin
spherical shell of radius about √n. How thin was a central question of asymptotic convex geometry for two decades, tied to the
central limit theorem for convex bodies and to Bourgain's slicing problem. Klartag and Lehec proved in 2025 that the width is
bounded by a universal constant. Chen and Klartag's paper gives the exact constant: the variance of |X|² is at most 8n, and
independent exponential coordinates attain it.

The authors are open about where the proof came from. Chen prompted GPT-5.6 Pro, pointing it to Klartag's 2014 paper on
log-concave moment measures and suggesting a bootstrap of a bound on a second trace moment; the model produced the initial proof.
The two authors then checked it, revised it and rewrote it for readers, which is what the title's "digesting" refers to. The
chat log is included with the arXiv submission.

#### Why it matters
This is a sharp result in a mature area, signed by people who built that area, with the model credited for the proof in the
abstract. It is also an early example of a practice that became common in the summer of 2026: specialists publish an AI-found
proof after rewriting it, and attach the transcript.

#### Not verified here
The ancillary chat log was not read for this entry. Eldan's role is taken from the acknowledgement only; his affiliation was not checked. The Kalai link is cited
from our archived summary of his post.

Sources: [arXiv 2607.23307: Digesting the proof of the sharp thin-shell inequality](https://arxiv.org/abs/2607.23307); [arXiv 2507.15495: Klartag & Lehec, Thin-shell bounds via parallel coupling (2025)](https://arxiv.org/abs/2507.15495); [Gil Kalai: “Be a Grothendieck!”—On AI and Mathematics (22 Sept 2026)](https://gilkalai.wordpress.com/2026/09/22/be-a-grothendieck-on-ai-and-mathematics/)

### 2026-07-27: Neurosurgery resident uses GPT-5.6 Sol to prove Crouzeix's conjecture in a 16-hour autonomous run

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-07-27-crouzeix-conjecture-proved/

A preprint posted 27 Jul 2026 proves Crouzeix's conjecture (2004): for every square matrix A and polynomial f, ‖f(A)‖ ≤ 2·max over the numerical range W(A) of |f|. The proof came from one uninterrupted 16-hour autonomous GPT-5.6 Sol run prompted by Shanmu Jin, a self-taught neurosurgery resident. Michel Crouzeix, Anne Greenbaum and Alex Townsend checked it.

- Previously best known constant: 1+√2 (Crouzeix–Palencia 2017); conjectured optimal constant 2
- Single 16-hour autonomous run of GPT-5.6 Sol
- Checked by Crouzeix himself, Anne Greenbaum and Alex Townsend (SIAM News essay)
- Independent proof: Emiel Lorist and Felix Schwenninger, 'A solution to Crouzeix's conjecture' (arXiv 2608.03841, 4 Aug 2026); their AI disclosure says GPT-5.6 Sol Pro in ChatGPT 'was used to explore proof strategies for this note'

#### What happened
A non-mathematician set GPT-5.6 Sol on the problem. The model produced a complete proof in one long run, which the conjecture's originator and other specialists confirmed.

#### Why it matters
Along with #1196, it showed that frontier models let amateurs resolve famous problems, which upended assumptions about who can do research mathematics.

Sources: [Alex Townsend: SIAM News essay on the Crouzeix conjecture (PDF)](https://alextownsend.net/essays/SIAMNews_CrouzeixConjecture.pdf); [SCMP: Chinese doctor stuns maths world cracking decades-old problem using ChatGPT](https://www.scmp.com/tech/tech-trends/article/3363966/chinese-doctor-stuns-maths-world-cracking-decades-old-problem-using-chatgpt); [arXiv 2608.03841: A solution to Crouzeix's conjecture (Lorist, Schwenninger)](https://arxiv.org/abs/2608.03841)

### 2026-07-29: Maxwell's conjecture on equilibria of point charges is false: five charges with at least 24 critical points, construction idea from GPT-5.6 Sol

***; Philip Arathoon, Gavin Ball, Matthew D. Kvalheim, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-07-29-maxwell-point-charge-conjecture-false/

Philip Arathoon, Gavin Ball and Matthew Kvalheim posted a configuration of five point charges whose electrostatic potential has at least 24 non-degenerate critical points (arXiv 2607.27197, 29 Jul 2026). This refutes the 'Maxwell conjecture', which traces to Maxwell's 1873 Treatise and bounds the count at (n−1)² = 16. Their disclosure: 'The idea behind this construction was suggested by an LLM (OpenAI's GPT5.6 Sol)'. The authors verified and wrote the argument.

- Maxwell (Treatise, 1873, §113) discussed the number of equilibria of n point charges; Gabrielov, Novikov and Shapiro formulated the conjecture that there are at most (n−1)² when all are non-degenerate; Morse and Cairns had posed the bound question in 1969
- Counterexample: 5 charges in R³ with ≥ 24 non-degenerate critical points, versus the conjectured maximum of 16
- Tool disclosure: 'The idea behind this construction was suggested by an LLM (OpenAI's GPT5.6 Sol). The authors have verified the mathematical details and have written the argument in their own words'; Mathematica and Maple used for verification
- Discussed on Hacker News as 'The Maxwell Conjecture Is False (GPT 5.6 Sol)' (157 points, 31 Jul 2026)

#### What happened
Three mathematicians took up a problem Maxwell discussed in his 1873 treatise: how many equilibrium points the field of n charges can have. They found a five-charge configuration with far more equilibria than the conjectured bound, based on an idea proposed by GPT-5.6 Sol.

#### Why it matters
It is a long-standing question at the boundary of physics and real algebraic geometry, and one of the few 2026 AI-assisted results in classical physics.

Sources: [arXiv 2607.27197: The Maxwell Conjecture is False (Arathoon, Ball, Kvalheim)](https://arxiv.org/abs/2607.27197)

### 2026-08-01: OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs

*****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/

On 1 Aug 2026 OpenAI published 'Ten advances in mathematics and theoretical computer science' by an internal model, Astra (released as GPT-6 Astra on 3 Sep). It came with a 249-page manuscript and Lean 4 proofs. The claims include the first explicit non-sofic group, a disproof of Connes's rigidity conjecture, the first improvement to the sphere-packing upper-bound exponent since 1978, and solutions to Erdős problems #146, #180 and #183.

- Claims: explicit non-sofic group (Gromov's question, ~1999); disproof of Connes's rigidity conjecture; quantum parallel repetition for general two-player entangled games
- Also: Ehrhart volume conjecture (partial per some sources); polynomial-factor NP-hardness of approximating the Closest Vector Problem; permanent circuit lower bound ~n⁴/log n
- Superexponential lower bound for multicolour Ramsey numbers (Erdős #183); Erdős #146 and #180; improved binary and spherical codes
- Sphere-packing upper-bound exponent ~0.5990558 → ~0.6044005, first improvement since Kabatiansky–Levenshtein (1978)
- Evidence: 249-page PDF, Lean 4 proofs (openai/ten-proofs); < $2,000 of tokens per solution at GPT-5.6 Sol prices; prompts not released
- Attribution dispute: Andreas Thom (11 Sep, guest post on Tao's blog) says the non-sofic proof relies crucially on his 2019 work with Gábor Kun (Prop. 2.3 of OpenAI's PDF) despite OpenAI's 'decade without progress' framing, and asks whether his own ChatGPT conversations about these techniques reached the model; Mark Sellke replied 'that did not happen'. Kun and Thom posted a follow-up, arXiv 2608.06222 (6 Aug)
- Independent audit (arXiv 2608.14673): 'No confirmed substantive mathematical error in a principal result remains'; one chapter needs major revisions, and some stronger results were not reproduced
- Follow-ups: Jihao Liu proved the equality case of Ehrhart's volume conjecture with GPT-5.6 Sol, Fable 5 and the Danus agent (arXiv 2608.01040); Raphael Steiner extended the superexponential multicolour Ramsey construction to odd cycles with a proof 'found autonomously by ChatGPT 5.6 Pro/Sol' (arXiv 2608.02537, later withdrawn and merged into arXiv 2608.02522)
- The Verge (10 Sep): Thom on Mastodon called Mark Sellke's categorical answer 'materially misleading; looking back it was plainly dishonest' and said 'De-identification may remove a name; it does not remove the intellectual content of a mathematical idea'. The Verge says OpenAI quietly amended its write-up to credit Kun and Thom without announcing the change
- Scientific American (Joseph Howlett, 6 Aug), 'OpenAI's latest math breakthroughs commit research misconduct, experts say': Stephen Miller (said the sphere-packing proof used his 2016 argument without proper citation): 'They are running roughshod over the work of others who came before them in a deliberate way. It seems completely systematic to me, and it points to research misconduct.' Francesco Fournier-Facio (Cambridge) on the soficity result: 'the big PR machine… does not care about being 100 percent accurate'. OpenAI: 'We take responsibility for the correctness of these results and are meeting the same standards generally expected of human mathematicians' and promised small updates

#### What happened
OpenAI released, in one announcement, ten research results produced by an internal model a month before its launch. Most came with machine-checked proofs.

#### Why it matters
It moved the frontier from individual AI-assisted results to a lab producing batches of significant theorems. An independent audit largely upheld them.

Sources: [Scientific American: OpenAI's latest math breakthroughs commit research misconduct, experts say (6 Aug)](https://www.scientificamerican.com/article/openais-latest-math-breakthroughs-commit-research-misconduct-experts-say/); [OpenAI: Ten advances in mathematics and theoretical computer science](https://openai.com/index/ten-advances-in-mathematics/); [OpenAI: ten proofs manuscript (PDF)](https://cdn.openai.com/pdf/ten-proofs-oai.pdf); [A Human Audit of OpenAI's AI-Generated Mathematical Proofs (arXiv 2608.14673)](https://arxiv.org/abs/2608.14673); [Simon Willison on the ten advances](https://simonwillison.net/2026/Aug/1/ten-advances-in-mathematics/); [Andreas Thom (guest post on Tao's blog): On the existence of non-sofic groups (attribution concerns)](https://terrytao.wordpress.com/2026/09/11/on-the-existence-of-non-sofic-groups/); [Kun & Thom: Nonsofic wreath products of residually finite groups (arXiv 2608.06222)](https://arxiv.org/abs/2608.06222); [MathOverflow: key new ideas in the non-soficity proof](https://mathoverflow.net/questions/513866/what-are-the-key-new-ideas-in-the-proof-of-nonsoficity-of-groups-in-openai-s-con); [Quanta: Why the legendary Erdős problems are falling to AI](https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/); [arXiv 2608.01040: The equality case of Ehrhart's volume conjecture (Liu)](https://arxiv.org/abs/2608.01040); [arXiv 2608.02522: Locally bipartite subgraphs via multicolor Ramsey numbers (Steiner)](https://arxiv.org/abs/2608.02522); [The Verge (11 Aug): The AI takeover of mathematics has begun](https://www.theverge.com/ai-artificial-intelligence/977273/the-ai-takeover-of-mathematics-has-begun); [The Verge Decoder (20 Aug): Welcome to the AI crisis in math (Robert Hart)](https://www.theverge.com/podcast/982434/ai-math-openai-astra-existential-crisis); [The Verge (10 Sep): Mathematicians want proof OpenAI didn't use their work (Thom)](https://www.theverge.com/ai-artificial-intelligence/993263/where-does-openai-get-mathematics-training-data)

### 2026-08-01: OpenAI hosts a closed-door summit of about 40 mathematicians on the future of mathematics

***; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-08-01-openai-future-of-mathematics-summit/

In early August 2026, shortly after its "ten advances in mathematics" post, OpenAI hosted about 40 mathematicians at its San Francisco offices to discuss what will be left for human mathematicians if AI surpasses them. Sébastien Bubeck asked Daniel Litt to describe "the future we'd all like to avoid"; Litt's talk "The End of Mathematics" became a widely read essay. The Washington Post reported on the summit on Aug 19.

- When/where: early August 2026, OpenAI offices in San Francisco; Litt posted on Aug 11 that he was returning from it
- About 40 mathematicians attended (36Kr summary of the Washington Post report)
- Named participants (WaPo via 36Kr and Litt): Sébastien Bubeck (OpenAI), Jacob Tsimerman, Daniel Litt (Toronto), Drew Sutherland (MIT), Bryna Kra (Northwestern), Melanie Matchett Wood (Harvard), Andreas Thom, David Bessis
- Litt's talk 'The End of Mathematics' described a scenario in which AI is superhuman at maths but the field stalls: an explosion of output of unknown quality, collapsing engagement, and mathematicians disconnected from the mathematics
- Litt's thread drew ~334k views; his follow-up essay 'A beginning for mathematics' (Sept 13) proposed reforms such as oral-defence PhDs
- Washington Post article, Aug 19, 2026: 'Mathematicians ask what's left for humans when AI can do math research'; reprinted as 'This may be the first academic profession to see its work taken over by AI'

#### What happened
OpenAI convened a private meeting with research mathematicians, co-organized by Bubeck and Tsimerman according to 36Kr's account, about how the profession
should respond to AI that can produce research results. The organizers deliberately asked for the pessimistic scenario to be presented.
OpenAI has published no official account of the summit; the details come from participants and press.

#### Why it matters
A lab whose models were producing new maths results was now directly engaging the profession about its future. Litt's talk became a
reference point in the 2026 debate on whether AI will empower or "disempower" mathematicians.

Sources: [Washington Post: Mathematicians ask what's left for humans when AI can do math research (Aug 19, 2026)](https://www.washingtonpost.com/technology/2026/08/19/mathematicians-ask-whats-left-humans-when-ai-can-do-math-research/); [Texarkana Gazette (WaPo reprint): This may be the first academic profession to see its work taken over by AI](https://www.texarkanagazette.com/news/2026/aug/22/this-may-be-the-first-academic-profession-to-see/); [36Kr: The End of Mathematics, 40 top mathematicians gathered at OpenAI's secret meeting](https://eu.36kr.com/en/p/3952735841860993); [Daniel Litt on X: returning from OpenAI's summit on the future of mathematics](https://x.com/littmath/status/2087302625423409549); [Daniel Litt: The End of Mathematics (Aug 11, 2026)](https://www.daniellitt.com/blog/2026/8/11/the-end-of-mathematics/); [Daniel Litt: A beginning for mathematics (Sept 13, 2026)](https://www.daniellitt.com/blog/2026/9/13/a-beginning-for-mathematics/)

### 2026-08-03: Planar Schiffer and Pompeiu conjectures disproved by two independent groups; one proof has a Lean certificate written by GPT-5.6

****; Matthew Colbrook, George Stepaniants, Gonzalo Cao-Labora, Jaume de Dios Pont; Science & math; confidence high; https://postcutoff.com/e/2026-08-03-schiffer-pompeiu-conjectures-disproved/

In August 2026 two groups independently disproved Schiffer's conjecture (Yau's Problem 80) and the planar Pompeiu problem (1929). Colbrook and Stepaniants (3 Aug) built one explicit ten-fold symmetric domain with a computer-assisted proof, using ChatGPT/Codex only to debug code. Cao-Labora and de Dios Pont (5 Aug) built infinitely many counterexamples by bifurcation theory. They used GPT-5.5/5.6, Claude Opus 4.8 and Claude Fable 5 for estimates and drafts, and GPT-5.6 wrote their Lean 4 certificate.

- Schiffer's conjecture: if a smooth domain has a Neumann eigenfunction that is constant on the boundary, the domain is a ball (Problem 80 in Yau's 1982 problem list). The planar Pompeiu problem dates to Pompeiu's papers of 1929
- Colbrook & Stepaniants (arXiv 2608.01579, 3 Aug 2026): one bounded, simply connected, non-circular domain with real-analytic boundary, close to an explicit degree-301 polynomial conformal map with ten-fold symmetry; eigenvalue k in (31.967007261, 31.967007293); existence certified by an interval-arithmetic contraction argument
- Colbrook & Stepaniants AI declaration: 'ChatGPT 5.5 in the form of codex was used to help debug an early version of some of the code. All of its outputs were checked line by line by the authors.'
- Cao-Labora & de Dios Pont (arXiv 2608.05114, 5 Aug 2026): infinitely many N-fold symmetric counterexamples for large N. They relax N to a real parameter and use bifurcation theory with branch size uniform in N. No computer assistance is needed in the proof
- Cao-Labora & de Dios Pont AI usage: 'LLMs, in the form of GPT 5.5 and 5.6, Claude Opus 4.8 and Claude Fable 5' were used to numerically verify asymptotic estimates, draft first versions of the Bessel-function estimate proofs and help with exposition; 'The Lean4 verification of the proof was written by GPT 5.6'
- Lean repository jaumededios/Schiffer (created 4 Aug 2026) contains challenge files for Schiffer's conjecture and for the Pompeiu statement in Google DeepMind's Formal Conjectures repository. It follows a slightly different route from the paper because Mathlib lacks elliptic regularity
- Follow-up (arXiv 2608.08953, 9 Aug): Colbrook, Sadeghi and Stepaniants disproved the unrestricted planar Berenstein conjecture (the Dirichlet counterpart). Their AI declaration says ChatGPT 5.6 'suggested ideas contributing to early versions of Lemma 3.2 and Proposition 3.5, as well as to a revised version of the code', starting from a warm start of the authors' own draft and code
- Colbrook presented the result as 'A Shortcake Counterexample to the Planar Pompeiu and Schiffer Conjectures' at Princeton (PACM seminar, 16 Sep 2026)
- Status: preprints, not yet peer-reviewed; the Cao-Labora–de Dios Pont proof is Lean-checked

#### What happened
On 3 August 2026 Matthew Colbrook and George Stepaniants posted a counterexample to the planar Pompeiu and Schiffer conjectures. They found a domain numerically, then proved that an exact counterexample exists near it with an interval-arithmetic contraction argument. Two days later Gonzalo Cao-Labora and Jaume de Dios Pont posted an independent construction of infinitely many counterexamples. Their method is bifurcation theory with a real-valued symmetry parameter. It is purely analytic, and their Lean certificate was written by GPT-5.6. Cao-Labora and de Dios Pont note in their paper that Colbrook and Stepaniants posted an independent construction two days earlier. On 9 August Colbrook's group used the same machinery to disprove the planar Berenstein conjecture, this time with ChatGPT 5.6 contributing ideas to some lemmas.

#### Why it matters
Schiffer's conjecture is a classic rigidity question and appears on Yau's list of open problems. The episode shows the range of AI roles in mid-2026 papers on the same problem. In one paper it only debugged code. In another it checked estimates, drafted proofs and wrote the full formal verification. In a third it proposed ideas that the authors then developed.

Sources: [arXiv 2608.01579: A computer-assisted counterexample to the planar Pompeiu and Schiffer conjectures (Colbrook, Stepaniants)](https://arxiv.org/abs/2608.01579); [arXiv 2608.05114: Counterexamples to Schiffer's Conjecture (Cao-Labora, de Dios Pont)](https://arxiv.org/abs/2608.05114); [GitHub: jaumededios/Schiffer (Lean 4 formalisation)](https://github.com/jaumededios/Schiffer); [Zenodo: Pompeiu–Schiffer validation certificate (Colbrook)](https://zenodo.org/records/21765287); [arXiv 2608.08953: A computer-assisted counterexample to the planar Berenstein conjecture](https://arxiv.org/abs/2608.08953); [Princeton PACM seminar: A Shortcake Counterexample to the Planar Pompeiu and Schiffer Conjectures](https://www.pacm.princeton.edu/events/shortcake-counterexample-planar-pompeiu-and-schiffer-conjectures); [Wikipedia: Pompeiu problem](https://en.wikipedia.org/wiki/Pompeiu_problem)

### 2026-08-04: Alexander Perry disproves the period-index conjecture (Colliot-Thélène, 2001); a flawed ChatGPT example was the starting point

****; Alexander Perry, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-08-04-period-index-conjecture-false/

Alexander Perry posted counterexamples to the period-index conjecture for Brauer classes: d-dimensional varieties (d ≥ 3) with a Brauer class of period 2 and index 2^d (arXiv 2608.03684; v1 4 Aug, v2 3 Sep 2026). His AI disclosure says the starting point was a flawed ChatGPT example of the right general shape. Perry found the correct construction himself (a generic K3 surface with a symplectic group action), with ChatGPT helping on literature and lattice calculations.

- Period-index conjecture: stated by Colliot-Thélène in 2001; predicts the index of a Brauer class on a d-dimensional variety divides period^(d−1)
- Result: failures in dimension 3 over Q and F_p(t), and in all dimensions d ≥ 3 over C and F_p((t)); the lower bound on the index uses a motivic obstruction, namely the non-existence of integral Hodge or Tate classes satisfying an equation mod 2
- Perry proposes a corrected period-index conjecture with corrections at small primes and proves its Hodge-theoretic counterpart
- AI disclosure, §1.7: prompted to use the paper's Hodge-theoretic strategy, ChatGPT 'produced a flawed example of a threefold X and Brauer class α' that 'had some features of our final solution'; 'Drawing inspiration from this failed example, the author eventually discovered Theorem 1.3'; ChatGPT then 'assisted with literature searches and lattice theory calculations'. The paper was written by the author
- Status: preprint, not peer-reviewed

#### What happened
Perry, who had earlier worked with de Jong on the Hodge theory of the period-index problem, asked ChatGPT to look for a counterexample along that strategy. The model's example was wrong, but its shape pointed Perry to a correct construction built from K3 surfaces with symplectic group actions.

#### Why it matters
The period-index conjecture was a central open question on Brauer groups of function fields. It is also an example of an AI error being productive: the model gave the idea and a human supplied the correct mathematics.

Sources: [arXiv 2608.03684: The period-index conjecture is false for motivic reasons (Perry)](https://arxiv.org/abs/2608.03684)

### 2026-08-05: Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro; Tao simplifies and formalises it

****; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-08-05-sendov-conjecture-proved/

Lech Mazur posted a computer-assisted proof, generated with GPT-5.6 Pro, of Sendov's conjecture for all degrees: if every root of a polynomial lies in the closed unit disk, each root is within distance 1 of a critical point. Terence Tao called it 'remarkably elementary', simplified it, and used AI agents to shrink the Lean proof from ~90k to ~15k lines.

- Conjecture from 1958; previously known for degree < 9 (Brown–Xiang) and for sufficiently large degree (Tao, 2020)
- Mazur's preprint 5 Aug 2026 (some lists say 3 Aug); Tao's digestion 12 Aug 2026
- Tao extended the method to the Phelps–Rodriguez conjecture

#### What happened
A non-academic used GPT-5.6 Pro to generate a computer-assisted proof covering the remaining degrees. Tao then digested it into a short argument based on the fundamental theorem of algebra and the Maclaurin inequality.

#### Why it matters
It is a classic, well-known conjecture closed by AI, with the leading expert on the problem verifying and formalising the result.

Sources: [Terence Tao: A digestion of the proof of Sendov's conjecture](https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/); [Lech Mazur: Sendov conjecture proof (PDF)](https://www.proofatlas.ai/papers/sendov-conjecture/SENDOV_CONJECTURE_PROOF_AUGUST_5_2026.pdf)

### 2026-08-05: HRT conjecture (1996) disproved: 12 time-frequency shifts of a Schwartz function are linearly dependent, found with ChatGPT-assisted guesswork

***; Faulhuber, Petersen, van Velthoven, Voigtlaender (academic mathematicians); Science & math; confidence high; https://postcutoff.com/e/2026-08-05-hrt-conjecture-disproved/

arXiv 2608.05044 (5 Aug 2026), by Markus Faulhuber, Philipp Petersen, Jordy Timo van Velthoven and Felix Voigtlaender, shows that finitely many time-frequency shifts of a Schwartz function can be linearly dependent. This disproves the Heil–Ramanathan–Topiwala (HRT) conjecture with an explicit 12-point example. ChatGPT helped with the initial strategy and parameter guesswork. The proof was written by hand and certified numerically, not in Lean.

- HRT conjecture (Heil, Ramanathan, Topiwala, 1996): any finite set of distinct time-frequency shifts of a nonzero L² function is linearly independent
- Counterexample: 12 time-frequency shifts of a nonzero Schwartz function with a nontrivial vanishing linear combination
- Key certified numerical step: an operator-norm distance below the 1/3 threshold (value 0.333032 per Tao's digest)
- AI role (per Tao): ChatGPT assisted with the initial proof strategy and 'AI-assisted guesswork' to choose parameters; final arguments handwritten with a readable overview
- v2 adds a separate, purely analytic proof of a qualitative counterexample; Python code in the arXiv ancillary files
- Follow-ups: Vignon Oussa proposed a four-point counterexample with Arb (interval arithmetic) verification

#### What happened
Four time-frequency analysts posted a counterexample to the HRT conjecture. Tao's next-day digest explains that ChatGPT helped them find a workable strategy and good parameter choices. The decisive estimate was then certified by traditional numerical computation, and the paper itself was written by hand.

#### Why it matters
It is a clean example of the "AI-assisted, human-written" mode of discovery. It sits alongside the autonomous, Lean-verified results of summer 2026 and settles a conjecture that had resisted proof for three decades.

Sources: [arXiv 2608.05044: Linear dependence of time-frequency shifts of a Schwartz function](https://arxiv.org/abs/2608.05044); [Terence Tao: A partial digestion of the HRT counterexample](https://terrytao.wordpress.com/2026/08/06/a-partial-digestion-of-the-hrt-counterexample/)

### 2026-08-06: DeepMind open-sources WeatherNext 2 and WeatherNext Cyclones with a Nature paper showing ~1 extra day of hurricane warning

***; Google DeepMind, Google Research; Science & math; confidence high; https://postcutoff.com/e/2026-08-06-weathernext-open-source/

On 6 Aug 2026 Google DeepMind released weights and code for WeatherNext Cyclones, WeatherNext 2 and WeatherNext 2-mini under commercial-use-friendly licences, alongside a Nature paper showing its cyclone model gives more than a day of extra lead time on track, intensity and size forecasts.

- Three-day WeatherNext Cyclones forecast about as accurate as prior systems at two days: '>24 hours lead time advantage'
- Released: WeatherNext Cyclones, WeatherNext 2, WeatherNext 2-mini (runs on a single TPU / free Colab)
- Licences: Apache 2.0 for code/notebooks, CC BY 4.0 for other materials — first DeepMind weather weights allowing commercial use
- Paper in Nature (s41586-026-10953-2)
- Partners: US National Hurricane Center, CIRA, UK Met Office; helped NHC forecast Hurricane Melissa's 2025 rapid intensification
- Google blog Q&A (Sept 1) with DeepMind's Ferran Alet: WeatherNext was trained on 50 years of historical weather data and runs on a single TPU; high-confidence AI forecasts helped the US National Hurricane Center warn of Hurricane Melissa's rapid intensification from Category 1 to Category 5
- 2026 season (Local 10 Miami, Sept 10): DeepMind's model was the most accurate hurricane track model of 2026 through early September in both the Atlantic and Pacific, beating NHC's official track forecasts by up to ~30% at five days, the second year in a row it topped NHC at most lead times
- Hurricane Isaias (Oct 7–10, 2026; the latest first Atlantic hurricane of a season on record): AI models, including WeatherNext Cyclones, which NHC uses operationally, favoured an eastern landfall on the western Florida Panhandle (Cat 1, early Sat Oct 10), while physics-based models favoured Mississippi/Louisiana; NHC's official track split the difference (Gizmodo, Oct 8). CNN (Andrew Freedman, Oct 8): NHC shaped its forecasts to closely match Google DeepMind's AI model (named in the syndicated text as WeatherNext 3) rather than the European and GFS physics models, forecasting Category 2 from the first advisory; NHC director Michael Brennan: 'The AI models have shown more consistency from one model run to the next'
- Isaias outcome: landfall near Destin, Florida (western Panhandle), 9:30 p.m. ET Fri Oct 9 as a Category 2 hurricane with 105 mph winds; at least 4 deaths in Florida and Alabama (FOX Weather live blog). FOX Weather hurricane specialist Bryan Norcross: 'The Google DeepMind AI forecast was once again the best'. The AI track (right side of the cone, farther east) verified; intensity was slightly overdone (over 60% of DeepMind ensemble members made Isaias a major hurricane, per secondary reports)

#### What happened
DeepMind published open weights and code for its WeatherNext family and a peer-reviewed Nature paper on WeatherNext Cyclones, co-developed with Google Research and operational forecasters.

#### Why it matters
An extra day of hurricane warning is roughly a decade of conventional meteorological progress; releasing the weights for commercial use lets national weather services and companies run state-of-the-art AI forecasting themselves.

Sources: [DeepMind: AI model achieves breakthrough in forecasting cyclones](https://deepmind.google/blog/weathernext-ai-model-achieves-breakthrough-in-forecasting-cyclones/); [GitHub: google-deepmind/weathernext](https://github.com/google-deepmind/weathernext); [Google blog: WeatherNext 2 cyclones](https://blog.google/innovation-and-ai/models-and-research/google-deepmind/weathernext-2-cyclones/); [Open Source For You: DeepMind open sources WeatherNext](https://www.opensourceforu.com/2026/08/google-deepmind-weathernext-ai/); [Google blog: How AI is helping forecast cyclones (WeatherNext Q&A, Sept 1)](https://blog.google/innovation-and-ai/models-and-research/google-deepmind/weathernext-extreme-weather-cyclone-predictions/); [Local 10: The AI hurricane model that continues to blow away the competition (Sept 10, 2026)](https://www.local10.com/weather/hurricane/2026/09/10/the-ai-hurricane-model-that-continues-to-blow-away-the-competition/); [CNN (via KSL): Why the National Hurricane Center bet big on AI to forecast Isaias (Oct 8)](https://www.ksl.com/article/51634059/why-the-national-hurricane-center-bet-big-on-ai-to-forecast-isaias); [CNN original: Why the National Hurricane Center bet big on AI to forecast Isaias](https://www.cnn.com/2026/10/08/weather/hurricane-isaias-ai-models-google); [FOX Weather live updates: Isaias landfall near Destin, Norcross on DeepMind forecast (Oct 9–10)](https://www.foxweather.com/live-news/live-updates-hurricane-isaias-threatens-the-gulf-coast-with-dangerous-landfall-expected-tonight); [Gizmodo: Tropical Storm Isaias is expected to slam the Gulf Coast as a hurricane this weekend (Oct 8)](https://gizmodo.com/tropical-storm-isaias-is-expected-to-slam-the-gulf-coast-as-a-hurricane-this-weekend-2000822829)

### 2026-08-10: Claude proves more than two-thirds of Riemann zeta zeros are simple and on the critical line (up from 41.6%)

*****; Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-08-10-claude-riemann-zeta-zeros-two-thirds/

Anthropic reported that an unreleased research version of Claude, running in Claude Code with ~60 subagents, raised the unconditional lower bound on the proportion of Riemann zeta zeros that are simple and on the critical line from ~41.6% to 67.2%. The previous 37 years had added only ~0.8 percentage points. Key results were formalised in Lean, reviewed by Brian Conrey and Dan Goldston, and independently re-proved by Youness Lamzouri.

- Paper: 'More than two thirds of the zeta zeros are simple and on the critical line' (arXiv 2608.13637)
- Prior record ~41.6% (Levinson–Conrey lineage); the last 37 years had gained ~0.8 points
- Run by Jarred Sumner with mostly encouragement-style prompting; checked by Levent Alpöge and Ralph Furman
- Lean formalisation of key results with Eric Easley; independent new proof by Lamzouri (arXiv 2609.02882)
- Further follow-ups: Biao Wang extended the results to short intervals using Lamzouri's method, with ChatGPT-6 Astra implementing the proof ideas (arXiv 2609.07918, 7 Sep); Kristian Muri Knausgård raised the distinct-zeros proportion to more than 83.69%, calling the paper 'primarily an experiment in AI-assisted mathematical research' done with OpenAI Codex and Anthropic Claude, with a Lean formalisation (arXiv 2609.33043, 27 Sep)
- 6 Oct 2026: Knausgård again (arXiv 2610.08965): more than 83.9% of zeros distinct (liminf ≥ 1645064/1960733) and more than 67.35% simple and on the critical line (≥ 0.67353; 'earlier work proves 0.67250'), improving Claude's two-thirds bound. 'The arguments, the computations, the Lean formalisation, the figures and much of the text were developed with substantial assistance from AI systems (OpenAI Codex and Anthropic Claude) under the author's direction.' Proved in Lean 4 with 'three axioms beyond those of Mathlib, one per inequality. Each records that a search program returned true; Lean's kernel does not check the run itself'
- 17 Oct 2026: Juan Arias de Reyna (Seville) posted 'Human Exposition of Claude's 2/3 Theorem on the zeros of zeta' (arXiv 2610.24612), a self-contained re-proof of N0(T) ≥ 0.6725 N(T) for a future book: 'This proof is one of the first important results obtained by a machine: Claude the AI by Anthropic.' He calls the original arXiv paper 'like the notes of a poor student. Filled with bad notations, wrong assertions and unneeded statements', but the argument 'actually very simple and much easier to follow than the previous ones'; 0.6725 matches Montgomery–Taylor's constant 'and not by coincidence'
- arXiv comment on Alpöge & Furman, arXiv 2608.13637: 'Proof discovered autonomously by Claude (Anthropic); verified and communicated by the listed authors. See §1 for provenance.'
- Expert survey on Terence Tao's blog (22 Oct 2026, guest post by Caroline Turnage-Butterbaugh, a co-author of the BGST-B pair-correlation papers Claude built on): 'On August 10, 2026, Anthropic announced that an unreleased research model of Claude had connected the BGST-B approach above with a 2000 paper by Bombieri to unconditionally prove' the 67.25% bound, 'soon after verified by Alpöge and Furman'; she calls Lamzouri's re-proof 'conceptually simpler' and stresses that even 100% would not prove RH. Tao notes the post 'was initially written in a different file format and converted using AI'

#### What happened
A large Claude agent swarm refined the mollifier method behind Levinson- and Conrey-style bounds far beyond the prior state of the art. The result was then checked formally and by leading experts.

#### Why it matters
It does not prove the Riemann hypothesis, but it is a dramatic quantitative advance on the most famous problem in mathematics, and it was independently confirmed.

Sources: [Farzad Aryan: On the Imaginary Part of ζ′/ζ, with 'An Expository Note on Claude's Argument' (arXiv 2610.12853, 9 Oct 2026)](https://arxiv.org/abs/2610.12853); [Arias de Reyna: Human Exposition of Claude's 2/3 Theorem on the zeros of zeta (arXiv 2610.24612, 17 Oct 2026)](https://arxiv.org/abs/2610.24612); [anthropics/formal-math: zeta23 Lean formalization](https://github.com/anthropics/formal-math); [Anthropic: Claude and the zeros of the Riemann zeta function](https://www.anthropic.com/research/riemann-zeta); [More than two thirds of the zeta zeros are simple and on the critical line (arXiv 2608.13637)](https://arxiv.org/abs/2608.13637); [Lamzouri: independent proof (arXiv 2609.02882)](https://arxiv.org/abs/2609.02882); [arXiv 2609.07918: Simple critical zeros and distinct zeros of the Riemann zeta-function in short intervals (Wang)](https://arxiv.org/abs/2609.07918); [arXiv 2609.33043: More than 83.69% of the zeros of the Riemann zeta function are distinct (Knausgård)](https://arxiv.org/abs/2609.33043); [arXiv 2610.08965: More than 83.9% of the zeros are distinct and more than 67.35% are simple and on the critical line (Knausgård)](https://arxiv.org/abs/2610.08965); [Tao's blog: Simple and critical zeros of the Riemann zeta-function (guest post by Turnage-Butterbaugh, 22 Oct)](https://terrytao.wordpress.com/2026/10/22/simple-and-critical-zeros-of-the-riemann-zeta-function/)

### 2026-08-12: Claude-assisted constructions complete Hadamard matrices for every order below 2000, including 668

***; Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-08-12-claude-hadamard-matrices-below-2000/

Claude-assisted searches constructed Hadamard matrices for the 12 remaining unknown orders below 2000 (668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964). Order 668 had been the smallest open case of the Hadamard conjecture for about 21 years.

- Orders constructed: 668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964
- Order 668 was the smallest unknown order since 428 was constructed in 2005
- People: Levent Alpöge, P. Voinov, S. Reynolds-Haertle; order 668 was an Epoch AI 'open problem' entry
- Independent verification (Egan, Kotsireas, London, Ó Catháin; arXiv 2610.27175, 19 Oct 2026): they 'verify and analyse the claimed matrices'; each is a Goethals–Seidel array over an abelian group (nine over cyclic groups, three over non-cyclic groups); 'A Hadamard matrix is therefore known at every order below 2000 divisible by 4'
- The same paper builds Hadamard matrices at sixteen further orders below 10,000 that were open in the published literature (e.g. 2076 via a symmetric BH(86, 4), 3732, 4492, 5020, 5932, 5788, 8476), and lists 95 orders ≤ 10,000 still open, the smallest 2396 = 4·599
- Its AI statement: 'The authors used Anthropic's Claude (Claude Opus 5.5, via Claude Code, September–October 2026) extensively in this work. Claude wrote the Python and C++ programs used to analyse the twelve matrices … and to construct the symmetric BH(86, 4), the T-matrices of orders 151 and 163, and the Hadamard matrices of orders 5788 and 8476'; every matrix is public at math74.com/Hadamard and 'verified by an exact computation of HH^T'

#### What happened
Guided searches built the missing matrices, verifiable by simple matrix multiplication.

#### Why it matters
It closed a famous "smallest unknown case" that had stood for two decades, with an easily verified result.

Sources: [Epoch AI open problems: Hadamard matrix of order 668](https://epoch.ai/frontiermath/open-problems/hadamard); [arXiv 2610.27175: Recent progress in the construction of Hadamard matrices (Egan, Kotsireas, London, Ó Catháin)](https://arxiv.org/abs/2610.27175); [John D. Cook: Constructing Hadamard matrices](https://www.johndcook.com/blog/2026/08/13/constructing-hadamard-matrices/)

### 2026-08-13: Banach's isometric conjecture (1932) completed in the real case with key steps from ChatGPT 5.5/5.6 Pro; complex and quaternionic cases follow five days later

****; Xinbao Lu, Kaiwen Yang, Antonio Acuaviva, Tomasz Kania, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-08-13-banach-isometric-conjecture-solved/

Xinbao Lu and Kaiwen Yang posted a proof of Banach's 1932 isometric conjecture for every odd n (arXiv 2608.13536, 13 Aug 2026). With Gromov's even-n theorem, this completes the real case: a real Banach space whose n-dimensional subspaces are all isometric must be a Hilbert space. They had reduced the problem to one theorem themselves. The approach to that theorem 'emerged through extensive interactions with ChatGPT 5.5 Pro and ChatGPT 5.6 Pro', and GPT-5.6 Sol drafted the key section. On 18 Aug Acuaviva and Kania extended the method to complex and quaternionic spaces with GPT-5.6 Sol's help.

- Banach (1932): if all n-dimensional subspaces of a real Banach space X (fixed 1 < n < dim X) are linearly isometric, is X a Hilbert space? Gromov proved it for even n; several odd cases were settled later; Lu–Yang cover every odd n
- Method: principal bundle theory combined with a Brouwer degree argument (arXiv 2608.13536, 22 pages)
- AI declaration (Lu–Yang): 'The authors had reduced the main problem to proving Theorem 3.10 before using generative AI tools. An approach to that theorem subsequently emerged through extensive interactions with ChatGPT 5.5 Pro and ChatGPT 5.6 Pro. The initial draft of Section 3 and the corresponding parts in Section 2 were generated by GPT 5.6 Sol … and subsequently checked and rewritten by the authors'
- Complex and quaternionic case: Acuaviva & Kania (arXiv 2608.18257, 18 Aug 2026) adapt 'the bundle-degree mechanism introduced by Lu and Yang'; 'ChatGPT 5.6 Sol assisted with certain technical details needed to carry out these extensions'
- Status: preprints, not peer-reviewed or formalised as of 30 Sep 2026

#### What happened
Lu and Yang closed the odd-dimensional cases of Banach's question about isometric subspaces, using ChatGPT Pro to find the approach to the decisive theorem. The complex and quaternionic versions followed from another team within a week, also with AI help.

#### Why it matters
Banach's conjecture is one of the oldest questions in the geometry of normed spaces, and Gromov's even-dimensional solution had left the remaining odd cases open. The human–AI division of labour is unusually well documented: human reduction, AI-found approach, AI-drafted proof, human verification.

Sources: [arXiv 2608.13536: A solution to Banach's isometric conjecture (Lu, Yang)](https://arxiv.org/abs/2608.13536); [arXiv 2608.18257: Banach's Isometric Conjecture over the Complex Field (Acuaviva, Kania)](https://arxiv.org/abs/2608.18257)

### 2026-08-17: AlphaEvolve helps lower the matrix multiplication exponent ω to below 2.371177

***; Google DeepMind, MIT; Science & math; confidence medium; https://postcutoff.com/e/2026-08-17-alphaevolve-matrix-multiplication-exponent/

A paper by Alman, Vassilevska Williams and co-authors including DeepMind researchers (arXiv 2608.16884) improved the bound on the matrix multiplication exponent from ω < 2.371339 to ω < 2.371177. AlphaEvolve refined the optimiser used in the laser-method analysis.

- ω < 2.371177 (previous: 2.371339)
- Humans reformulated the optimisation problem; AlphaEvolve improved the numerical optimisation

#### What happened
Leading researchers on fast matrix multiplication used AlphaEvolve inside their laser-method pipeline to squeeze out a new record bound.

#### Why it matters
Progress on ω comes in tiny, hard-won steps. AI now contributes to the asymptotic theory as well as to small concrete algorithms.

Sources: [arXiv 2608.16884](https://arxiv.org/abs/2608.16884); [AI Weekly: AlphaEvolve helps push matrix multiplication to 2.371177](https://aiweekly.co/alerts/alphaevolve-helps-push-matrix-multiplication-to-2371177); [Pushmeet Kohli on X announcing ω < 2.371177](https://x.com/pushmeet/status/2089717134129565763)

### 2026-08-18: Claude autonomously designs protein binders that work in the lab against 14 of 15 targets

****; Anthropic, Adaptyv Bio, Twist Bioscience; Science & math; confidence high; https://postcutoff.com/e/2026-08-18-claude-protein-binders-wet-lab/

On Aug 18, 2026 Anthropic reported that Claude (Mythos Preview and Opus 4.8), running autonomously in Claude Science, designed de novo protein binders against 15 targets and produced confirmed binders for 14 of them in outside wet-lab tests by Adaptyv Bio and Twist Bioscience. Hit rates were 22.6–35.1%, against the 10–15% typical of protein design campaigns. The same post showed Opus 5 processing raw NMR and LC-MS files in about 25 minutes, matching a contract lab's results.

- 354 confirmed binders from 1,320 designs, against 14 of 15 targets (30 designs requested per target)
- Hit rates: Mythos Preview 26.7% and Opus 4.8 22.6% in multi-target mode (48 h, up to 12,500 H100 hours); Mythos Preview 35.1% in single-target mode (24 h, up to 2,500 H100 hours per target). Anthropic cites 10–15% as typical
- Wet-lab production and testing done independently by Adaptyv Bio and Twist Bioscience
- High-affinity binders against at least six targets; binders matching or exceeding the best reported affinity against at least four
- RBX1: Mythos Preview reached a 40% hit rate vs 3.7% among Adaptyv competition participants; its top design outperformed the winning entry (245 designs entered)
- Opus 4.8, not Mythos Preview, succeeded on the hard target TNFα, including binders cross-reactive to human, cynomolgus monkey and mouse TNFα
- Failures: no confirmed binder against maltose-binding protein (0 of 90 designs); only three modest binders against the de novo β-barrel BBF-14
- Humans only granted access approvals and monitored infrastructure after the initial prompt; prompts and all in vitro and in silico data released
- Chemistry test: Opus 5 in Claude Science returned processed NMR and LC-MS results in 23 and 19 minutes from raw vendor files; purity 96.4% vs the lab's 96.33%. It decoded an undocumented LC-MS format, checked against all 2,664 scans
- Protein design and other dual-use biology stay blocked in generally available Claude Fable 5; Anthropic said an access program for scientists was a top priority

#### What happened
Anthropic ran a multi-arm binder design campaign in Claude Science. Claude chose binding sites on each target, orchestrated open-source
structure-design, sequence-design and co-folding models, ran several rounds of in silico optimization and screened candidates. After the
first prompt nobody gave it scientific guidance. Adaptyv Bio and Twist Bioscience then made and tested the designs. Anthropic described the
354 binders as a sizeable addition to public de novo binder data. By comparison, it counted about 770 binders from 5,700 designs across 40
targets in the two largest existing public collections.

The post also tested a generally available model, Opus 5, on routine analytical chemistry. Working only from a contract lab's raw instrument
files, it reproduced the lab's NMR and purity results. It also proposed the same heavy-water follow-up test that the lab had run on its own.

#### Why it matters
It was among the first wet-lab-validated demonstrations of a general-purpose LLM agent running a whole protein design campaign at or above
expert level. It was the basis for Anthropic's later work to make the pipeline cheaper (Sept 17) and for its Life Sciences Verification Program.
The results are the company's own, and the hit-rate comparisons use Anthropic's baseline figures. Anthropic says it will characterize the
binders further to confirm hit rates and affinities.

Sources: [Anthropic: How Claude is accelerating protein design and analytical chemistry](https://www.anthropic.com/research/Claude-accelerates-protein-design); [The Next Web: Anthropic says Claude designed working protein binders, and beat human experts on some](https://thenextweb.com/news/anthropic-claude-protein-design-chemistry); [Dataconomy: Claude designed protein binders for 14 of 15 targets](https://dataconomy.com/2026/08/20/claude-ai-protein-binders-14-of-15-targets/); [pharmaphorum: Claude Science outperforms experts in protein binder task](https://pharmaphorum.com/news/claude-science-outperforms-experts-protein-binder-task)

### 2026-08-18: Palomar launches: a registry of Lean-verified mathematics to curb misrepresented AI proof claims

***; Lean FRO, ICARM; Science & math; confidence high; https://postcutoff.com/e/2026-08-18-palomar-lean-registry/

On 18 Aug 2026 the Lean FRO and ICARM launched Palomar (palomar-registry.org), "the analogue of a preprint server for Lean proofs". It indexes GitHub repositories whose formal results are checked mechanically with Lean's Comparator tool and checked with an LLM for semantic alignment with the informal statement. It was built in response to the flood of AI-generated proofs, and explicitly does not claim peer-review status.

- Each entry: a human-readable challenge file, a solution module with the formal proof, and a formalization.yaml with informal description and metadata
- Automated checks: mechanical verification via leanprover/comparator plus LLM-based semantic-alignment check
- Scientific advisory board incl. Jeremy Avigad, Matthew Ballard, Jaume de Dios, Nestor Guillen, Bryna Kra, Kim Morrison, Terence Tao, Ravi Vakil, Akshay Venkatesh
- First entry PALOMAR-2026-08-13-000001 (teorth/sendov, the Sendov conjecture formalisation)
- Aim: minimal safeguard against misrepresentation of AI claims, not a judgement of novelty or significance

#### What happened
As AI systems produced Lean proofs of old and new results at a growing rate, the Lean community set up a registry that makes formal claims inspectable and checks mechanically that a formal statement matches what is claimed informally.

#### Why it matters
Formal verification became the main way to trust AI mathematics in 2026. Palomar supplies the missing public infrastructure: a place where "proved in Lean" can be checked rather than asserted.

Sources: [Palomar registry](https://palomar-registry.org/); [Terence Tao: Palomar, a registry of Lean-verified mathematics](https://terrytao.wordpress.com/2026/08/18/palomar-a-registry-of-lean-verified-mathematics/); [Palomar statement](https://palomar-registry.org/statement); [GitHub: leanprover/comparator](https://github.com/leanprover/comparator); [GitHub: mathlib-initiative/formalization.yaml](https://github.com/mathlib-initiative/formalization.yaml)

### 2026-08-19: Peking University preprint claims an AI-found disproof of the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics

****; Peking University, Jihao Liu, Anthropic, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-08-19-yau-tian-donaldson-cscK-disproof-claim/

On 19 Aug 2026 Jihao Liu (Peking University) posted a 79-page preprint describing a polarized smooth projective fivefold that is K-polystable but has no constant scalar curvature Kähler (cscK) metric. That would disprove the cscK form of the Yau–Tian–Donaldson conjecture. The paper says 'the main result of this paper was obtained using generative AI', namely Claude Code (Fable 5), Codex (GPT-5.6 Sol) and the Danus research agent. A later solo Danus run re-derived counterexample and proof in 5 h 29 min.

- arXiv 2608.19301 (19 Aug 2026), 79 pages, single author, with an appendix on AI use written jointly with Bin Dong and Guoxiong Gao (Peking University)
- Claim: a K-polystable polarized smooth projective fivefold with no cscK metric. The Kähler–Einstein (Fano) case of Yau–Tian–Donaldson was proved by Chen–Donaldson–Sun in 2015; the cscK form was open
- Process (appendix): exploring open problems with Claude Code, the author saw 'initial signs of a possible breakthrough', then had Claude Code (Fable 5), Codex (GPT-5.6 Sol) and Danus work on it together; 'the three systems together produced the counterexample and its proof'
- Human input 'was crucial at one point': the author noticed the example must refute either the Codogni–Stoppa conjecture or the cscK YTD conjecture and told the agents to settle which
- Replication: an improved Danus working alone from the bare problem 'settled the problem 5 hours and 29 minutes into its run … and produced a counterexample and a complete proof'
- Comparison (per the appendix): QED, ProofCouncil, MechMath, Codex (GPT-5.6 Sol), Claude Code (Fable 5) and GPT-5.6 Sol Pro all failed in 12 hours, even when given the counterexample and asked only to prove it
- Danus is described as a specialised research agent built on the Rethlas system (Liang Xiao and Bin Dong's group); the same author used it for the equality case of Ehrhart's volume conjecture (arXiv 2608.01040)
- Status: preprint; no formal verification; no independent expert confirmation found as of 30 Sep 2026

#### What happened
A Peking University algebraic geometer released a counterexample to the cscK form of the Yau–Tian–Donaldson conjecture. He credits a combination of Claude Code, Codex and his group's Danus agent for the example and its proof. The appendix also records a control experiment: a stronger Danus reproduced the result alone, while six other agent systems could not solve it or even verify the given counterexample.

#### Why it matters
The Yau–Tian–Donaldson programme is central to modern complex geometry, and the paper is explicit that its main result is AI-generated. The appendix argues that such a counterexample is itself a proof, with no finite certificate, because it asserts something about every possible degeneration. If the result is confirmed, it would be one of the deepest AI-derived results of 2026.

Sources: [arXiv 2608.19301: Disproof of the Yau–Tian–Donaldson conjecture (Liu)](https://arxiv.org/abs/2608.19301); [arXiv 2608.01040: The equality case of Ehrhart's volume conjecture (Liu; same AI setup)](https://arxiv.org/abs/2608.01040)

### 2026-08-23: Claude-assisted construction claims a complex structure on the 6-sphere, answering Hopf's 1947 problem (pending verification)

*****; Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-08-23-hopf-problem-s6-complex-structure/

On 23 Aug 2026 Anthropic's Levent Alpöge posted a 100+ page document, produced with an internal Claude model, claiming that the 6-sphere S⁶ admits an integrable complex structure. This would answer Hopf's 1947 question. A Lean formalisation was reported on 27 Aug. Experts describe an emerging consensus that the construction is plausible, but independent verification is not complete.

- Construction from the (3,4,∞) modular family of 2-tori, completed at its three special points; yields uncountably many non-biholomorphic Oka complex structures
- Boris Alexeev (OpenAI) reported a Lean formalisation on 27 Aug 2026
- Robert Bryant: 'emerging consensus that the construction is plausible'
- Ilka Agricola: 'You don't know how many prompts were needed to arrive at the result, how much human fine-tuning was required.'
- Jeff Viaclovsky constructed a two-parameter family of complex structures on S^6 that are 'not biholomorphic to the original Alpöge–Claude examples', following Philip Engel's construction and using ChatGPT 6 Astra for drafting and most computations in Sections 4–5 (arXiv 2609.33785, 27 Sep 2026)

#### What happened
Weeks after the Jacobian counterexample, Alpöge released a long construction of complex structures on S⁶, followed by a reported formalisation.

#### Why it matters
The existence of a complex structure on S⁶ is one of the best-known open problems in geometry. Confirmation would make this among the biggest AI-assisted pure-maths results. Status: pending.

Sources: [Scientific American: AI solves 79-year-old math mystery of six-dimensional spheres](https://www.scientificamerican.com/article/ai-solves-79-year-old-math-mystery-of-six-dimensional-spheres/); [OfficeChai: Anthropic researcher says Claude helped build a complex structure on S⁶](https://officechai.com/ai/anthropic-researcher-says-claude-helped-build-a-complex-structure-on-s%E2%81%B6-taking-aim-at-the-unsolved-hopf-problem/); [Follow-up paper (arXiv 2609.26706)](https://arxiv.org/abs/2609.26706); [arXiv 2609.33785: A two-parameter family of complex structures on S^6 (Viaclovsky)](https://arxiv.org/abs/2609.33785)

### 2026-08-23: Claude-assisted search breaks the elliptic curve rank record: rank 30, then 31

***; Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-08-23-elliptic-curve-rank-record-31/

An elliptic curve over Q with rank at least 30 was reported on 20 Aug 2026 and one with rank ≥31 on 23 Aug. These broke the Elkies–Klagsbrun rank-29 record from 2024. The rank-31 curve has 31 explicit independent rational points, so the bound is unconditional. The ICARM record page credits Claude with L. Alpöge and A. Howell.

- Previous record: rank ≥ 29 (Elkies–Klagsbrun, 2024); earlier ≥ 28 (Elkies, 2006)
- Rank 30 on 20 Aug; rank 31 on 23 Aug 2026; first submitted under the name 'ranksunbounded'
- 31 independent rational points given explicitly

#### What happened
AI-directed searches through families of elliptic curves found new record-rank examples twice in one week.

#### Why it matters
Rank records move very rarely (2006, 2024). Two in a week signal AI's strength at large, structured searches in number theory.

Sources: [ICARM: new record-breaking elliptic curve reported](https://icarm.io/news/new-record-breaking-elliptic-curve-reported/); [Andrej Dujella: history of elliptic curve rank records](https://web.math.pmf.unizg.hr/~duje/tors/rankhist.html); [Epoch AI open problems: elliptic curve rank](https://epoch.ai/frontiermath/open-problems/elliptic-curve-rank)

### 2026-08-26: GPT-5.6 improves the Erdős–Rankin / Ford–Green–Konyagin–Maynard–Tao bound for large prime gaps

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-08-26-erdos-rankin-large-prime-gaps-improved/

On 26 Aug 2026 the user "DottedCalculator" posted to erdosproblems.com (problem #4) a proof, generated with GPT-5.6, that there are infinitely many prime gaps larger than C·log n·log log n / log log log log n. This removes a log log log n factor from the 2018 Ford–Green–Konyagin–Maynard–Tao bound. Thomas Bloom wrote an exposition calling the ideas elementary. A fuller proof by GPT-6 Astra with a Lean formalization followed on 4 Sep 2026.

- New bound: p_{n+1} − p_n > C·log n·log log n / log log log log n for infinitely many n
- Previous record: FGKMT 2018 (Ford, Green, Konyagin, Maynard, Tao), which had an extra log log log n factor in the denominator
- Method: a new weighting function to filter residue subsets, combined with the FGKMT18 machinery; Bloom notes neither ingredient alone improves the record
- Model naming differs: erdosproblems.com says 'GPT 5.6 Pro (prompted by DottedCalculator)', while Wikipedia's AI-discoveries list says GPT-5.6 Sol
- Follow-up: GPT-6 Astra full proof submitted 4 Sep 2026 with a Lean formalization (openai/LongGapsBetweenPrimes)
- Traictory (1 Sep 2026): no independent human verification yet at that point
- Follow-ups by Tristan Freiberg: an improvement to long runs of smooth integers and the divisor function of n!, whose 'core new argument … was generated during a private interaction with Claude Fable 5.1' on 8 Sep 2026 (arXiv 2609.15597); and an elementary exposition of the 'tilted sieve introduced by GPT-5.6 Sol' (arXiv 2609.28253, which notes the Lean formalisation by B. Alexeev)
- Downstream (Sep–Oct 2026): Artyom Radomskii (HSE) used the preprint's weight / truncated divisor-sum construction to improve the Kalmynin–Konyagin bound for gaps between Romanoff numbers (arXiv 2609.38408) and then to answer the $50 O(log N) part of Erdős Problem #32 on additive complements to the primes (arXiv 2610.14127); no AI-use statement in either paper

#### What happened
A pseudonymous user got a GPT-5.6 model to combine new sieve weights with the Ford–Green–Konyagin–Maynard–Tao construction. The result improved the long-standing record for how large prime gaps can be. Thomas Bloom wrote it up on erdosproblems.com (last edited 31 Aug 2026). OpenAI's GPT-6 Astra then produced a complete proof with a Lean formalization.

#### Why it matters
Large prime gaps were famously advanced by Maynard and by Ford–Green–Konyagin–Tao in 2014–2018, and experts treated the FGKMT bound as hard to beat. This came four days before GPT-6 Astra's bounded-gaps record (246 → 186), so both ends of the prime-gap problem moved within a week.

Sources: [Erdős problem #4](https://www.erdosproblems.com/4); [erdosproblems.com forum: problem #4 proof claims](https://www.erdosproblems.com/forum/thread/4/proof-claims); [Traictory: GPT-5.6 claims a prime-gap record. Who checks the proof?](https://traictory.com/news/2026-09-01-gpt-5-6-prime-gap-math-proofs); [Wikipedia: List of mathematical discoveries by artificial intelligence](https://en.wikipedia.org/wiki/List_of_mathematical_discoveries_by_artificial_intelligence); [arXiv 2609.15597: Long runs of integers with small prime factors and the divisor function of n! (Freiberg; Claude Fable 5.1)](https://arxiv.org/abs/2609.15597); [arXiv 2609.28253: A tyro's approach to the tilted sieve (Freiberg)](https://arxiv.org/abs/2609.28253)

### 2026-08-27: Google's Antigravity 'Teamwork' multi-agent framework with Gemini 3.7 Flash solves seven open CS/math problems, incl. part of Knuth's cycles problem

***; Google, Google DeepMind; Science & math; confidence medium; https://postcutoff.com/e/2026-08-27-antigravity-teamwork-open-problems/

On Aug 27, 2026 Google announced updates to Teamwork, a multi-agent orchestration framework in its Antigravity coding environment. Using Gemini 3.7 Flash (with 3.1 Pro), its "Long Proof" pattern produced results on seven open problems in theoretical computer science and math, including the first proofs for two simpler constructions in the even case of Knuth's "Claude's Cycles" problem (40+ and 70+ page proofs, one formally verified in Lean). It scored 71% on Google's TCSBench, built a cycle-accurate RISC-V CPU simulator that boots xv6, and had optimizations merged into Eigen and ParlayHash.

- Teamwork: agents propose, critique and refine each other's work for hours or days; patterns chosen automatically (Iterative Coding, Distributed Coding, Long Proof, Self-Verification inspired by DeepMind's Aletheia, Document Review); available as /teamwork-preview on all paid Antigravity plans
- Seven results (Google): improved coreset bounds for lp subspace approximation (FOCS 2025 open problem, arXiv:2608.26047); conditional lower bound for sparse least-squares (JMLR 2021 problem, arXiv:2608.02588); near-closing the complexity gap for Chamfer similarity (arXiv:2607.20393); provable Hadamard quantization with ~5.93x smaller leading constant (arXiv:2608.02564); independent reproduction of the Erdős unit-distance breakthrough without internet access; near-optimal lower bound for prefix-matrix factorizations (arXiv:2608.08238); Knuth's cycles, even case (GitHub dpwoodru/knuthCycles)
- Verification: Google says all results were reviewed and confirmed by human experts except the Knuth result, where the 40-page proof was formally verified in Lean; five papers are on arXiv
- TCSBench (Google-internal benchmark of open/challenging TCS problems): 71% with Gemini 3.7 Flash + 3.1 Pro, up from 67.7% with 3.6 Flash + 3.1 Pro
- Systems: cycle-accurate out-of-order RISC-V simulator built from scratch; boots xv6 to a shell; 0.71% cycle alignment error vs. BOOM hardware ground truth; runs 100+ standard benchmarks
- Open source: SIMD fast path for single-row/column GEMV merged into Eigen; 'Swiss Parlay' ideas for ParlayHash (2x insert throughput at 64 threads, 25% less memory per element) landed upstream

#### What happened
Google updated **Teamwork**, the multi-agent framework in its Antigravity IDE first shown at Google I/O, and published results meant to show
that orchestrating many cheap Flash-model agents can match much larger models on hard, long tasks. Its **Long Proof** pattern generates
competing proof strategies, pairs each with a "falsifier" agent that tries to break it, splits the chosen strategy into subproblems, and
runs tournaments of improved drafts. It keeps a registry of pitfalls across rounds.

The headline math result touches the problem from Donald Knuth's February 2026 "Claude's Cycles" note: decomposing the 3D torus digraph
into three Hamiltonian cycles. Claude had found the construction for odd m, and the even case was left largely open. Teamwork produced the
first proofs for two simpler even-case constructions, and Google says the 40-page proof was checked in Lean. Other results appear as arXiv
preprints on problems posed at FOCS and in JMLR. In engineering, Teamwork wrote a RISC-V out-of-order CPU simulator that boots an operating
system and contributed performance patches that maintainers accepted into Eigen and ParlayHash.

#### Why it matters
It is another data point in the summer-2026 wave of AI-assisted results on open problems. This one comes from a cheap "Flash" model plus
orchestration rather than a frontier flagship, and it lands on a problem first opened up by a rival lab's model. Caveats: the problems are
mostly specialized TCS questions, the preprints are not peer-reviewed, and TCSBench is Google's own benchmark.

Sources: [Google Antigravity blog: Teamwork: When AI becomes a research partner](https://antigravity.google/blog/teamwork-when-ai-becomes-a-research-partner); [Google blog: Antigravity Teamwork with Gemini 3.7 Flash](https://blog.google/innovation-and-ai/technology/developers-tools/antigravity-teamwork-multi-agent/); [Antigravity docs: Teamwork](https://antigravity.google/docs/teamwork); [Knuth cycles even-case proofs (GitHub dpwoodru/knuthCycles)](https://github.com/dpwoodru/knuthCycles); [arXiv:2608.26047: coresets for lp subspace approximation](https://arxiv.org/abs/2608.26047); [TCSBench paper (arXiv:2608.09538)](http://arxiv.org/pdf/2608.09538v1)

### 2026-08-28: Matrix Spencer conjecture proved; authors credit GPT-5.6 Sol Pro with 'the heavy-lifting' for the key lemma

****; Emrullah Akbas, Suvrit Sra, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-08-28-matrix-spencer-conjecture-proved/

Emrullah Akbas and Suvrit Sra posted a proof of the Matrix Spencer conjecture (arXiv 2608.28816, 28 Aug 2026). For symmetric n×n matrices A_1..A_n with operator norm at most 1, one can efficiently find signs x in {±1}^n with ||Σ x_i A_i|| = O(√n). They credit GPT-5.6 Sol (Pro) with finding the full proof of the key hereditary small-ball lemma, after a human suggestion on the route, and with 'almost all the calculations'.

- Result: for symmetric n×n matrices with ||A_i|| ≤ 1, an efficiently computable coloring x with ||Σ x_i A_i|| = O(√n), the matrix analogue of Spencer's 'six standard deviations' theorem
- Technique: matrix small-ball estimates with a log-barrier determinantal weight, matrix-weighted Poincaré inequalities and dimension-free constants; the new hereditary small-ball estimate for Gaussian series is of independent interest
- AI statement: 'This work made extensive use of GPT-5.6 Sol (Pro) over a period of months'; the route (log-barriers, Schur complements) was suggested to the model by Sra; 'GPT found a full proof after some failed attempts … we attribute the heavy-lifting for that lemma and for almost all the calculations presented in this writeup to GPT'; GPT also wrote the first rough draft, which the authors polished and verified
- Sra also wrote 'GPT, the Counterexample Machine' (arXiv 2608.29595), a catalogue of 15+ counterexamples he found with GPT Pro over 12 months
- Status: preprint (47 pages), not peer-reviewed or formalised

#### What happened
After months of working with GPT-5.6 Sol Pro, Akbas and Sra obtained the hereditary small-ball lemma that completed their approach to Matrix Spencer. They credit the model with most of the technical work.

#### Why it matters
Matrix Spencer was one of the headline open problems in discrepancy theory. The paper is candid that the model did the "heavy-lifting".

Sources: [arXiv 2608.28816: A Proof of the Matrix Spencer Conjecture (Akbas, Sra)](https://arxiv.org/abs/2608.28816); [arXiv 2608.29595: GPT, the Counterexample Machine (Sra)](https://arxiv.org/abs/2608.29595)

### 2026-08-30: GPT-6 Astra lowers the bounded prime gaps record from 246 to 186

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-08-30-bounded-prime-gaps-186/

An OpenAI preprint (30 Aug 2026) claims lim inf (p_{n+1} − p_n) ≤ 186, improving Polymath8b's bound of 246, which had stood since 2014. It uses 'triply densely divisible' conditions feeding a multidimensional Selberg sieve and was announced with a Lean formalisation. Julia Stadlmann independently reached 240 at about the same time.

- Previous record: 246 (Polymath8b, 2014), building on Zhang (2013) and Maynard (2013)
- New claimed bound: 186
- Lean formalisation announced (Weijie Su); independent human verification not complete
- Human counterpart: Julia Stadlmann (UIUC), arXiv 2608.31126 (submitted 31 Aug 2026), proves 240 alone, 'with the assistance of traditional numerical computation, but not modern AI tools' (Tao); key idea: Motohashi–Pintz–Zhang estimates for only 'partly smooth' moduli

#### What happened
OpenAI's model found a refinement of the Maynard–Tao sieve set-up that substantially improves the gap bound.

#### Why it matters
Bounded prime gaps were one of the celebrated stories of 2013–14. An AI improving the collaborative record is a striking, if still pending, result.

Sources: [OpenAI: short gaps between primes (PDF)](https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf); [Julia Stadlmann: Bounded gaps between primes (arXiv 2608.31126; human-only, bound 240)](https://arxiv.org/abs/2608.31126); [Terence Tao on Mathstodon: Stadlmann shaves 246 to 240 without modern AI tools](https://mathstodon.xyz/@tao/117197525544971208); [Weijie Su on X (Lean formalisation)](https://x.com/weijie444/status/2095600108956262911)

### 2026-09-02: NVIDIA's Nemotron-3-Ultra-CC outscores every human at IOI 2026 (535.4/600)

***; NVIDIA; Science & math; confidence medium; https://postcutoff.com/e/2026-09-02-nvidia-nemotron-ioi-2026/

NVIDIA reported that its fine-tuned Nemotron-3-Ultra-CC (550B total / 55B active MoE) scored 535.4 of 600 on the IOI 2026 problem set, graded by the IOI team. The top human scored 498.27, making it the first AI claimed to beat the best human contestant at the IOI. The model ran unofficially, offline, under contest limits.

- Score 535.4/600 vs top human 498.27; human gold cutoff 361.12
- Nemotron-3-Ultra-CC: 550B total, 55B active parameters; also a 30B Nano-CC variant
- Trained with SFT and RL on ~22,000 curated competitive-programming problems (arXiv 2609.02849)
- Unofficial participation in Uzbekistan with no internet access and the same time and submission limits
- Context: at IOI 2025, OpenAI's system scored 533.29 and placed 6th among humans

#### What happened
NVIDIA's post-trained open model family competed alongside IOI 2026 under supervision and beat every human's score.

#### Why it matters
Top-human performance in olympiad programming, previously only approached by closed frontier models, came from NVIDIA's Nemotron family rather than from a chatbot-focused frontier lab.

Sources: [NVIDIA AI on X: IOI 2026 result](https://x.com/NVIDIAAI/status/2096032566310789528); [Post-Training Language Models for Gold-Medal Performance in Coding Competitions (arXiv 2609.02849)](https://arxiv.org/abs/2609.02849); [AI Weekly: Nvidia's 550B Nemotron beats top human coder at IOI 2026](https://aiweekly.co/alerts/nvidias-550b-nemotron-beats-top-human-coder-at-ioi-2026); [IOI 2026 statistics](https://stats.ioinformatics.org/olympiads/2026)

### 2026-09-02: Nature: 'Designing physics experiments with artificial intelligence' (Krenn group) on AI-found setups that beat human designs

**; University of Tübingen, TU Wien, University of Vienna; Science & math; confidence medium; https://postcutoff.com/e/2026-09-02-krenn-nature-designing-physics-experiments-ai/

Published in Nature around Sept 2–3, 2026, Klimesch, Arlt, Ruiz-Gonzalez et al. (Mario Krenn's group, Tübingen, with TU Wien and Vienna) describe how search and optimization algorithms explore huge spaces of lab components to propose experimental setups that give more precise results or new measurement capabilities than human designs, across quantum optics, electron microscopy, fusion, particle detectors and gravitational-wave detectors.

- Citation (Tübingen AI Center): Klimesch, J., Arlt, S., Ruiz-Gonzalez, C. et al. 'Designing physics experiments with artificial intelligence', Nature 657, 47–58 (2026), doi 10.1038/s41586-026-10898-6
- University of Vienna news dated Sept 3, 2026; Tübingen AI Center news dated Sept 2, 2026
- Method: optimization over mathematical models of available components (not a chatbot/LLM); humans set goals and constraints
- Application areas: quantum experiments, electron microscopy with entanglement, fusion reactors, particle detectors, gravitational-wave detector sensitivity
- Krenn: 'Human work is simply shifting to a higher level'; some AI-found designs are provably better but lack an intuitive explanation

#### What happened
Krenn began this line of work as a student in Vienna, when an algorithm found a quantum-optics setup his group could not design by hand. The Nature
paper extends the approach across physics; TU Wien's Philipp Haslinger said the AI proposed microscope designs "a human would probably never have come up with".

#### Why it matters
It shows AI in science moving from analysing data to designing the instruments and experiments themselves. Confidence is medium on details because
the Nature text is paywalled; the press releases give no quantitative improvement figures.

Sources: [Nature: Designing physics experiments with artificial intelligence](https://www.nature.com/articles/s41586-026-10898-6); [University of Vienna: Artificial intelligence suggests new physics experiments](https://physik.univie.ac.at/en/news/news-detail/news/artificial-intelligence-suggests-new-physics-experiments/); [Tübingen AI Center: AI could help scientists design experiments humans would never think of](https://tuebingen.ai/news/ai-could-help-scientists-design-experiments-humans-would-never-think-of); [Phys.org: AI suggests new physics experiments that could outperform human-designed setups](https://phys.org/news/2026-09-ai-physics-outperform-human-setups.html)

### 2026-09-03: Claude-written Lean proof claims the dying percolation conjecture θ(p_c)=0 in every dimension

*****; Anthropic, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-09-03-dying-percolation-theta-pc-zero/

In early September 2026 a Lean 4 formalization written by Anthropic's Claude models (directed by Justin Leder, published in anthropics/formal-math) claimed to prove that critical Bernoulli bond percolation on Z^d has no infinite cluster for every d ≥ 2. It does this by proving a gluing inequality from Kozma–Nitzan (2024) that implies θ(p_c)=0. Gil Kalai called it "a remarkable breakthrough" if verified. Days later Ahmed Bou-Rabee, using GPT-5.6 Sol and Claude Fable 5.1, posted Lean proofs of stronger Kozma–Nitzan conjectures. No human referee has signed off yet.

- Problem: θ(p_c)=0 (no percolation at criticality); previously known only for d = 2 and high dimensions (d ≥ 11). Open for 3 ≤ d ≤ 10
- Route: Kozma & Nitzan (arXiv 2401.12397, 2024) showed their Conjecture 3 ('near-one gluing') implies θ(p_c)=0 on Z^d for all d ≥ 2
- anthropics/formal-math percolation README: 247 Lean files, ~86,900 lines; axioms only propext, Classical.choice, Quot.sound; 'no human wrote or edited the Lean code'
- README caveat: 'has not yet been refereed by human mathematicians or by anyone independent of the author'
- Gil Kalai blog, 3 Sep 2026: 'If verified, this is a remarkable breakthrough'; he flags missing details in the written proof and the need to check the formalization
- Hugo Duminil-Copin had used θ(p_c)=0 as his main example in an essay on AI and mathematics a few days earlier ('Care for a little more AI?', proofsandprompts.com, Aug 30, 2026: AI use 'does not empower us, it petrifies us')
- Ahmed Bou-Rabee's verification page (updated 5 Sep 2026): Kozma–Nitzan Conjectures 1, 2, 4, 6 and Questions 5, 7, 9 proved in stronger form by 'ChatGPT 5.6 Sol and Claude Fable 5.1, prompted by Ahmed Bou-Rabee'; Question 8 fails under one reading
- Scientific American (Manon Bischoff, Sept 30, 2026) covered it as AI solving a 'holy grail' of probability theory: Duminil-Copin had written on Aug 30 that it was 'only a matter of time before the most famous conjecture in our field... also falls to the bulldozers'; Benedikt Jahnel described 'ambivalent feelings'
- Quanta (Konstantin Kakaes, Sept 2026): Justin Leder, an Anthropic member of technical staff on its data science team with no prior percolation papers, posted the proof on GitHub on Aug 28. Gady Kozma: 'We are still waiting for Anthropic to publish a human readable version.' Bou-Rabee: 'clever and technically brilliant, but… does not contain substantial new ideas'
- Related human work: Lucas Flammant, arXiv 2609.03492 (Sept 3, 2026), proposes a finite-graph conjecture that also implies θ(p_c)=0 and proves it for planar graphs; it does not discuss the AI proof
- Human exposition (Jean-Baptiste Gouéré, Université de Tours; arXiv 2610.30063, 21 Oct 2026): 'we give an exposition of the recent proof of the Kozma-Nitzan conjecture provided by OpenAI' (Conjecture 1, from OpenAI's 24 Sep manuscript 'Critical bond and site percolation on the cubic lattice'); 'Theorem 1 was proved shortly beforehand in [8]' (Justin Leder's Lean 4/Mathlib development in anthropics/formal-math); 'The proofs in [8] and [9] are very similar and rely on the same ideas'; cites Blanc-Renaudie's human exposition of the Anthropic proof; no AI disclosure in the note

#### What happened
Kozma and Nitzan reduced the θ(p_c)=0 problem to an inequality about gluing connection events on finite graphs. In early September 2026 a Claude-written Lean development proved an additive form of that inequality. It went through a "conditioned slack hierarchy" of covariance inequalities, then applied Kozma–Nitzan's Theorem 6 to get θ(p_c)=0 in all dimensions d ≥ 2. Gil Kalai heard about it from Itai Benjamini and wrote it up on 3 Sep 2026. Separately, Ahmed Bou-Rabee published Lean proofs of several stronger Kozma–Nitzan conjectures, produced with GPT-5.6 Sol and Claude Fable 5.1.

The Wikipedia list credits the result to "Claude + Ahmed Bou-Rabee". The Anthropic repository itself credits Justin Leder as the director of the Claude run. Anthropic had not put out a press release as of late September 2026.

#### Why it matters
If the formal statement matches the intended theorem, a famous problem in mathematical physics is settled by machine-written formal mathematics. Commentators stress that Lean confirms the proof is correct but does not confirm the statement is the right one. Human experts still have to check that the formal definitions capture percolation on Z^d.

Sources: [arXiv 2610.30063: Gouéré, On the proof of the Kozma-Nitzan conjecture](https://arxiv.org/abs/2610.30063); [Hugo Duminil-Copin: Care for a little more AI? (Proofs and Prompts, Aug 30, 2026)](https://proofsandprompts.com/2026/08/30/care-for-a-little-more-ai/); [anthropics/formal-math: percolation README (commit 795efb8)](https://github.com/anthropics/formal-math/blob/795efb86f191735c5481675763537cfb4ff37e55/percolation/README.md); [Gil Kalai: Amazing: There is no Percolation at the Critical Probability in all Dimensions](https://gilkalai.wordpress.com/2026/09/03/amazing-there-is-no-percolation-at-the-critical-probability-in-all-dimensions-solved-by-ai-via-a-conjecture-of-gady-kozma-and-shahaf-nitzan/); [Ahmed Bou-Rabee: Kozma–Nitzan conjectures verification page](https://nitromannitol.github.io/kn1-verification-b80e9/); [Kozma & Nitzan: A reduction of the θ(p_c)=0 problem to a conjectured inequality (arXiv 2401.12397)](https://arxiv.org/abs/2401.12397); [Proofs and Prompts: Applied mathematics has met the machine before (on verification vs validation)](https://proofsandprompts.com/2026/09/28/applied-mathematics-has-met-the-machine-before/); [Wikipedia: Dying percolation conjecture](https://en.wikipedia.org/wiki/Dying_percolation_conjecture); [Scientific American: AI solves a 'holy grail' problem from probability theory](https://www.scientificamerican.com/article/ai-solves-a-holy-grail-problem-from-probability-theory/); [Quanta Magazine: Transformation (dispatches on AI mathematics, Sept 2026)](https://www.quantamagazine.org/updates/transformation/); [Flammant: A finite-graph conjecture related to θ(p_c)=0 (arXiv 2609.03492)](https://arxiv.org/abs/2609.03492); [Anthropic: How Anthropic enables self-service data analytics with Claude (Leder co-author)](https://claude.com/blog/how-anthropic-enables-self-service-data-analytics-with-claude)

### 2026-09-03: GPT-6 Astra proves the Erdős–Sós conjecture (1962) with a short counting argument; mathematicians race to simplify and extend it

*****; OpenAI, Epoch AI; Science & math; confidence high; https://postcutoff.com/e/2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra/

In Epoch AI's FrontierMath Erdős runs, a pre-release GPT-6 Astra autonomously proved the Erdős–Sós conjecture (Erdős problem #548): every graph with average degree greater than k−2 contains every tree on k vertices. The proof is Lean-verified. Its short, elementary argument counts vertex orderings. Within three weeks, leading combinatorialists published simplified versions, and other authors used Astra to extend the method to hypergraphs (Kalai's conjecture) and to digraphs.

- Conjecture posed by Erdős and Sós around 1962–63; Chung's collection of Erdős's graph problems called it 'one of the most tantalizing problems in extremal graph theory'; erdosproblems.com listed a $100 prize
- Found in 1 of 3 FrontierMath Erdős attempts ($363, 20 hours of working time); Lean proof in tadamcz/erdos548 (created 3 Sep 2026), written autonomously by the model
- Idea (per Bloom's summary): count pairs (π, j) where π orders the vertices and v1–vj is an edge. This equals 2m(n−1)!, while for a fixed tree T an induction bounds it by C(T) + (k−2)·n!, and C(T) = 0 if G has no copy of T
- Human expositions and simplifications: Thomas Bloom (erdosproblems.com), Riordan & Scott (arXiv 2609.15893, which also find the extremal graphs), David Wood (arXiv 2609.17877), Bryce Frederickson (arXiv 2609.21159, random cyclic orderings), Jay Cummings (arXiv 2609.32011, visual exposition), and Ben Golub (erdosproblems.com, written with AI assistance)
- Riordan & Scott: 'In a startling development, the Erdős–Sós conjecture was recently proved in full by GPT-6 Astra … with a short and ingenious argument'
- Extensions found by Astra when prompted by Mubayi & Verstraëte: Kalai's conjecture for tight trees in hypergraphs (arXiv 2609.08012, r = 2 is Erdős–Sós) and an Erdős–Sós analogue for Eulerian digraphs (arXiv 2609.10987); both papers say the proofs were found by GPT-6 Astra
- Parallel human work: Reed & Stein proved the dense case (k ≥ γn) 'without any use of AI', with a version ready in early August (arXiv 2609.05417); Santos, Stein & Williams adapted the Astra argument to antidirected trees

#### What happened
The proof came out of the same Epoch AI / Thomas Bloom benchmark runs that disproved Erdős problem #1. Once the Lean proof and Bloom's exposition were public, combinatorialists quickly posted cleaner human versions. They include Oliver Riordan and Alex Scott, who also determined the extremal graphs, and David Wood. Dhruv Mubayi and Jacques Verstraëte prompted Astra to extend the method to hypergraphs and digraphs and published the results with the proofs credited to the model.

#### Why it matters
Erdős–Sós is one of the best-known conjectures in extremal graph theory, and this is among the clearest cases of an AI finding a genuinely new, short idea that experts call "ingenious" and "surprising". The follow-up papers show the method being absorbed into the field within weeks.

Sources: [GitHub: tadamcz/erdos548 (Lean proof found by GPT-6 Astra)](https://github.com/tadamcz/erdos548); [arXiv 2609.25050: FrontierMath Erdős, Appendix B.4](https://arxiv.org/abs/2609.25050); [erdosproblems.com #548](https://www.erdosproblems.com/548); [arXiv 2609.15893: A short proof of the Erdős–Sós Conjecture (Riordan, Scott)](https://arxiv.org/abs/2609.15893); [arXiv 2609.17877: The Erdős–Sós Theorem (Wood)](https://arxiv.org/abs/2609.17877); [arXiv 2609.21159: Erdős–Sós via random cyclic orderings (Frederickson)](https://arxiv.org/abs/2609.21159); [arXiv 2609.32011: A visual exposition of the proof discovered by GPT-6 Astra (Cummings)](https://arxiv.org/abs/2609.32011); [arXiv 2609.08012: Kalai's Conjecture for Tight Trees (Mubayi, Verstraëte; proof by GPT-6 Astra)](https://arxiv.org/abs/2609.08012); [arXiv 2609.10987: Erdős–Sós for digraphs (Mubayi, Verstraëte; proof by GPT-6 Astra)](https://arxiv.org/abs/2609.10987); [arXiv 2609.05417: The Erdős–Sós conjecture in dense graphs (Reed, Stein; no AI)](https://arxiv.org/abs/2609.05417)

### 2026-09-03: Pre-release GPT-6 Astra disproves Erdős's 'first serious problem' (1931, $500) and proves the rational-exponents conjecture, all Lean-verified, in Epoch's FrontierMath Erdős runs

*****; OpenAI, Epoch AI; Science & math; confidence high; https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/

In Epoch AI's FrontierMath Erdős runs (announced 3 Sep 2026), a pre-release GPT-6 Astra autonomously resolved five of 68 hand-picked open Erdős problems with Lean-checked proofs. They include a disproof of Erdős problem #1 on distinct subset sums, which Erdős dated to 1931 and called 'perhaps my first serious problem', and a proof of the Erdős–Simonovits rational-exponents conjecture for bipartite Turán numbers (#571). The Erdős–Sós conjecture (#548) is in a separate entry.

- Benchmark: 68 open Erdős problems chosen by Thomas Bloom from 652 open problems on erdosproblems.com; a resolution counts only if it is proved in Lean; default budget $300 and 72 hours per problem
- Official score (one attempt each): pre-release GPT-6 Astra 3% (#74 disproved for $218 in 15 h; #126 proved for $247 in 16 h); GPT-5.6 Sol, GPT-5.5, Claude Fable 5.1 and Claude Fable 5 all 0%
- Extra, non-benchmark attempts with larger budgets resolved 5 problems in total (#1, #74, #126, #548, #571), using over $220,000 of compute versus about $20,000 for the benchmark run
- #1 (Erdős, dated 1931, $500 prize): if A ⊆ {1..N} has n elements with all subset sums distinct, must N ≫ 2^n? Astra showed that for every ε > 0 there are arbitrarily large n with N ≤ ε·2^n, so the conjecture is false. The previous best construction was N ≤ 0.22002·2^n (Bohman). The proof is ineffective (no explicit n for a given ε). Found in 2 of 4 attempts ($405 / 27 h and $1,384 / 84 h)
- #571 (Erdős–Simonovits): for every rational α in [1,2) there is a bipartite graph G with ex(n; G) ≍ n^α. Bloom calls it 'the most difficult of the five solutions'. Found in 1 of 3 attempts ($617, 41 h)
- #74 (Erdős–Hajnal–Szemerédi): disproved in the unexpected direction. If every n-vertex subgraph can be made bipartite by deleting at most f(n) edges with f growing slowly enough, the graph has chromatic number ≤ 3
- #126 (Erdős–Turán): |S(A)| ≫ n^(1/2) primes divide the pairwise sums of an n-element set, improving the classical log n bound; Astra gave three distinct proofs (exponents 1/8, 1/3 and 1/2)
- erdosproblems.com now marks #1 as 'DISPROVED (LEAN)'
- The Lean repositories (tadamcz/erdos1, erdos74, erdos126, erdos571), created 3 Sep 2026, say the proofs were found autonomously in the benchmark harness; the report calls the informal write-ups 'placeholders' until human experts prepare proper papers
- Also published (repo tadamcz/erdos1207, created 7 Sept 2026): Erdős problem #1207 (isosceles-free subsets, attributed to Riddell) proved: there is c > 0 with P_2(n) < n^(1−c), i.e. for large n some n-point planar sets have no isosceles-free subset larger than n^(1−c) (a power saving that Erdős asked for); 'found autonomously by a pre-release version of GPT-6 Astra … running in the harness of the FrontierMath Erdős benchmark'; Lean 4 proof with an audit script; repository 'machine-written by AI assistants at the direction of Tom Adamczewski'; not counted in the five problems above and not independently reviewed

#### What happened
Epoch AI and Thomas Bloom (erdosproblems.com) built FrontierMath Erdős, a benchmark of 68 open Erdős problems formalised in Lean, and ran five frontier models on it. Under the fixed $300 budget only the pre-release GPT-6 Astra solved anything (2 of 68). In further runs with larger budgets, which the authors report separately and do not count as benchmark results, Astra also disproved problem #1, proved the Erdős–Sós conjecture (#548) and proved the rational-exponents conjecture (#571). Every resolution comes with a Lean proof checked against a small challenge statement.

#### Why it matters
Problem #1 is probably the longest-standing open Erdős problem. The #571 result settles a central question on Turán numbers of bipartite graphs. The paper also gives a denominator that most AI-math announcements lack. The fixed-budget run solved only 2 of 68 problems. Of the 63 unsolved problems, 56 were retried two to five times each (172 attempts) with no success, and the five resolutions took over $220,000 of compute. That makes it a sober counterweight to OpenAI's later claim of "100+ open problems".

#### Follow-up: expert write-up of #571 (13 Oct)
On 13 Oct 2026 eight extremal combinatorialists (Conlon, Janzer, Jiang, Kim, Lee, Liu, Longbrake, Yepremyan) posted [arXiv 2610.18713](https://arxiv.org/abs/2610.18713), which explains and situates the #571 proof "claimed by Epoch AI, obtained during benchmarking for the recently released model GPT-6 Astra". It replaces the placeholder write-up with a human paper; see [its own entry](/e/2026-10-13-rational-exponents-conjecture-proof-paper-conlon/).

Sources: [GitHub: tadamcz/erdos1207 (Lean proof of Erdős #1207 by GPT-6 Astra)](https://github.com/tadamcz/erdos1207); [arXiv 2609.25050: FrontierMath Erdős (Adamczewski, Bloom)](https://arxiv.org/abs/2609.25050); [erdosproblems.com #1 (status: disproved, Lean)](https://www.erdosproblems.com/1); [GitHub: tadamcz/erdos1 (Lean disproof)](https://github.com/tadamcz/erdos1); [GitHub: tadamcz/erdos571 (rational exponents, Lean proof)](https://github.com/tadamcz/erdos571); [arXiv 2610.18713: A proof of the rational exponents conjecture (Conlon, Janzer et al., expert write-up of #571)](https://arxiv.org/abs/2610.18713); [GitHub: tadamcz/erdos74](https://github.com/tadamcz/erdos74); [GitHub: tadamcz/erdos126](https://github.com/tadamcz/erdos126); [Thomas Bloom on X: A big day for AI and mathematics (3 Sep 2026)](https://x.com/thomasfbloom/status/2095630765035864260); [Quanta: Why Erdős Problems Are Falling to AI (3 Aug 2026)](https://www.quantamagazine.org/why-the-legendary-erdos-problems-are-falling-to-ai-20260803/)

### 2026-09-03: GPT-6 Astra autonomously disproves Smale's mean value conjecture (constant 1) in Lean; HKU mathematicians build on it to show Smale's constant exceeds 1.1018

****; OpenAI, Epoch AI, University of Hong Kong; Science & math; confidence medium; https://postcutoff.com/e/2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra/

In an Epoch AI LeanOpenProblems evaluation run in September 2026, a pre-release GPT-6 Astra autonomously produced a Lean 4 proof that Smale's mean value conjecture with constant 1 is false: there is a polynomial p with p(0) = 0, p'(0) = 1 and |p(c)/c| > 1 at every critical point c. Tom Adamczewski published the proof (GitHub repository created 3 Sep 2026); MathWorld lists it as a proposed counterexample. On 15 Oct 2026 Aneesh Jatar and Tuen-Wai Ng (University of Hong Kong) posted arXiv 2610.21472, which builds on the Astra construction, recasts Smale's question in univalent function theory and proves Smale's constant M > 1.1018, with polynomials of arbitrarily large degree beating 1.1018.

- Conjecture: Smale (1981) proved that every polynomial has a critical point c with |p(z) − p(c)|/|z − c| ≤ 4|p'(z)|, and asked whether 4 can be replaced by 1 (or by (d−1)/d); one of Smale's problems for the 21st century. Known for degree d ≤ 4 and special classes
- Claim: the K = 1 conjecture is false; the witness has 'very large, unspecified degree and violates the bound by a small margin' (repository README)
- How found (README): Epoch AI evaluation run 'wikipedia-vega-1000usd' of its LeanOpenProblems harness; a pre-release GPT-6 Astra attempted all 222 research-open statements of the Wikipedia collection of Google DeepMind's Formal Conjectures, once each, with $1,000 and 96 hours per statement, in a network-isolated container; 'No human saw or steered the proof search'
- Cost (README): about 2.2M output and 68M cache-read tokens, $265 at GPT-6 Astra list prices; 15.0 hours of agent working time
- Verification: Lean 4 + Mathlib proof checked by Comparator against the Formal Conjectures statement, using only propext, Quot.sound and Classical.choice; the README's prose account of the proof is machine-generated and 'has not been checked by a human mathematician'
- Construction (README): a conformal map of the four-petal set {|z⁴ + 64i| ≤ √4097}, a Blaschke factor, polynomial approximation, then adding the primitive of q²ⁿ pushes all critical points into a thin collar where |g| > 1 + δ
- MathWorld (last updated 2 Oct 2026): 'Adamczewski (2026) released a proposed counterexample to the constant 1, constructed autonomously by GPT-6 Astra'; independent review 'had not been reported as of Sep. 7, 2026'
- Human follow-up: Jatar & Ng, 'Smale's mean value constant and admissible univalent Blaschke pairs', arXiv 2610.21472 (15 Oct 2026): 'Motivated and based upon the GPT-6 Astra construction'; they show M = A = B (two new extremal constants) and 1.1018586788… = √(8081/6656) ≤ B₁ < √(7/5), so M > 1.1018. The original argument can give S(P) > 1 + 2⁻³⁵ but is 'hard to push' further
- AI use in the follow-up: 'GPT-6Astra and Grok 4.6 were utilized to conduct numerical experiments and generate preliminary draft proofs via targeted prompting. The authors then collaboratively verified, significantly simplified, and rewrote all proofs in their own words.' Tuen-Wai Ng co-authored the 2002 Beardon–Minda–Ng bound on the same problem

#### What happened
Smale proved in 1981 that every complex polynomial p of degree at least 2 has, for every point z, a critical point c with
|p(z) − p(c)|/|z − c| ≤ 4|p'(z)|. He asked whether 4 can be replaced by 1, or even by (d−1)/d, which the polynomial
z − z^d/d attains. In 45 years the constant came down only to 4 − O(1/d). The problem is on Smale's list of problems for the
21st century.

In September 2026 Epoch AI ran a pre-release GPT-6 Astra over the 222 open statements in the "Wikipedia" collection of Google
DeepMind's Formal Conjectures, once each, with $1,000 and 96 hours per statement. For the mean value problem the model chose the
negation and proved it in Lean: there is a polynomial with p(0) = 0 and p'(0) = 1 whose value ratio |p(c)/c| exceeds 1 at every
critical point. Tom Adamczewski published the proof with a Comparator check (repository created 3 Sep 2026; the README says the
whole repository was machine-written by AI assistants at his direction). The witness has huge, unspecified degree and beats the
bound only slightly, which fits all known partial results. MathWorld added it as a "proposed counterexample" and noted that no
independent review had been reported as of 7 Sep.

On 15 Oct 2026 Aneesh Jatar and Tuen-Wai Ng of the University of Hong Kong posted [arXiv 2610.21472](https://arxiv.org/abs/2610.21472).
They describe the Astra construction step by step, say their work is "motivated and based upon" it, and turn Smale's question
into an extremal problem for pairs of a univalent map and a Blaschke product. They prove that Smale's constant M equals the new
constants A and B, and that M > 1.1018 (from B₁ ≥ √(8081/6656)), with polynomials of arbitrarily large degree beating 1.1018.
They used GPT-6 Astra and Grok 4.6 for numerical experiments and draft proofs, then "verified, significantly simplified, and
rewrote all proofs in their own words".

No Hacker News thread on the disproof was found (Algolia, 16 Oct); X was not searched.

#### Why it matters
The conjecture of constant 1 had been the expected answer for decades. Here it was refuted by an unattended benchmark run at
about $265 of tokens, with a machine-checked proof, and the result sat in a GitHub repository for six weeks before specialists
(one of them an author of a classic bound on the problem) built on it. The Lean proof settles correctness of the formal
statement; the follow-up paper is the first expert treatment and is not yet refereed. The stronger (d−1)/d form is also refuted,
since it implies the constant-1 form.

Unverified: the exact public release date of the repository (3 Sep is its creation date on GitHub) and whether Epoch AI or OpenAI
announced the result.

Sources: [GitHub: tadamcz/mean-value-problem (Lean 4 disproof found by GPT-6 Astra)](https://github.com/tadamcz/mean-value-problem); [arXiv 2610.21472: Smale's mean value constant and admissible univalent Blaschke pairs (Jatar & Ng)](https://arxiv.org/abs/2610.21472); [MathWorld: Smale's Mean Value Conjecture](https://mathworld.wolfram.com/SmalesMeanValueConjecture.html); [Epoch AI: LeanOpenProblems harness](https://github.com/epoch-research/LeanOpenProblems); [Formal Conjectures: MeanValueProblem.lean](https://github.com/google-deepmind/formal-conjectures/blob/9cbe1d3c12998c786b7c2cd99ce28a21b6631f66/FormalConjectures/Wikipedia/MeanValueProblem.lean)

### 2026-09-03: Google DeepMind's WeatherNext 3 learns from live satellite data: hourly 5 km global forecasts and up to 60% better precipitation skill

****; Google DeepMind, Google Research; Science & math; confidence high; https://postcutoff.com/e/2026-09-03-weathernext-3/

On Sept 3, 2026 Google DeepMind and Google Research released WeatherNext 3. It ingests live geostationary satellite mosaics and trains directly on station observations, producing a new global forecast every hour at up to 5 km resolution (about 5x sharper than WeatherNext 2). Precipitation CRPS improves by up to 60% against IMERG. Google calls it the most accurate global weather model on Brightband's independent live leaderboard, and it powers Search, Gemini, Maps and Earth Engine.

- Announced Sept 3, 2026; paper arXiv 2609.03582
- Inputs: live 1-hour geostationary satellite mosaics plus historical analysis, into a single Functional Generative Network (FGN) mesh transformer
- Hourly forecasts; 5 km for key surface variables, 10 km for other surface variables, 25 km for atmospheric variables (WeatherNext 2: 25 km, 6-hourly)
- Precipitation CRPS improvement up to 60% vs IMERG, 30% vs MRMS, 10% vs rain gauges at early lead times; up to 50% more accurate precipitation forecasts a day or more ahead for users
- New clean-energy variables: 100 m wind speed, cloud cover, surface solar radiation
- Ranked top on Brightband's live Operational WeatherBench, per Google; deployed in Search, Gemini, Maps, Maps Platform Weather API, Earth Engine, BigQuery

#### What happened
DeepMind's weather model stopped depending only on physics-model reanalysis and learns directly from real-time observations, which removes the six-hour data lag of numerical weather prediction.

#### Why it matters
It is a step from AI emulating weather simulators to AI forecasting from raw observations, with global 5 km detail that regions without supercomputing budgets have lacked.

Sources: [Google: Introducing WeatherNext 3](https://blog.google/innovation-and-ai/models-and-research/google-deepmind/introducing-weathernext-3/); [arXiv 2609.03582: WeatherNext 3](https://arxiv.org/abs/2609.03582); [Brightband Operational WeatherBench live leaderboard](https://owb.brightband.com/); [TechCrunch: Google's latest AI weather model](https://techcrunch.com/2026/09/03/googles-latest-ai-weather-model-gives-you-no-excuse-to-forget-your-umbrella/); [Google DeepMind: Introducing WeatherNext 3](https://deepmind.google/blog/introducing-weathernext-3-our-most-advanced-and-accurate-global-weather-ai-model/)

### 2026-09-03: Complete male fruit fly nervous-system connectome (166,000 neurons), reconstructed with Google AI, published in Cell

***; Google Research, HHMI Janelia, MRC Laboratory of Molecular Biology, University of Cambridge; Science & math; confidence high; https://postcutoff.com/e/2026-09-03-male-fruit-fly-connectome/

On Sept 3, 2026 Google Research, HHMI Janelia and Cambridge (MRC LMB) announced peer-reviewed publication in Cell of the first complete connectome of an adult male fruit fly's central nervous system (brain plus ventral nerve cord). It maps more than 166,000 neurons and about 125 million synapses in 11,691 cell types. Google calls it the largest brain map by neuron count to date. Google's AI segmented the electron microscopy data, and Janelia and Cambridge teams proofread and annotated it. The map allows the first whole-CNS comparison of male and female wiring in an animal with complex social behavior.

- >166,000 neurons and ~125 million synapses; 11,691 neuron types in the male CNS (Google blog/Google Research)
- First complete male fruit fly CNS map; builds on the 2024 complete female fruit fly brain connectome (FlyWire)
- Sex differences: most neurons are the same in both sexes; a minority are sex-specific, and 'dimorphic' neurons exist in both sexes but wire differently, mainly in courtship- and aggression-related circuits
- Method: thin sections imaged by focused ion beam scanning electron microscopy; Google's flood-filling networks and its PATHFINDER reconstruction system (trained partly on synthetic neurons) turned millions of 2D images into 3D neurons; human experts at Janelia and Cambridge proofread and labeled them
- Papers: Cell, 'Sexual dimorphism in the complete connectome of the Drosophila male central nervous system', with companion papers; the Google blog links doi 10.1016/j.cell.2026.08.014, 10.1016/j.cell.2026.08.016 and Current Biology 10.1016/j.cub.2026.08.013; a preprint appeared on bioRxiv in Oct 2025
- Data: Janelia 'male-cns' dataset, explorable in Neuroglancer
- Google Research's post also cites new Nature papers on an elephantnose fish cerebellum-like circuit and a zebrafish whole-brain connectome (not separately verified here)

#### What happened
A years-long collaboration between HHMI Janelia, Google Research, the MRC Laboratory of Molecular Biology and the University of Cambridge
published the first complete wiring diagram of an adult **male** fruit fly's central nervous system, covering both the brain and the ventral
nerve cord. Janelia imaged the tissue with focused ion beam electron microscopy. Google's machine-learning segmentation reconstructed the 3D
shapes of more than 166,000 neurons from millions of 2D images, and human teams proofread and classified them into 11,691 types.
Comparing it with the female brain map from 2024 shows which neurons are sex-specific or wired differently between sexes, mostly in circuits
for courtship and aggression.

#### Why it matters
Connectomics is one of the clearest cases where AI turns an impossible manual task into a feasible one. This is the largest complete brain
map by neuron count, and the first that allows a whole-nervous-system comparison of the two sexes. Google says the same pipeline is being
scaled toward vertebrate brains.

Sources: [Google Research: A connectomics milestone: mapping the complete male fruit fly brain](https://research.google/blog/a-connectomics-milestone-mapping-the-complete-male-fruit-fly-brain/); [Google blog: 5 amazing visuals show how the male fruit fly's brain map is advancing neuroscience](https://blog.google/innovation-and-ai/technology/research/male-fruit-fly-brain-map/); [Cell paper (doi 10.1016/j.cell.2026.08.014)](https://doi.org/10.1016/j.cell.2026.08.014); [Cell companion paper (doi)](https://doi.org/10.1016/j.cell.2026.08.016); [Current Biology companion paper (doi)](https://doi.org/10.1016/j.cub.2026.08.013); [Janelia: Researchers reveal connectome of the male fruit fly central nervous system](https://www.janelia.org/news/researchers-reveal-connectome-of-the-male-fruit-fly-central-nervous-system); [Janelia FlyEM: male CNS connectome dataset](https://www.janelia.org/project-team/flyem/male-cns-connectome); [Neuroscience News: Microscopy and AI decipher fruit fly neural architecture](https://neurosciencenews.com/ai-fly-connectome-31217/)

### 2026-09-03: PPPL's PACMAN framework lets multiple AI models control a tokamak in ~20 ms, preventing a tearing mode

**; Princeton Plasma Physics Laboratory, General Atomics; Science & math; confidence medium; https://postcutoff.com/e/2026-09-03-pppl-pacman-fusion-ai-control/

PPPL reported PACMAN, a modular framework that plugs several ML models directly into a tokamak's control system, reading plasma data and issuing commands in about 20 ms. In five DIII-D experiments an RL model took full control of the heating systems, and the framework predicted edge bursts (ELMs), controlled fast-particle-driven waves, and predicted and prevented a tearing mode.

- ~20 ms decision loop; multiple ML models run simultaneously
- 5 DIII-D demonstrations incl. full RL control of heating and pre-emptive tearing-mode suppression
- Humans set goals and safety limits; published in Nuclear Fusion

#### What happened
PPPL moved from single-purpose AI controllers to a framework where several models share control of one machine in real time.

#### Why it matters
It is a step toward the AI-supervised operation that future power-plant tokamaks such as SPARC and ITER are expected to need.

Sources: [PPPL: PACMAN AI framework makes key fusion decisions in milliseconds](https://www.pppl.gov/news/2026/pacman-ai-framework-controlling-fusion-systems-safely-makes-key-decisions-milliseconds); [ScienceDaily: PACMAN AI framework for fusion](https://www.sciencedaily.com/releases/2026/09/260903064215.htm); [Phys.org: PACMAN AI framework controls fusion systems safely](https://phys.org/news/2026-09-pacman-ai-framework-fusion-safely.html)

### 2026-09-04: Claude produces the first complete machine-checked proof of Fermat's Last Theorem in Lean, in 11 days

*****; Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-09-04-claude-formalizes-fermats-last-theorem/

Anthropic reported that a Claude model (roughly comparable to Claude Fable 5.1), running for 11 days (7–18 Aug 2026) using the Prove2Me multi-agent platform, produced a complete Lean formalisation of Fermat's Last Theorem using only Lean's three standard axioms: about 13 million lines and 30,300 theorems, over 5× the size of Mathlib.

- Run 7–18 Aug 2026; published 4 Sep 2026
- ~13M lines of Lean; 30,300 theorems (29,500 used); ~6 billion output tokens
- Only occasional high-level instructions from Anthropic researcher Tianyi Peng (e.g. 'Jacobian as a scheme sounds high priority')
- Checked against Mathlib's statement of FLT with a comparator; no axioms beyond Lean's standard three
- Kevin Buzzard (who leads the human FLT formalisation project): 'This extraordinary autoformalization achievement ... proves Fermat's Last Theorem with no assumptions other than the axioms of mathematics.'

#### What happened
An agentic Claude, orchestrated through the Prove2Me platform, wrote the missing chain of Lean on top of Mathlib, through the modularity-lifting machinery of the Wiles–Taylor proof, up to FLT itself.

#### Why it matters
Formalising FLT had been a flagship multi-year human project. Its completion by AI shows that even the deepest modern proofs can now be machine-checked at AI speed.

Sources: [Anthropic: Formalizing Fermat's Last Theorem](https://www.anthropic.com/research/formalizing-fermats-last-theorem); [AI Weekly: Claude formalized Fermat's Last Theorem in 11 days](https://aiweekly.co/alerts/claude-formalized-fermats-last-theorem-in-11-days-anthropic); [Anthropic on X: first formalized proof of Fermat's Last Theorem](https://x.com/AnthropicAI/status/2095947707605266436); [Kevin Buzzard (Xena Project): FLT: Anthropic has beaten me to it](https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-has-beaten-me-to-it/)

### 2026-09-04: Reed–Solomon codes list-decoded up to capacity and proximity gaps settled, with GPT-5.6 Sol and ChatGPT 5.6 Pro (Brakensiek–Chen–Putterman–Zhang–Zheng; Jeronimo)

****; Simons Institute, OpenAI, Ethereum Foundation; Science & math; confidence medium; https://postcutoff.com/e/2026-09-04-reed-solomon-list-decoding-capacity-ai/

In early September 2026 two preprints settled long-standing problems about Reed–Solomon codes, the most widely used error-correcting codes. Brakensiek, Chen, Putterman, Zhang and Zheng (ECCC TR26-164, 4 Sept; arXiv 2609.08005) gave a deterministic polynomial-time algorithm that list-decodes RS codes over prime fields up to capacity, far past the Guruswami–Sudan/Johnson barrier. Jeronimo (arXiv 2609.05870, 5 Sept) extended it to all rates and proved near-optimal proximity gaps (mutual correlated agreement), the property behind the Ethereum Foundation's $1M Proximity Prize. Both disclose substantial AI help: GPT-5.6 Sol "to help produce the rest of the results" and ChatGPT 5.6 Pro interactions "crucial".

- Brakensiek, Chen, Putterman, Zhang, Zheng, 'Algorithmic List Decoding of Reed–Solomon Codes up to Capacity': ECCC TR26-164 (4 Sept 2026), arXiv 2609.08005 (7 Sept). Deterministic poly-time decoding over prime fields from a 1−R−δ fraction of errors, every evaluation set; first for low constant rate, extended to all constant rates with a padding reduction from Alrabiah, Goyal and Guruswami
- Their AI statement: inspired by a Proximity Prize submission on better.codes by user 'nasqret' that went slightly beyond the Johnson radius for one rate-1/2 code of length 2^18; the authors 'relied on AI interaction, specifically GPT-5.6 Sol, to help produce the rest of the results of this paper'; all statements 'independently verified by the authors'
- They also note that after their posting 'a number of groups contemporaneously realized (with LLM assistance)' how to extend the proof white-box to all rates
- Jeronimo, 'Algorithmic List Decoding at Capacity and Optimal Proximity Gaps for Reed–Solomon Codes': arXiv 2609.05870 (5 Sept), ECCC TR26-169. List decoding at capacity for every rate with list size n^{O(1)} independent of q, plus mutual correlated agreement (MCA) with error n^{O(1)}/q and no proximity loss; it gives fixed-slack certificates for the MCA and list-size targets of the preliminary Proximity Prize statement, but 'does not determine their largest safe radius'
- Jeronimo's AI disclosure: 'The interactions with ChatGPT 5.6 Pro were crucial in obtaining the main results of this paper.'
- Proximity Prize: a $1,000,000 Ethereum Foundation programme on RS proximity gaps and list decoding (judges Dan Boneh, Giacomo Fenzi, Gal Arnon; deadline 31 Dec 2027; 'AI-aided submissions are allowed'); companion live Lean-scored challenge at better.codes. No award announced as of 7 Oct 2026
- Follow-ups: Dao, Kominers and Thaler (IACR ePrint 2026/2056, 16 Sept) give quantitative versions over cryptographic fields and report proof-size savings of 4.6–11.1% in ProveKit, ZisK and LambdaVM; Harsha, Kumar and Saptharishi (TIFR) posted a unified exposition (arXiv 2610.08610, 6 Oct) saying both problems 'were resolved in the last few weeks with AI assistance' and that they used Claude (Opus and Sonnet) to understand the dense originals
- Verification status: unrefereed preprints, but already re-derived independently in an expository paper by three coding theorists

#### What happened
Reed–Solomon (RS) codes encode a message as the values of a low-degree polynomial. They are used in storage, communications and, more and more, in the hash-based proof systems (STARKs, FRI and relatives) behind blockchain scaling. Two questions about them had been open for decades. One was whether RS codes can be decoded efficiently from errors beyond the Johnson radius, the limit of the Guruswami–Sudan algorithm (1999), all the way up to the information-theoretic capacity. The other was whether RS codes have "proximity gaps" up to capacity: if many points on a line of words are close to the code, all of them are, with correlated agreement. The second property sets the soundness and therefore the proof size of FRI-style proof systems. In 2025 the Ethereum Foundation put up a $1M **Proximity Prize** for it, with a live companion challenge at better.codes.

On 4 September 2026, Joshua Brakensiek, Yeyuan Chen, Aaron Putterman, Zihan Zhang and Kai Zhe Zheng posted a deterministic polynomial-time algorithm that list-decodes RS codes over prime fields up to capacity for low constant rates (ECCC TR26-164; arXiv 2609.08005). Within days a revision extended it to all constant rates, using a padding observation from Alrabiah, Goyal and Guruswami. On 5 September Fernando Granha Jeronimo posted a unified "hidden-derivative" framework (arXiv 2609.05870). It gives capacity list decoding with list size independent of the field and near-optimal mutual correlated agreement, which certifies the asymptotic targets of the prize statement but not the largest safe radius.

Both papers credit AI heavily. Brakensiek et al. say they started from a better.codes submission by the user "nasqret", which pushed one specific code slightly past the Johnson radius. They "relied on AI interaction, specifically GPT-5.6 Sol, to help produce the rest of the results of this paper". They also note that several groups "contemporaneously realized (with LLM assistance)" how to extend the proof to all rates. Jeronimo writes that "the interactions with ChatGPT 5.6 Pro were crucial in obtaining the main results".

Follow-ups came quickly. Dao, Kominers and Thaler (a16z crypto) made the bounds quantitative over cryptographic fields and reported 4.6–11.1% smaller proofs in existing systems (IACR ePrint 2026/2056). On 6 October, Prahladh Harsha, Mrinal Kumar and Ramprasad Saptharishi (TIFR) posted a unified exposition (arXiv 2610.08610). They write that the problems "have recently met their fate, thanks to some heavy lifting by AI tools" and that understanding the dense originals took "several extended conversations with Claude (Opus and Sonnet)". The exposition itself is human-written.

The Proximity Prize page lists no award as of 7 October 2026. Its rules allow AI-aided submissions if they are human-verified.

#### Why it matters
This is one of the clearest cases so far of AI-assisted research changing a central result in theoretical computer science with direct engineering consequences: proximity-gap bounds feed straight into the parameters and proof sizes of deployed zero-knowledge systems. It also shows a new research pipeline: a crypto bounty, a crowd-sourced Lean-scored leaderboard, an anonymous submission and chatbot-assisted generalisation, all within weeks. Earlier update runs did not catch it (the papers sit on ECCC and in cs.IT, not in the maths listings the sweep scans); the TIFR exposition's AI note surfaced it a month later.

Sources: [ECCC TR26-164: Algorithmic List Decoding of Reed–Solomon Codes up to Capacity](https://eccc.weizmann.ac.il/report/2026/164/); [arXiv 2609.08005 (Brakensiek, Chen, Putterman, Zhang, Zheng)](https://arxiv.org/abs/2609.08005); [arXiv 2609.05870: Jeronimo, list decoding at capacity and optimal proximity gaps](https://arxiv.org/abs/2609.05870); [ECCC TR26-169 (Jeronimo)](https://eccc.weizmann.ac.il/report/2026/169/); [arXiv 2610.08610: Harsha, Kumar, Saptharishi, Reed–Solomon codes at capacity (exposition)](https://arxiv.org/abs/2610.08610); [IACR ePrint 2026/2056: Dao, Kominers, Thaler, Reed–Solomon codes beyond Johnson](https://eprint.iacr.org/2026/2056); [Ethereum Foundation: The Proximity Prize](https://proximityprize.org/)

### 2026-09-05: The I3322 Bell inequality needs infinite dimensions: Pál–Vértesi conjecture (2010) proved from an approximate proof by GPT-5.5 Pro

***; Andrea Coladangelo, University of Washington, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-09-05-i3322-bell-inequality-infinite-dimensions/

Andrea Coladangelo (University of Washington) proved the Pál–Vértesi conjecture that no finite-dimensional quantum strategy reaches the maximal violation of the I3322 Bell inequality, while an infinite-dimensional one does (arXiv 2609.06038, 5 Sep 2026). I3322 is one of the simplest Bell inequalities. His AI statement says 'An approximate proof (containing all of the main ideas) was produced by GPT 5.5 Pro'. He then substantially revised it for completeness and correctness, with help from GPT-5.6 Sol.

- Conjecture: Pál and Vértesi (Physical Review A, 2010); I3322 has three questions and two answers per party, the simplest Bell scenario where this phenomenon could occur
- Previously a 5-question, 3-answer correlation with this property was known (Coladangelo and Stark, Nature Communications 2020)
- Proof tool: a Bellman-function-style certificate from two families of inequalities that bounds the value of any finite-dimensional strategy
- AI usage statement: 'An approximate proof (containing all of the main ideas) was produced by GPT 5.5 Pro after some rounds of interaction with the author. That first approximate proof was much shorter, and arguably quite difficult to parse by a human'; the final text is the author's revision, with help from GPT 5.6 Sol
- Status: preprint (78 pages)

#### What happened
Coladangelo, who had earlier found a larger Bell scenario with the same property, worked with GPT-5.5 Pro on the simplest candidate, I3322. The model produced a terse approximate proof containing the key ideas. The author expanded it into a 78-page rigorous paper.

#### Why it matters
It settles a basic question about how much entanglement quantum correlations can require, in the smallest possible Bell scenario. It is also one of the clearest 2026 cases of an AI supplying the core idea in quantum foundations.

Sources: [arXiv 2609.06038: The I3322 Bell inequality requires infinite dimensions (Coladangelo)](https://arxiv.org/abs/2609.06038)

### 2026-09-06: Collatz: a positive fraction of integers provably reach 1 in 10.46 ln n steps; AI-agent proof (ProofAtlas + Codex) by Lech Mazur, Lean-formalized and re-formalized by Axiom

****; ProofAtlas, OpenAI, Axiom Math; Science & math; confidence medium; https://postcutoff.com/e/2026-09-06-collatz-positive-density-log-time-mazur-codex/

On Sept 6, 2026 Lech Mazur posted (X, ~204k views) "Explicit Positive-Density Collatz Convergence in Logarithmic Time": for explicit c > 0 and X₀, every X ≥ X₀ has at least cX positive integers n < X that reach 1 within (523/50) ln n ordinary Collatz steps. Previous counting bounds for convergent integers were sublinear (Krasikov–Lagarias X^0.84; Mazur's own July 2026 X^0.90). The paper's disclosure says generative AI "contributed substantially to mathematical exploration, proof development, computational testing, formalization, and exposition", using the ProofAtlas.ai harness with OpenAI GPT models through Codex. The theorem has a Lean 4 formalization (~42k added lines), and on Oct 9 Axiom Math published an independent Lean formalization (AxiomMath/CollatzPositiveDensity) checked with the Lean comparator. The constants are astronomically small/large and the conjecture remains open. No peer review yet.

- Theorem 1.1: explicit rational c > 0 and integer X₀ with #{1 ≤ n < X : τ(n) ≤ (523/50) log n} ≥ cX for every X ≥ X₀; so convergence to 1 within 10.46 log n steps holds on a set of positive lower natural density
- Prior bounds: Krasikov & Lagarias (Acta Arith. 2003) X^0.84 convergent integers ≤ X; Mazur, 'Certified x^0.90 Lower Bounds for Collatz Predecessor Sets' (ProofAtlas, July 17, 2026); Tao (Forum Math. Pi 2022): almost all orbits (log density) attain almost bounded values, which does not give convergence to 1
- Paper's own limits: 'The density bound is extremely small and the cutoff extremely large; no practical numerical range is claimed'; 'The result does not settle the Collatz conjecture'. Axiom's README states the thresholds as N ≥ 2^2^2^2^87640 and fraction 1/2^2^2^87640
- AI disclosure (quoted): 'Generative AI contributed substantially to mathematical exploration, proof development, computational testing, formalization, and exposition, not only to language editing. Lech Mazur directed the AI-assisted research and is the author responsible for the manuscript and its claims. The ProofAtlas.ai harness was used for the AI-assisted research and verification workflow, with OpenAI GPT models accessed through Codex. Anthropic Claude was also used for manuscript review.'
- Mazur on X: 'Developed primarily by AI agents using the ProofAtlas.ai harness and Codex, with some high-level guidance from me (which may or may not have helped)'
- Lean (paper §9): main theorem formalized with no admitted proof or project axiom, only propext, Classical.choice, Quot.sound; builds on earlier formalizations of Tao's almost-boundedness and Mazur's natural-density extension (July 21, 2026). ProofAtlas page: 'Lean checked', 'Unfinished proof steps: None', 'Publication: Review pending'
- Axiom Math repo AxiomMath/CollatzPositiveDensity (created 2026-10-09) formalizes the theorem plus variants for other targets s coprime to 6 and t not divisible by 3, with a challenge file verified by the Lean comparator
- Audit note (19 Oct): the Axiom repository's README calls it a formalization 'refining Lech Mazur's Theorem 1.1'; its formalization.yaml says it is a formal proof of the main results of Kenny Lau's paper 'Explicit positive lower density for the Collatz Conjecture' (hexagonmath.org/2610.00249), that 'The Lean files were generated by AxiomProver', and gives review status 'author-verified' (the challenge was checked against the paper by Lau); sorry_count 0, axioms propext, Quot.sound, Classical.choice

#### What happened
Lech Mazur (who posted the GPT-5.6 Pro Sendov proof in August) announced on Sept 6, 2026 that a fixed positive fraction of all
integers below every large enough cutoff reach 1 under the Collatz map within 10.46 ln n ordinary steps. Until then, unconditional
counting results only gave sublinear numbers of convergent integers: X^0.84 (Krasikov–Lagarias 2003) and X^0.90 (Mazur's own
AI-assisted ProofAtlas paper of July 2026). Tao's 2019 theorem is a different kind of statement: almost every orbit gets below any
slowly growing function, which does not give convergence to 1. The paper combines finite ternary mixing, a weighted inverse-orbit
construction from one fixed convergent seed, and a counting estimate. Both constants are given by finite expressions, but they are
astronomically extreme.

The disclosure credits generative AI with the exploration, proof development, testing, formalization and exposition, through the
ProofAtlas.ai harness with OpenAI GPT models in Codex, and Claude for manuscript review. ProofAtlas lists the Lean build as passed
with no unfinished steps and the publication as "review pending". On Oct 9, Axiom Math published a separate Lean formalization
that also covers other targets, with a comparator-checked challenge file.

We missed this for six weeks. It was not on arXiv and had no Techmeme or press coverage. The sweep found it through Axiom's GitHub
repo (which cites the X post) while checking the AxiomMath account on Oct 18.

#### Why it matters
Collatz is one of the best-known open problems. This does not prove it, but it is the first result showing that convergent
integers make up a positive proportion, with a logarithmic time bound. It was produced mainly by AI agents directed by a
non-academic, and checked in Lean twice. No expert review or published reaction was found by Oct 18, so its standing in the
field is unknown.

Sources: [Lech Mazur on X: announcement (Sept 6, 2026)](https://x.com/LechMazur/status/2096464472973975782); [Paper: Explicit Positive-Density Collatz Convergence in Logarithmic Time (v2.1, PDF)](https://www.proofatlas.ai/papers/positive-density-log-time-collatz/Mazur_Explicit_Positive_Density_Collatz_Convergence_in_Logarithmic_Time_v2.pdf); [ProofAtlas formalization page](https://www.proofatlas.ai/formalizations/positive-density-log-time-collatz/); [AxiomMath/CollatzPositiveDensity (independent Lean formalization)](https://github.com/AxiomMath/CollatzPositiveDensity)

### 2026-09-06: Preprint credits the 'Odin' AI research agent with a positively curved metric on S³×S³, which would disprove Hopf's Euler-characteristic sign conjecture (unverified)

****; Odin Automatic AI Research Agent, Harvard T.H. Chan School of Public Health; Science & math; confidence medium; https://postcutoff.com/e/2026-09-06-odin-ai-agent-positive-curvature-s3xs3-hopf-sign-conjecture/

On 6 Sept 2026 Shengtao Guo, Ethan X. Fang and Junwei Lu posted arXiv 2609.11980, which constructs a smooth metric with strictly positive sectional curvature on S³×S³. Since S³×S³ is even-dimensional with Euler characteristic 0, this would disprove the positive-curvature case of Hopf's sign conjecture. The abstract says "The metric and the proof were discovered by the Odin Automatic AI Research Agent." It follows the same team's S²×S³ metric (2608.22133, also credited to Odin) and Brendle and Hung's human disproof of Hopf's product conjecture on S²×S² (2608.19068). Specialists Nienhaus and Wulle call the 6-dimensional claim "entirely AI and not verified by geometers".

- Paper: 'A Metric with Positive Sectional Curvature on S³×S³', Shengtao Guo, Ethan X. Fang, Junwei Lu (Department of Biostatistics, Harvard T.H. Chan School of Public Health), arXiv 2609.11980 [math.DG], v1 6 Sept 2026, still v1 on 16 Oct
- AI statement (abstract): 'The metric and the proof were discovered by the Odin Automatic AI Research Agent.' Introduction: 'Odin Automatic AI Research Agent was used to construct the metric and the proof.' No model behind Odin is named
- Claim: since χ(S³×S³) = 0, the result 'disproves the positive-curvature case of Hopf's sign conjecture' (closed even-dimensional manifolds of positive sectional curvature have χ > 0)
- Method (paper's outline): perturbations of a nonnegatively curved metric with height functions selecting a torus of zero-curvature planes, a smooth global correction tensor and a 'conformal Poisson correction'; finite algebraic identities checked exactly over Q(z) by code at github.com/shengtaoguo/s3xs3-curvature-calculations ('python3 verification/verify.py' prints 'VERIFIED:'), while the passage to positive curvature on the whole Grassmann bundle is a written proof (Sections 4–7), not a formal one
- Earlier step by the same authors: 'A Metric with Positive Sectional Curvature on S²×S³' (arXiv 2608.22133, 22 Aug 2026), whose PDF also says the results 'are discovered by the Odin Automatic AI Research Agent'
- Human context: Simon Brendle and P. K. Hung, 'A metric on S²×S² with positive sectional curvature' (arXiv 2608.19068, 19 Aug 2026), disproving Hopf's product conjecture with a third-order perturbation of a Cheeger–Müter metric and Mathematica calculations; no AI credit in the abstract
- Specialist reaction: Nienhaus and Wulle's Hopf T³ paper (arXiv 2610.22173, 15 Oct) footnotes 'an announcement of a counterexample to the Hopf conjecture in dimension 6, but to the knowledge of the authors this work is entirely AI and not verified by geometers' (no citation given; this preprint matches the description)
- The same agent is credited in Fang, Lu et al.'s forest-unimodality paper (arXiv 2609.20961): 'The Odin Automatic AI Research Agent was used to find the initial proof' (see 2026-09-27-erdos-993-forest-independence-polynomials-unimodal)
- No Hacker News thread, press coverage or expert confirmation found as of 16 Oct 2026

#### What happened
Heinz Hopf asked two famous questions about positive sectional curvature: whether S²×S² admits such a metric (the product
conjecture) and whether every even-dimensional closed positively curved manifold has positive Euler characteristic (the sign
conjecture). In August 2026 Simon Brendle and P. K. Hung posted a metric of positive curvature on S²×S², disproving the product
conjecture. Within days a team at Harvard's Department of Biostatistics posted a positively curved metric on S²×S³ (22 Aug), and on
6 Sept a metric on S³×S³. Because χ(S³×S³) = 0, the second would be a counterexample to the sign conjecture in dimension 6, where
Geroch's 1976 algebraic example had shown that the four-dimensional Gauss–Bonnet argument fails.

Both papers credit the "Odin Automatic AI Research Agent" with discovering the metric and the proof. The S³×S³ paper runs to 30
pages, ships Python code that checks its finite algebraic identities exactly, and leaves the global positivity argument as a written
proof.

#### Why it matters
If it holds, one of the central conjectures of Riemannian geometry fails, and an AI agent is credited with the full construction. The
claim is unverified: on 15 Oct the geometers Nienhaus and Wulle, whose own AI-assisted paper proves the conjecture under T³
symmetry, wrote that the dimension-6 announcement is "entirely AI and not verified by geometers". The case shows how AI-produced
claims about famous problems can sit unexamined for weeks.

Sources: [arXiv 2609.11980: A Metric with Positive Sectional Curvature on S³×S³ (Guo, Fang, Lu)](https://arxiv.org/abs/2609.11980); [Verification code: shengtaoguo/s3xs3-curvature-calculations](https://github.com/shengtaoguo/s3xs3-curvature-calculations); [arXiv 2608.22133: A Metric with Positive Sectional Curvature on S²×S³ (Guo, Fang, Lu)](https://arxiv.org/abs/2608.22133); [arXiv 2608.19068: Brendle and Hung, A metric on S²×S² with positive sectional curvature](https://arxiv.org/abs/2608.19068); [arXiv 2610.22173: Nienhaus and Wulle, The Hopf conjecture with T³-symmetry (footnote on the dimension-6 claim)](https://arxiv.org/abs/2610.22173); [MathWorld: Hopf Conjecture (lists the Brendle–Hung announcement)](https://mathworld.wolfram.com/HopfConjecture.html)

### 2026-09-07: Pre-release GPT-6 Astra disproves the Köthe conjecture (1930) with a Lean-verified counterexample

****; OpenAI, Epoch AI; Science & math; confidence high; https://postcutoff.com/e/2026-09-07-koethe-conjecture-disproved/

During an Epoch AI run over the Formal Conjectures collection, pre-release GPT-6 Astra autonomously found an explicit 2×2 matrix counterexample over a nil algebra (Krempa's matrix form) with a Lean 4 proof, disproving the Köthe conjecture of 1930. Mathematicians wrote it up in arXiv 2609.07996.

- Köthe conjecture (1930): if a ring has no nonzero nil two-sided ideals, it has no nonzero nil one-sided ideals
- Counterexample via Krempa's equivalent matrix formulation; Lean 4 proof
- Found inside Epoch AI's LeanOpenProblems evaluation (222 research-open formal problems); repository README: 'No human saw or steered the proof search'
- Write-up by Adamczewski, Böhmler and Marczinzik; a second counterexample by Greenfeld, King and Vendramin with some Astra help

#### What happened
Epoch AI ran pre-release Astra against a library of formalised open conjectures. The model returned a Lean-checked counterexample to Köthe's conjecture, which human algebraists then confirmed and wrote up.

#### Why it matters
If it survives review, it resolves one of the most famous open problems in ring theory, found autonomously and verified formally.

Sources: [arXiv 2609.07996 (write-up)](https://arxiv.org/abs/2609.07996); [GitHub: tadamcz/koethe (Lean proof)](https://github.com/tadamcz/koethe); [Wikipedia: List of mathematical discoveries by artificial intelligence](https://en.wikipedia.org/wiki/List_of_mathematical_discoveries_by_artificial_intelligence)

### 2026-09-07: Caltech team (Anandkumar) reports a stable self-similar singularity candidate for the unforced 3D Euler equations on R³, found with PINNs and LLM help

***; Caltech; Science & math; confidence medium; https://postcutoff.com/e/2026-09-07-anandkumar-euler-singularity-r3/

On 7 Sep 2026, the evening before OpenAI's Navier–Stokes announcement, Anima Anandkumar's Caltech group posted a self-similar singular profile for the unforced incompressible 3D Euler equations on all of R³. Physics-informed neural networks found it, and LLMs helped simplify the bounds and formalise derivations in Lean. The arXiv papers (2609.10867, 2609.10860) describe "evidence" and a stability framework that is conditional on certifying explicit constants, so this is not yet a complete proof.

- Authors: Adarsh Ganeshram, Valentin Duruisseaux, Anima Anandkumar (+ Robert J. George on the stability paper)
- Setting: incompressible Euler on unbounded R³, no forcing; axisymmetric self-similar ansatz at blow-up rate 0.5 (matching a prediction by Constantin et al., arXiv 2602.17570)
- Method: PINN finds approximate profile; second-order optimisers (SS-eSOAP, SS-Broyden); certified via spline representation with interval arithmetic
- AI use (guest post): 'we used the OpenAI and other models extensively to simplify our bounds as well as formalize the derivations in Lean'
- arXiv 2609.10867 (111 pp.) abstract: 'We provide evidence of a finite-time singularity'; 2609.10860 (113 pp.): stability proof closes 'conditional on rigorous certification of the estimates and constants'
- The authors complain that mainstream media followed OpenAI's press release and did not acknowledge their work

#### What happened
In the same week as the Buckmaster–Alpöge forced blow-up results (7 Sep) and OpenAI's forced Navier–Stokes claim (8 Sep), a third group posted a singularity for the *unforced* Euler equations on the whole space. They used AI-driven numerical discovery followed by computer-assisted proof techniques. Their guest post on Tao's blog stresses AI as a "complementary" tool, "built to propose solutions that did not compete with humans".

#### Why it matters
Unforced Euler blow-up on R³ is a famous open problem in its own right, and it is a stepping stone toward the unforced Navier–Stokes question. The claim is still partly conditional, so its status should be tracked.

Sources: [Anima Anandkumar (guest post on Tao's blog): Stable singularity of the Euler equations on R³](https://terrytao.wordpress.com/2026/09/10/stable-singularity-of-the-euler-equations-on-r3/); [arXiv 2609.10867: Self-Similar Singularity of the Euler Equations on R³](https://arxiv.org/abs/2609.10867); [arXiv 2609.10860: Stability Framework for the Singularity of the Euler Equations on R³](https://arxiv.org/abs/2609.10860); [Anandkumar group page on the Euler result](https://tensorlab.cms.caltech.edu/users/anima/euler.html)

### 2026-09-07: Erdős's 1946 conjecture on almost-monotone additive functions (Erdős problem #1122) proved in an anonymous Zenodo preprint whose proofs and Lean code were generated by GPT-6 Astra; Mangerel builds on it to settle Sárközy's 2001 conjectures

***; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-09-07-erdos-1122-monotone-additive-gpt-6-astra/

On Sept 7, 2026 an anonymous author posted "Almost everywhere monotone additive functions" on Zenodo. It claims a proof of Erdős's 1946 conjecture (Erdős problem #1122) that an additive function which decreases only on a set of density zero must be c·log n. The paper says GPT-6 Astra proposed the proofs and generated a Lean 4 formalization (cited results taken as hypotheses). On Oct 6 Alexander Mangerel, who had made the earlier partial progress, called it "an AI-assisted proof" and used its method to resolve two 2001 conjectures of Sárközy (arXiv 2610.08424).

- Problem (Erdős 1946, Ann. of Math. 47; erdosproblems.com #1122): if f is additive and #{n ≤ X : f(n+1) < f(n)} = o(X), must f(n) = c log n? Erdős proved it when f never decreases or when f(n+1) − f(n) → 0
- Earlier partial progress: Mangerel (arXiv 2108.12351, published 2022) proved it when the decreases number ≪ X/(log X)^(2+c) and f(p) is not too large
- Preprint: Anonymous, 'Almost everywhere monotone additive functions', Zenodo 10.5281/zenodo.22651918 (v1 Sept 7, 2026; v2 Sept 22 adds erdos-1122-lean.zip). About 64 views and 32 downloads by Oct 10
- Method (abstract): deduce finite concentration without a growth assumption on higher prime powers, via a truncated normalization and bounded clipping that reduce to Mangerel's first-moment short-interval theorem, Ruzsa's second-moment estimate and Elliott's fourth-moment inequality
- AI disclosure: 'GPT-6 Astra was used to propose the mathematical proofs and generate the Lean formalization. GPT-5.6 Sol and Claude Opus 5 were used for editorial review of the exposition. The author finished the final manuscript and takes full responsibility for its content.'
- Lean 4: the formalization treats the cited results of Mangerel, Ruzsa, Erdős and Hildebrand as hypotheses, so it is not a full formal proof from Mathlib. We have not compiled it
- Oct 6, 2026: Alexander P. Mangerel, 'On Sárközy's Local Extrema Conjectures' (arXiv 2610.08424): 'Recently, an AI-assisted proof of problem (ii') was obtained [1] that relied crucially on this work'. He says his paper is 'inspired by the method in [1]' and that the AI paper does not adequately describe its context. He resolves, in a strong form, Sárközy's two 2001 conjectures on multiplicative functions with finitely many local maxima or minima
- erdosproblems.com/1122 (checked Oct 10): one proof claim listed, but the page was last edited April 1, 2026 and the problem is not yet marked solved. New proof claims there were suspended in early October

#### What happened
In 1946 Erdős proved that an additive function (f(ab) = f(a) + f(b) for coprime a, b) that never decreases must be a constant
multiple of log n. He conjectured that the same holds if it decreases only on a set of natural density zero. The question is
listed as problem #1122 on Thomas Bloom's erdosproblems.com. Alexander Mangerel made the best partial progress (2021–22).

On Sept 7, 2026 an anonymous author posted an 8-page proof on Zenodo, tagged "Erdős Problem 1122". A second version on Sept 22
added a Lean 4 project. The paper's "Declaration of generative AI use" says OpenAI's GPT-6 Astra proposed the proofs and
generated the Lean formalization. The record drew little attention (about 60 views by Oct 10).

It surfaced on Oct 6, when Mangerel posted "On Sárközy's Local Extrema Conjectures" on arXiv. He cites the Zenodo paper as "an
AI-assisted proof" of Erdős's conjecture that "relied crucially" on his earlier work. He describes its two-step strategy (finite
concentration, then rigidity), says his own paper is inspired by it, and uses the method to classify all positive multiplicative
functions with rare local maxima or minima. This settles two 2001 conjectures of András Sárközy in strong form. Mangerel's
paper itself has no AI-use statement that we found.

We found this through the Oct 10 arXiv sweep. It flagged Mangerel's paper only as "claiming to settle conjectures", because the
AI mention sits in his introduction, not in a disclosure section. We found no press or social-media discussion.

#### Why it matters
It fits the autumn 2026 pattern of Erdős problems and old conjectures settled by frontier models and posted outside arXiv,
anonymously or by unknown authors. It is also an early case of a specialist openly building new human research on an AI-generated
proof. The proof has not been peer-reviewed, and the Lean code assumes the cited analytic theorems as hypotheses.

Sources: [Zenodo: Almost everywhere monotone additive functions (Anonymous, Erdős Problem 1122)](https://doi.org/10.5281/zenodo.22651918); [Erdős Problem #1122 (T. F. Bloom, erdosproblems.com)](https://www.erdosproblems.com/1122); [arXiv 2610.08424: A. P. Mangerel, On Sárközy's Local Extrema Conjectures](https://arxiv.org/abs/2610.08424); [arXiv 2108.12351: A. P. Mangerel, Additive functions in short intervals, gaps and a conjecture of Erdős](https://arxiv.org/abs/2108.12351)

### 2026-09-08: OpenAI claims a Millennium Prize problem: 10,000 AI agents prove forced Navier–Stokes blow-up; priority dispute erupts

*****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/

On 8 Sep 2026 OpenAI released a 166-page paper and a Lean formalisation proving that the 3D incompressible Navier–Stokes equations with a smooth external force can develop a finite-time singularity from smooth initial data. This fits option (C) of Fefferman's official Clay problem statement. About 10,000 agents on an internal model worked for 88 hours. Experts say the unforced problem that matters physically remains open. The result builds on Córdoba and Martínez-Zoroa's techniques, and a bitter priority dispute with Tristan Buckmaster (NYU) and Levent Alpöge (Anthropic) followed.

- Autoformalisation critique (arXiv 2610.08144, 6 Oct; HN 334 points): Alexander Bastounis, Fabian Circelli and Anders C. Hansen argue that resolving ambiguity in natural-language mathematics for a 'semantically faithful' translation into Lean sits arbitrarily high in the SCI/arithmetical hierarchy, so Lean checks of AI autoformalisations 'may offer no confidence in the original NL argument'; they give examples including this release and state that 'the formalised Lean proof does not correspond to the NL proof of blow-up of solutions to the Navier-Stokes equations'. New Scientist covered it on 8 Oct ('OpenAI mistranslated mathematics into code for its Navier-Stokes proof'). OpenAI had not responded publicly as of 8 Oct
- Scale: ~10,000 agents, 88 hours, ~2.7M messages (some reports ~5M), ~130B tokens; Lean formalisation in 17 more hours; led by Sébastien Bubeck
- Claim: a smooth fluid initially at rest, under smooth forcing, develops a singularity in finite time (velocity unbounded, energy bounded)
- Clay Institute (11 Sep): problem has 'apparently been settled' but its process is 'deliberately unhurried'; no prize awarded
- Luis Silvestre: 'The Clay problem is settled, but the main problem for the Navier-Stokes equations is not.'
- Charles Fefferman: 'The heroes of the story… are Córdoba and Martínez-Zoroa'
- Buckmaster and Alpöge (with Matei Coiculescu) released forced blow-up results for IPM, 2D Boussinesq and 3D Euler on 7 Sep, obtained with Claude and Codex and Lean-verified on 22 Aug
- TechCrunch (8 Sep): Buckmaster says Bubeck pressed him to drop Alpöge's credit ('Why would you ruin your career?'); OpenAI says it 'did not see any of their work through any means until released publicly' but could not rule out that de-identified Codex usage data helped indirectly; the week-long OpenAI effort used ~300B output tokens (~$22.5M at Astra list rates)
- Understanding AI (Kai Williams, 10 Sep) and MIT Technology Review (Grace Huckins, 8 Sep): OpenAI offered Buckmaster co-authorship only without Alpöge; he refused
- Córdoba and Martínez-Zoroa (guest post on Tao's blog, 4 Oct): OpenAI's announcement made 'two separate claims', forced Navier–Stokes blow-up and unforced Euler blow-up from smooth data; the first, 'if verified under the stated hypotheses', settles an allowed Millennium alternative but 'is distinct from unforced Navier–Stokes blow-up'; they are 'not in a position to say anything substantive' about how it compares with their cascade method
- Valerio Capraro's viral X post 'OpenAI might have stolen another major proof' (~9 Sep, HN front page) relayed Andreas Thom's separate claim about the Astra soficity result (see 2026-08-01-openai-astra-ten-advances)
- Buckmaster alleged OpenAI may have benefited from his Codex sessions; OpenAI's statements shifted from 'cannot rule out' to denial ('no user inputs past July 3rd')
- The Verge (12 Sep): OpenAI spokesperson Laurance Fauconnet said 'it is impossible for Dr. Buckmaster's Codex prompts over the last two months to have influenced the system in any way, including training'; Buckmaster says Bubeck offered him near 'unlimited compute' and sole authorship of OpenAI's paper if Alpöge (an Anthropic employee) was left out, which he called a 'bribe'; Bubeck told the NYT he offered resources and asked 'how can we have an internal OpenAI project with an Anthropic employee?'
- The Verge (12 Sep): Fauconnet also said 'since the completion of Navier-Stokes we have made substantial progress on another Millennium Prize problem'; social media speculated about the Hodge conjecture (unconfirmed), and rumours said Anthropic was close to one too
- IEEE Spectrum (Oct 5, 'OpenAI's Navier–Stokes claim sparks a reckoning among mathematicians'), reporting from the Heidelberg Laureate Forum (from Sept 13): Peter Scholze: 'They're really just solving these difficult mathematical problems as benchmarks, as some kind of PR stunt'; Geordie Williamson: 'I think that OpenAI behaved extremely poorly, and that we should acknowledge that in the community'; Michael Harris: reports 'depict OpenAI as bullying and disrupting disciplinary norms'; Jacob Tsimerman: capabilities rose 'even faster than many people, including myself, expected'. The claim was still described as pending verification
- New Scientist (8 Oct) covered the Bastounis–Circelli–Hansen critique under the headline 'OpenAI mistranslated mathematics into code for its Navier-Stokes proof' (paywalled, not read). See also 2026-07-28-lean-kernel-soundness-bug-collatz for summer 2026 Lean kernel bugs that let false proofs pass
- Scientific American (Joseph Howlett, Oct 13): several groups of experts working through the paper told the magazine that the proof does not follow the 'Spanish program' of Diego Córdoba and Luis Martínez-Zoroa, which OpenAI officials had 'clearly and repeatedly stated' it did on Sept 8. Gonzalo Cao-Labora (EPFL): 'When you go to the details, it is a different approach.' Both use the same loophole in the Clay problem statement (an unphysical external force), but OpenAI's proof builds the force 'in a totally distinct way', so its similarities to Buckmaster and Alpöge's progress are 'superficial'. Some experts say a smaller team of agents did follow the program for an intermediate result. Camillo De Lellis (IAS) on the dispute: 'I honestly don't see a factual resolution of this.' The article gives the swarm as about 10,000 copies, 88 hours and 130 billion tokens
- The Decoder (Manuel Uth, Oct 17): relays Scientific American and adds OpenAI researcher Noam Brown on Dwarkesh Patel's channel: nobody knows what the ~10,000-agent swarm actually contributed; Brown estimates less than 10% of the success came from the multi-agent setup, crediting mainly the base model's ability to work long and in parallel

#### What happened
OpenAI ran a massive swarm of agents on the forced Navier–Stokes blow-up problem, starting 1 Sep on a model in training since 28 Aug. It announced a complete proof with Lean code on 8 Sep. A day earlier, Buckmaster and Alpöge had released related forced-blow-up results for Euler-type equations using Claude and Codex, and Buckmaster accused OpenAI of rushing after learning of their work. Critics note that the official problem statement allows forcing (option C), but that experts regard the unforced question as the real open problem. Wikipedia now hosts a separate article on the priority controversy.

#### The authorship dispute (added 2026-09-29, verification pass)
Per Fortune's timeline and Scientific American:
- **2026-08-15:** Tristan Buckmaster and Levent Alpöge (Anthropic) privately proved that the Euler equations (Navier–Stokes without viscosity) can blow up. They built on the forcing methods of Diego Córdoba and Luis Martínez-Zoroa.
- **2026-08-15 to 08-22:** word of this unpublished work reached OpenAI. Sébastien Bubeck's math team then produced the proof extending it to the forced Navier–Stokes equations.
- **2026-09-03 to 09-06:** according to Buckmaster, OpenAI offered him sole authorship of a paper crediting OpenAI's model, on condition that Alpöge be removed because of his Anthropic affiliation. Buckmaster says Bubeck asked "Why would you ruin your career?" Buckmaster also raised the possibility that OpenAI's model had seen his Codex session drafts.
- **2026-09-08:** Buckmaster posted his statement the night OpenAI announced its result. Bubeck replied "We did not use their prompt or models or proof" and called the account "false and inflammatory". OpenAI pledged not to claim the Clay prize, and later recruited nine mathematicians to referee its math claims.
This is the first public priority and misconduct dispute between frontier labs over a mathematical result.

#### Why it matters
It is the first credible AI claim on a Clay Millennium Prize problem, even if only a technically permitted variant. It also crystallised disputes over credit, data provenance from AI products, and how AI labs announce results, which culminated in the Fields Medallists' open letter three days later.

Sources: [IEEE Spectrum: OpenAI's Navier–Stokes claim sparks a reckoning among mathematicians (Oct 5)](https://spectrum.ieee.org/millennium-prize-ai); [TokenPost: OpenAI announces Navier-Stokes proof as Clay Prize review continues](https://www.tokenpost.com/news/technology/26880); [Sebastien Bubeck on X: allegations are "false and inflammatory"](https://x.com/SebastienBubeck/status/2097214122471432349); [OfficeChai: Bubeck says he tried to coordinate release with Buckmaster & Alpöge](https://officechai.com/ai/openais-sebastien-bubeck-says-he-tried-to-coordinate-release-of-navier-stokes-related-proofs-with-buckmaster-alpoge-but-was-rebuffed/); [OpenAI: Navier–Stokes solution](https://openai.com/index/navier-stokes-solution/); [Quanta: AI has solved one of math's $1 million Millennium Prize problems](https://www.quantamagazine.org/ai-has-solved-one-of-maths-1-million-millennium-prize-problems-20260908/); [Scientific American: Did OpenAI solve the wrong Navier–Stokes problem?](https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/); [Terence Tao: finite-time blowup with smooth forcing (Buckmaster–Alpöge–Coiculescu)](https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/); [Fortune: OpenAI says it cracked Navier–Stokes; Buckmaster accusation](https://fortune.com/2026/09/08/openai-says-it-cracked-navier-stokes-math-grand-challenge-buckmaster-accusation-cheating-intimidation-tao-lament/); [CNBC: OpenAI claims to have solved 90-year-old Navier–Stokes problem in 88 hours](https://www.cnbc.com/2026/09/09/openai-navier-stokes-math-problem-solved.html); [Wikipedia: Navier–Stokes priority controversy](https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_priority_controversy); [Scientific American: OpenAI claims blockbuster math breakthrough amid swirl of controversy](https://www.scientificamerican.com/article/openai-claims-blockbuster-math-breakthrough-amid-swirl-of-controversy/); [Alexander Gamburd: The Siren Call of Silicon Leviathan (arXiv 2609.28591, reflective essay)](https://arxiv.org/abs/2609.28591); [London Mathematical Society statement on the Navier–Stokes developments (9 Sep)](https://www.lms.ac.uk/news/navier-stokes-equations-breakthrough); [Córdoba & Martínez-Zoroa (guest post on Tao's blog): On classical solutions and singularity formation in incompressible fluids](https://terrytao.wordpress.com/2026/10/04/on-classical-solutions-and-singularity-formation-in-incompressible-fluids/); [Anima Anandkumar: Stable singularity of the Euler equations on R³ (concurrent unforced result)](https://terrytao.wordpress.com/2026/09/10/stable-singularity-of-the-euler-equations-on-r3/); [Techmeme cluster, 2026-09-08](https://www.techmeme.com/260908/p26); [Startup Fortune: OpenAI recruits nine mathematicians to referee its AI math claims](https://startupfortune.com/openai-recruits-nine-mathematicians-to-referee-its-ais-math-claims/); [OpenAI on X: Navier-Stokes solution announcement](https://x.com/OpenAI/status/2097374640582668336); [Noam Brown on X: OpenAI mathematicians' 'Lee Sedol moment'](https://x.com/polynoamial/status/2097375272387613183); [Tristan Buckmaster on Mastodon: three blow-up results and statement](https://mastodon.social/@tristanbuckmaster/117233413705701198); [Terence Tao on Mathstodon: Alpöge–Buckmaster, a remarkable achievement](https://mathstodon.xyz/@tao/117233527638291447); [Terence Tao on Mathstodon: open problems as a non-renewable resource (thread)](https://mathstodon.xyz/@tao/117204929023813310); [TechCrunch: OpenAI fought dirty on career-making math problem, says NYU mathematician](https://techcrunch.com/2026/09/08/openai-fought-dirty-on-career-making-math-problem-says-nyu-mathematician/); [MIT Technology Review: What OpenAI's latest controversy tells us about the future of math](https://www.technologyreview.com/2026/09/08/1143747/what-openais-latest-controversy-tells-us-about-the-future-of-math/); [Understanding AI: OpenAI spent millions to solve this famous math problem — mathematicians are furious](https://www.understandingai.org/p/openai-spent-millions-to-solve-this); [Valerio Capraro on X: 'OpenAI might have stolen another major proof' (Thom/soficity)](https://x.com/ValerioCapraro/status/2097791836269977996); [Hacker News: OpenAI might have stolen another major proof](https://news.ycombinator.com/item?id=49638353); [The Verge (8 Sep): Drama swirls around OpenAI's legendary mathematical milestone](https://www.theverge.com/ai-artificial-intelligence/991710/openai-navier-stokes-solution); [The Verge (9 Sep): OpenAI's sly mathematical breakthrough sends a chill through academia](https://www.theverge.com/ai-artificial-intelligence/992953/openai-math-millennium-prize-navier-stokes); [The Verge (12 Sep): OpenAI just wants to win (Buckmaster and Thom interviews)](https://www.theverge.com/ai-artificial-intelligence/994255/openai-millennium-prize-problem-tristan-buckmaster-competition); [The Verge: All the drama around AI's takeover of mathematics (storystream, updated Oct 5)](https://www.theverge.com/ai-artificial-intelligence/1004933/ai-math-openai-breakthrough-solution); [Quanta (Oct 5): Is AI the End of Math As We Know It?](https://www.quantamagazine.org/is-ai-the-end-of-math-as-we-know-it-20261005/); [Bastounis, Circelli & Hansen: Navier–Stokes lost in translation (arXiv 2610.08144)](https://arxiv.org/abs/2610.08144); [HN discussion: Navier–Stokes Lost in Translation (334 pts)](https://news.ycombinator.com/item?id=49994145); [New Scientist: OpenAI mistranslated mathematics into code for its Navier-Stokes proof (8 Oct; paywalled, not read)](https://www.newscientist.com/article/2592824-openai-mistranslated-mathematics-into-code-for-its-navier-stokes-proof/); [Scientific American: OpenAI's blockbuster Navier-Stokes proof isn't what we thought it was (Joseph Howlett, Oct 13)](https://www.scientificamerican.com/article/openais-blockbuster-navier-stokes-proof-isnt-what-we-thought-it-was/); [The Decoder: Mathematicians say OpenAI doesn't understand its own Navier-Stokes proof (Oct 17)](https://the-decoder.com/mathematicians-say-openai-doesnt-understand-its-own-navier-stokes-proof/); [Dwarkesh Patel: OpenAI researcher on agent swarms & recursive self-improvement (Noam Brown; cited at 5:35)](https://youtu.be/6AgOfiZOWiY?t=335)

### 2026-09-08: AlphaGenome Atlas predicts the effect of all ~9 billion possible single-letter human DNA variants

***; Google DeepMind; Science & math; confidence medium; https://postcutoff.com/e/2026-09-08-alphagenome-atlas/

On 8 Sep 2026 DeepMind released AlphaGenome Atlas: predictions for all ~9 billion possible single-nucleotide variants in the human genome (~1 PB of data). A new variant-impact score reportedly 'more than doubles' rare-disease variant identification versus the previous standard, and collaborators experimentally confirmed variants in unsolved rare-disease cases.

- ~9 billion variants, ~1 petabyte of predictions
- New AVI score: 'more than doubles' rare-disease variant identification (company claim)
- Collaborators verified variants in previously unsolved rare-disease cases

#### What happened
DeepMind pre-computed AlphaGenome predictions for every possible single-letter change in the human genome and released them as an atlas for clinicians and researchers.

#### Why it matters
Like the AlphaFold database for proteins, it turns a model into a lookup resource that could speed up rare-disease diagnosis.

Sources: [Fortune: Google DeepMind AI predictions for 9 billion mutations in the human genome](https://fortune.com/2026/09/08/google-deepmind-ai-predictions-9-billion-mutation-human-genome/); [DeepMind: AlphaGenome](https://deepmind.google/blog/alphagenome-ai-for-better-understanding-the-genome/); [Google DeepMind: AlphaGenome Atlas: a predictive map of every possible DNA letter change in the human genome](https://deepmind.google/blog/alphagenome-atlas-a-predictive-map-of-every-possible-dna-letter-change-in-the-human-genome/); [Google blog: AlphaGenome Atlas](https://blog.google/innovation-and-ai/models-and-research/google-deepmind/alphagenome-atlas/)

### 2026-09-09: Pierce–Birkhoff conjecture (1956) disproved by a multi-agent GPT + Claude harness; counterexamples Lean-verified

****; University of Chicago, Lek-Heng Lim, Zehua Lai, Junyu Ren, OpenAI, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-09-09-pierce-birkhoff-conjecture-disproved/

On 9 Sep 2026 Zehua Lai, Lek-Heng Lim and Junyu Ren posted a counterexample to the Pierce–Birkhoff conjecture (Birkhoff and Pierce, 1956). It is a continuous piecewise-quadratic function on R^30 that is not a finite max–min combination of polynomials. The counterexample came out of a multi-agent harness that chained GPT-5.6, GPT-6, Claude Opus 5 and Claude Fable 5.1. The authors say no single model found it. Counterexamples in dimensions 5, 6, 7, 30 and 72 are formalised in Lean.

- Pierce–Birkhoff conjecture (1956): every continuous piecewise-polynomial function on R^n is a finite lattice (max–min) combination of polynomials; previously known only for n ≤ 2
- arXiv 2609.10420 (7 pages): an explicit counterexample on V = S²(R⁵) × S²(R⁵), dimension 30, piecewise quadratic, chosen because it 'can be readily checked by hand'
- The same setup also produced counterexamples in dimensions 5, 6, 7, 22, 24 and 72; those for 5, 6, 7, 30 and 72 are Lean-formalised (GitHub 7pocheR/Pierce-Birkhoff)
- The authors had earlier proved the conjecture for splines (hyperplane partitions) 'as a purely human endeavor with no AI usage'
- AI setup: an orchestrator agent in Codex and Claude Code assigned proof searches, counterexample searches and reviews across GPT (5.6, 6) and Claude (Opus 5, Fable 5.1) models; 'The counterexample in this article is notably not found with any single large language model (we tried doing so without success)'
- Per Section 5, the dimension-72 matrix-pair construction and the argument excluding representations of arbitrary degree 'emerged from GPT investigations'; the 30-dimensional version and its simplification came with further ChatGPT help
- Side result: combined with the authors' earlier work, splines ⊊ pure transformers ⊊ semialgebraic splines, so some semialgebraic splines cannot be computed by a ReLU pure transformer
- AI disclosure (abstract, arXiv 2609.10420): 'The counterexample was found with the assistance of our multi-agent, multi-model setup that chains together GPT 5.6, GPT 6, Claude Opus 5, and Claude Fable 5.1.'

#### What happened
Lek-Heng Lim's group had proved the conjecture for splines by hand. They then pointed a custom orchestrated team of GPT and Claude agents at the general case. The agents found counterexamples in several dimensions. The authors picked a 30-dimensional one that is easy to verify, formalised several in Lean and wrote the paper.

#### Why it matters
Pierce–Birkhoff is a classic problem in real algebraic geometry and ordered rings, open for 70 years. It is also one of the first documented cases where the authors say a result needed several labs' models working together, and it has a direct consequence for the expressivity of transformers.

Sources: [arXiv 2609.10420: Pierce-Birkhoff conjecture is false (Lai, Lim, Ren)](https://arxiv.org/abs/2609.10420); [GitHub: 7pocheR/Pierce-Birkhoff (Lean formalisation)](https://github.com/7pocheR/Pierce-Birkhoff)

### 2026-09-09: Google and NASA JPL release MAPL-EMIT, an AI model that maps methane plumes worldwide from the EMIT space instrument (PNAS)

***; Google Research, NASA JPL; Science & math; confidence high; https://postcutoff.com/e/2026-09-09-google-nasa-jpl-mapl-emit-methane/

On Sept 9, 2026 Google Research and NASA's Jet Propulsion Laboratory published MAPL-EMIT (Methane Analysis and Plume Localization with EMIT) in PNAS. The deep-learning model detects, quantifies and localizes methane plumes in hyperspectral data from NASA's EMIT instrument on the ISS. It finds 50% more plumes than human experts and more than 23,000 additional plumes globally, and the database, models and code are open.

- Trained on 3.6 million physics-simulated methane plumes
- Detects 50% more plumes than human experts; 23,000+ additional plumes identified globally
- Found plumes at 24 of the world's 25 largest-emitting landfills
- Paper: PNAS, DOI 10.1073/pnas.2612145123
- Released: global plume database and app on Google Earth Engine, models on Kaggle, inference code on GitHub (google-research/mapl)

#### What happened
EMIT was built to map surface minerals in dust-source regions, but its hyperspectral bands also pick up methane's spectral signature.
Google trained a detector on simulated plumes and ran it over the whole archive.

#### Why it matters
Methane is a strong short-term warming gas (Google cites about 30x the warming potential of CO2 over 100 years). Automated global
plume mapping with open data lets regulators and companies find and fix leaks, landfills and super-emitters at scale.

Sources: [Google blog: A new deep learning model maps global methane emissions from space](https://blog.google/innovation-and-ai/models-and-research/google-research/mapping-global-methane-emissions-from-space/); [PNAS paper (DOI 10.1073/pnas.2612145123)](https://www.pnas.org/doi/10.1073/pnas.2612145123); [GitHub: google-research/mapl](https://github.com/google-research/mapl); [Earth Engine app: MAPL-EMIT plumes](https://nature-trace.projects.earthengine.app/view/mapl-emit); [Unite.AI: Google and NASA JPL unveil AI model mapping global methane plumes](https://www.unite.ai/google-and-nasa-jpl-unveil-ai-model-mapping-global-methane-plumes/)

### 2026-09-10: First Phase III trial of a generative-AI-discovered drug doses first patient (Insilico's rentosertib)

****; Insilico Medicine; Science & math; confidence high; https://postcutoff.com/e/2026-09-10-insilico-rentosertib-phase-3/

On 2026-09-10 Insilico Medicine dosed the first patients in GENESIS-IPF-3, billed as the world's first Phase III trial of a drug whose target and molecule were discovered with generative AI: rentosertib, a TNIK inhibitor for idiopathic pulmonary fibrosis, tested in 320 patients at 47 Chinese centers over 52 weeks.

- First patients dosed 2026-09-10 at Peking Union Medical College Hospital and Shanghai Pulmonary Hospital
- Randomized, double-blind, placebo-controlled; 320 participants; 47 centers in China; once daily for 52 weeks
- Primary endpoint: annual rate of FVC decline over 52 weeks; key secondary: time to first disease-progression event
- Phase IIa (Nature Medicine, 2025): 60 mg QD arm showed mean FVC +98.4 mL at 12 weeks, dose-dependent trend
- Mechanism: TNIK inhibition (target also identified by Insilico's AI platform)
- Pipeline context: on 2026-01-23 the FDA cleared Insilico's IND for ISM8969, an oral brain-penetrant NLRP3 inhibitor for Parkinson's disease discovered and optimized with its Chemistry42 generative-chemistry engine (Phase I in healthy volunteers; co-developed 50/50 with Hygtia Therapeutics, which also won China NMPA IND clearance)

#### What happened
Insilico's rentosertib is the furthest-advanced drug in which both the target and the molecule came from generative AI. Phase III is the final stage before regulatory approval.

#### Why it matters
If positive (results likely 2027+), it would be the first approved generative-AI-discovered drug — the key proof point for AI drug discovery's promise to cut time and cost.

Sources: [Insilico: first patient dosed in GENESIS-IPF-3](https://insilico.com/news/isn1009261-insilico-medicine-doses-first-patient-genesis-ipf-3); [PR Newswire: Insilico initiates Phase III trial for rentosertib](https://www.prnewswire.com/news-releases/insilico-initiates-phase-iii-clinical-trial-for-rentosertib-its-ai-empowered-tnik-inhibitor-for-idiopathic-pulmonary-fibrosis-302819553.html); [EurekAlert: Nature Medicine publishes rentosertib Phase IIa results (June 2025)](https://www.eurekalert.org/news-releases/1086096); [Drug Target Review: Insilico begins Phase III of AI-designed drug](https://www.drugtargetreview.com/insilico-medicine-launches-phase-iii-trial-of-ai-designed-rentosertib-drug/2135890.article); [Insilico: IND approval from FDA for ISM8969 (Jan 23, 2026)](https://insilico.com/news/az7xmeem81-insilico-medicine-receives-ind-approval); [EurekAlert: ISM8969/HT-001 receives NMPA IND clearance](https://www.eurekalert.org/news-releases/1138579)

### 2026-09-10: GPT-6 Astra's Epoch AI run adds more Lean-checked results: Dittert conjecture proved, Ibragimov–Iosifescu and eternal-domination conjectures disproved

***; OpenAI, Epoch AI; Science & math; confidence medium; https://postcutoff.com/e/2026-09-10-astra-leanopenproblems-september-results/

After the Köthe disproof, the same September 2026 Epoch AI run of pre-release GPT-6 Astra over the Formal Conjectures collection produced more machine-written Lean results, published by Tom Adamczewski: a proof of the full Dittert permanent conjecture, a counterexample to the Ibragimov–Iosifescu φ-mixing CLT conjecture, a disproof of the strong n-conjecture for n=4, and a 243-vertex graph refuting the Gamma–Theta eternal-domination conjecture (arXiv 2609.11500, with William Klostermeyer). Most results have not had independent expert review.

- Setting: Epoch AI's LeanOpenProblems harness; pre-release GPT-6 Astra tried each research-open Formal Conjectures statement once, autonomously (see the Köthe entry)
- Dittert conjecture: φ(A) ≤ 2 − n!/n^n for nonnegative n×n matrices with entries summing to n, with equality only for the all-1/n matrix. Lean proof passed the Comparator check (repo tadamcz/dittert); the exposition is not independently reviewed. Humans had earlier proved n ≥ 17 (arXiv 2606.01531) and n = 16 (arXiv 2607.19439, GPT-5.6 Sol-assisted)
- Ibragimov–Iosifescu conjecture (Ibragimov, 1971): disproved with a strictly stationary φ-mixing counterexample; a 13,047-line Lean proof, 'Lean-checked, statement unaudited', announced 5 Sep 2026 (repo tadamcz/phi-mixing-clt)
- Strong n-conjecture, n = 4: disproved in Lean with extra SymPy arithmetic checks (repo tadamcz/n-conjecture-strong)
- Eternal domination: 243-vertex graph with γ(G) = γ∞(G) < θ(G), refuting the Gamma–Theta conjecture. Tom Adamczewski & William F. Klostermeyer, arXiv 2609.11500, 10 Sep 2026
- The repositories say they were 'machine-written by AI assistants at the direction of Tom Adamczewski'
- Same run, Wikipedia collection ('wikipedia-vega-1000usd', all 222 research-open statements, $1,000 and 96 h each): disproof of Smale's mean value conjecture for K = 1 (repo tadamcz/mean-value-problem; own entry 2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra). Epoch's LeanOpenProblems README publishes no overall solved count (checked 16 Oct)

#### What happened
Epoch AI ran pre-release GPT-6 Astra once on each research-open statement in the Formal Conjectures collection. Besides Köthe, several more outputs were packaged as Lean repositories by Tom Adamczewski in the first half of September 2026. For the graph-theory counterexample, domination expert William Klostermeyer co-wrote an arXiv paper.

#### Why it matters
Autonomous formal proof search now turns out a steady stream of mid-level resolved conjectures, not one-off headlines. The bottleneck is shifting to human auditing of whether the formal statements are the intended ones.

Sources: [arXiv 2609.11500: A Counterexample to an Eternal Domination Conjecture](https://arxiv.org/abs/2609.11500); [GitHub: tadamcz/dittert](https://github.com/tadamcz/dittert); [GitHub: tadamcz/phi-mixing-clt (Ibragimov–Iosifescu)](https://github.com/tadamcz/phi-mixing-clt); [GitHub: tadamcz/n-conjecture-strong](https://github.com/tadamcz/n-conjecture-strong); [VibeMathed: Ibragimov–Iosifescu conjecture status](https://vibemathed.com/problem/ibragimov-iosifescu-varphi-mixing-clt-conjecture); [Wikipedia: List of mathematical discoveries by artificial intelligence](https://en.wikipedia.org/wiki/List_of_mathematical_discoveries_by_artificial_intelligence)

### 2026-09-10: SAIR's IGP24 challenge: all 25,000 transitive groups of degree 24 realized as Galois groups; Klüners and Malle win without AI against mostly AI-assisted teams

***; SAIR Foundation, LMFDB, Paderborn University, RPTU Kaiserslautern; Science & math; confidence medium; https://postcutoff.com/e/2026-09-10-igp24-inverse-galois-degree-24-humans-win/

The third SAIR Foundation competition, IGP24 (announced by Terence Tao on 16 June 2026, run with the LMFDB), asked for explicit degree-24 polynomials realizing each of the ~25,000 transitive permutation groups of degree 24 as a Galois group over ℚ. Paderborn University reported on 10 Sep 2026 that all 25,000 groups had been solved by July and that Jürgen Klüners (Paderborn) and Gunter Malle (RPTU Kaiserslautern) won, finding more than 143,000 relevant polynomials "without using AI", while almost all other participants used it. The final status is from the university's press release; the organizers' own summary was not read.

- Announcement: Terence Tao's blog, 16 June 2026, 'Third SAIR competition: inverse Galois challenge'; organizers John Jones, Jen Paulhus, David Roe, Andrew Sutherland and Tao (per search summaries of the post); verification by Magma and PARI/GP rather than Lean
- Scope: 25,000 groups and 165,836 signatures; at the start the LMFDB held concrete polynomials for only 286 groups and 622 signatures (Paderborn)
- Outcome: 'By July, all 25,000 groups had been solved', so the degree-24 inverse Galois problem was fully resolved for these groups; participants submitted millions of integer polynomials (Paderborn)
- Winners: Jürgen Klüners (Paderborn) and Gunter Malle (RPTU Kaiserslautern), more than 143,000 relevant polynomials
- AI: 'Almost all the other participants relied on the technology. Nevertheless, the team from Paderborn and Kaiserslautern managed to win the competition without using AI'; they used AI only to help write submission scripts (Paderborn)
- Context: Tao's 9 Oct Math 2.0 lecture cites IGP24 as an example of new experimental mathematics

#### What happened
In June 2026 Terence Tao announced the third SAIR Foundation competition, run with the L-functions and Modular Forms Database (LMFDB): find explicit polynomials whose Galois groups are each of the transitive permutation groups of degree 24, almost the first open degree. Submissions were verified by computer algebra (Magma, PARI/GP).

Paderborn University's press release of 10 Sep 2026 says the problem was finished: "By July, all 25,000 groups had been solved", from a starting point of polynomials for only 286 groups in the LMFDB. Jürgen Klüners and Gunter Malle won with more than 143,000 relevant polynomials. The release frames it as humans against AI: "Almost all the other participants relied on the technology. Nevertheless, the team from Paderborn and Kaiserslautern managed to win the competition without using AI." They used AI only for scripts that eased submission.

The organizers' final report and the competition leaderboard were not read for this entry, so the 'all solved' claim rests on the university release.

#### Why it matters
IGP24 is one of the large crowdsourced experiments Tao has promoted as a new way to do mathematics with AI in the loop. Its result is a useful counterpoint to the many AI-credited results of autumn 2026: in a computational search problem, deep human expertise in Galois theory beat AI-assisted teams.

Sources: [Paderborn University: German mathematicians come out on top against AI in an international competition (10 Sep 2026)](https://www.uni-paderborn.de/en/news-item/161090); [Terence Tao: Third SAIR competition: inverse Galois challenge (16 June 2026)](https://terrytao.wordpress.com/2026/06/16/third-sair-competition-inverse-galois-challenge/); [SAIR competition page: IGP24](https://competition.sair.foundation/competitions/igp24/overview)

### 2026-09-10: Irrationality measure of π lowered to 7.101862832357, the first improvement on Zeilberger–Zudilin (2019), with GPT-5.6 Sol and GPT-6 Astra; four weeks later OpenAI claimed the exact value 2

***; OpenAI, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-09-10-pi-irrationality-measure-bai-gpt/

On 10 Sep 2026 Yufei Bai posted arXiv 2609.11276, proving that the irrationality measure of π is below 7.101862832357, down from the 7.103205334137 of Zeilberger and Zudilin (arXiv 1912.06345), by introducing two independent exponents into their contour integral and setting both to 1857/2785. The abstract carries a "Generative-AI disclosure": OpenAI's GPT-5.6 Sol and GPT-6 Astra "assisted throughout the research and preparation of this article, including parameter exploration, proof development and checking, computation, and manuscript drafting and revision". The gain is small (about 0.019%), and the paper is an unrefereed preprint. A French independent researcher, David Niedbala Giraudin, then carried the same "displaced parameter" mechanism to ζ(3) and ζ(2), with Claude as technical assistant. OpenAI's 6 Oct math release then claimed μ(π) = 2 exactly (family 017, Lean-formalized, not yet independently reviewed).

- Result: μ(π) < 7.101862832357 (Bai, arXiv 2609.11276, v1 10 Sep, v2 11 Sep 2026); previous record μ(π) ≤ 7.103205334137… (Zeilberger & Zudilin, arXiv 1912.06345), which refined Salikhov's 7.606308…
- Improvement: 'more than 0.001342501780', about 0.01890%; the point A1 = A2 = 1857/2785 is 'a strict two-dimensional local minimizer' of the auxiliary bound function in its chamber, 'not a claim that the point is a global optimizer among all constructions'
- Disclosure (abstract): 'OpenAI's GPT-5.6 Sol and GPT-6 Astra assisted throughout the research and preparation of this article, including parameter exploration, proof development and checking, computation, and manuscript drafting and revision. Responsibility for all mathematical claims and for the final text rests with the author.' Acknowledgments thank Bin Dong and Haocheng Ju of Peking University 'for their guidance'
- Follow-up 1: Niedbala Giraudin, arXiv 2609.37459 (28 Sep, v3 5 Oct): one closed-form law μ(a0,a1,b) for the Salikhov–Zeilberger–Zudilin–Bai family that reproduces all three records, with computational evidence that Bai's point is the global optimum of the family (statements labelled PROVED / VERIFIED / CONJECTURE); 'Technical assistance: Claude (Anthropic)'
- Follow-up 2 (ζ(3)): Niedbala Giraudin, arXiv 2609.26980 (22 Sep): μ(ζ(3)) < 5.5138800, the first improvement on Rhin–Viola's 5.513891 (2001), by moving their parameter point 'the mechanism … recently used by Bai for π'; the author calls the gain of 1.07·10⁻⁵ 'by itself … of little interest'. Acknowledgements: 'This work was carried out with the assistance of an AI system (Claude, Anthropic); every claim in it has been checked by independent computation'; ancillary C verifier (verify_zeta3.c)
- Follow-up 3 (ζ(2)): Niedbala Giraudin, arXiv 2610.02912 (2 Oct): μ(ζ(2)) < 5.0193784, below the Lean-checked 5.0495243 of Sept 2026; 'Technical assistance: Claude (Anthropic)' (see 2026-09-25-zeta2-irrationality-measure-claude-lean)
- Overtaken by a stronger claim: OpenAI's 6 Oct 2026 catalogue, family 017 'The irrationality exponent of π is 2' (preprint dated 24 Sep 2026, Lean formalization of the exponent), proves 'the conjecture that the irrationality exponent of π is 2' and hence convergence of the Flint–Hills series Σ 1/(n³ sin² n); not yet independently reviewed (see 2026-10-06-openai-math-release-722-manuscripts)
- Spin-off: a pseudonymous 'Dan Brown' preprint (dated 7 Oct, footnote 'Author of The Da Vinci Code', posted on GitHub Pages) adapts the interpolation-determinant argument of the μ(π) = 2 paper to claim μ(log α) = 2 for every positive rational α ≠ 1, so μ(log 2) = 2 (previous record 3.5746, Marcovecchio), formalized in Lean 4 over Mathlib; HN thread 21 points; author identity unknown, unreviewed
- Verification: unrefereed preprints; the Niedbala Giraudin papers ship standalone verification programs; no expert reaction found as of 11 Oct 2026

#### What happened
The irrationality measure of a number says how well it can be approximated by fractions. For π only upper bounds are known, and they
come from carefully tuned integrals: Hata, then Salikhov (7.606…), then Zeilberger and Zudilin (7.103205…, posted in December 2019).
Yufei Bai added two free exponents to the Zeilberger–Zudilin integral and found a better parameter point, giving μ(π) < 7.101862832357.
The abstract itself carries a generative-AI disclosure naming GPT-5.6 Sol and GPT-6 Astra.

Within weeks David Niedbala Giraudin, an independent researcher in France, applied the same idea, displacing a classical parameter
point instead of changing the construction, to ζ(3). That gave the first improvement on Rhin and Viola's 2001 record, though by only
about 10⁻⁵. He did the same for ζ(2) and also wrote a note unifying the three π records in one closed-form law. All three papers
credit Claude (Anthropic).

#### Why it matters
Records for classical constants such as π and ζ(3) had stood for years. They fell to small, computation-heavy parameter searches of the
kind AI assistants are good at, and the authors say so. The gains are tiny, and the authors themselves say they matter less than what
they show: the classical parameter choices were not optimal. For π the episode was short: four weeks after Bai's bound, OpenAI claimed the exact value 2, which, if it holds up, ends the race of upper bounds altogether.

Sources: [Bai: The irrationality measure of π is at most 7.101862832357 (arXiv 2609.11276)](https://arxiv.org/abs/2609.11276); [Zeilberger & Zudilin: The irrationality measure of π is at most 7.103205334137… (arXiv 1912.06345)](https://arxiv.org/abs/1912.06345); [Niedbala Giraudin: A closed-form law for the Salikhov–Zeilberger–Zudilin–Bai family (arXiv 2609.37459)](https://arxiv.org/abs/2609.37459); [Niedbala Giraudin: A new upper bound for the irrationality exponent of ζ(3) (arXiv 2609.26980)](https://arxiv.org/abs/2609.26980); [Niedbala Giraudin: A new upper bound for the irrationality exponent of ζ(2) (arXiv 2610.02912)](https://arxiv.org/abs/2610.02912); [OpenAI math catalogue (CONTENTS.md): 017 The irrationality exponent of π is 2](https://github.com/openai/math/blob/main/CONTENTS.md); ['Dan Brown': The irrationality measure of the logarithm of a rational number is 2 (PDF, 7 Oct)](https://jdb19937.github.io/log-irrationality-measure/logmeasure.pdf); [Hacker News: The logarithms of rational numbers have irrationality exponent 2](https://news.ycombinator.com/item?id=50027884)

### 2026-09-11: Fields Medallists' open letter 'A Severe Misalignment of AI in Mathematics' criticises labs' race for famous problems

***; mathandai.org; Science & math; confidence high; https://postcutoff.com/e/2026-09-11-fields-medalists-letter-ai-mathematics/

On 11 Sep 2026 about 25 Fields Medallists, including Terence Tao, Peter Scholze, Maryna Viazovska and Pierre Deligne, published an open letter criticising AI labs for treating famous open problems as marketing targets. It cited the Navier–Stokes announcement and the Jacobian-conjecture tweet. It does not call for a ban on AI in mathematics.

- Signatories: 25 Fields Medallists per Scientific American (Wikipedia lists 26)
- Concerns: announcement by press release or tweet, credit to prior human work, data provenance, and incentives distorting mathematics
- Signatures grew to 7,000+ by 19 Sep 2026 (Po-Shen Loh); the separate Leiden Declaration (June 2026) had 4,000+
- Context: an Aug 2026 arXiv essay 'The crisis of AI-generated mathematics' (2608.02859) argued for total opposition; the letter is more moderate

#### What happened
Three days after OpenAI's Navier–Stokes announcement, the mathematical establishment's most decorated members publicly objected to how AI companies pursue and publicise famous problems.

#### Why it matters
It marked open tension between AI labs and the mathematical community at the moment AI began producing major results, and shaped norms for credit and verification.

Sources: [Grant Sanderson (guest post on Terence Tao's blog, Sept 18, 2026): If math is more than proof, we need to better celebrate the rest of it](https://terrytao.wordpress.com/2026/09/18/if-math-is-more-than-proof-we-need-to-better-celebrate-the-rest-of-it/); [David Bessis: The fall of the theorem economy (Apr 21, 2026)](https://davidbessis.substack.com/p/the-fall-of-the-theorem-economy); [Terence Tao: A severe misalignment of AI in mathematics](https://terrytao.wordpress.com/2026/09/11/a-severe-misalignment-of-ai-in-mathematics/); [Scientific American: 25 winners of math's Nobel decry the AI invasion of their discipline](https://www.scientificamerican.com/article/25-winners-of-maths-nobel-prize-decry-the-ai-invasion-of-their-discipline/); [The crisis of AI-generated mathematics (arXiv 2608.02859)](https://arxiv.org/abs/2608.02859); [mathandai.org: A Severe Misalignment of AI in Mathematics (declaration text, signatories)](https://mathandai.org/); [Terence Tao on Mathstodon announcing the declaration](https://mathstodon.xyz/@tao/117253629967855195); [Timothy Gowers: Why I didn't sign the Fields medallists' letter](https://terrytao.wordpress.com/2026/09/17/why-i-didnt-sign-the-fields-medallists-letter/)

### 2026-09-11: Medvedev's logic of finite problems (1962) shown undecidable; key idea from ChatGPT Sol 5.6, central argument checked in Lean by Claude Opus 5

***; Rodrigo Nicolau Almeida, Søren Brinck Knudstorp, OpenAI, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-09-11-medvedev-logic-undecidable/

Rodrigo Nicolau Almeida and Søren Brinck Knudstorp proved that Medvedev's logic of finite problems, a well-known superintuitionistic logic, is undecidable, which settles what they call a longstanding open problem (arXiv 2609.13359, 11 Sep 2026). They reduce the periodic tiling problem to it. The authors 'do not take any credit' for the key ideas: ChatGPT Sol 5.6 obtained them on 4 Sep 2026, and Claude Opus 5 produced a Lean verification of the central argument. All prompts and model outputs are published.

- Medvedev introduced the logic of finite problems in 1962 (Dokl. Akad. Nauk 142); whether it is decidable was a longstanding open question
- Main theorem: Medvedev's logic ML is undecidable, via a reduction from periodic tiling to non-theoremhood; similarly Skvortsov's logic of infinite problems is undecidable, and the two logics are separated by any aperiodic tiling of the plane
- Methodology §7: 'the key idea for the undecidability of ML – namely the construction of the tiling posets P_W, and the encoding of the reduction – was obtained by prompting ChatGPT Sol 5.6 … whilst the authors assume responsibility for the correctness of the results, they do not take any credit in such ideas'
- The authors rewrote the text themselves and invoke the Leiden Declaration; Claude Opus 5 was used for proofreading and for a Lean verification of the main central argument; prompts, preliminary documents and verification ledgers are public on the first author's website
- Status: preprint (32 pages)
- Follow-up: Han Xiao (arXiv 2609.39577, 30 Sep 2026) proves the logic Cheq is not finitely axiomatizable (a human result 'established before the rise of AI') and that Medvedev's undecidability implies Cheq's; the paper notes Theorem 6.2 'does use the AI-generated proof that Medvedev logic is undecidable'
- AI disclosure (abstract, arXiv 2609.13359): 'The core idea and technical work of the undecidability proof were obtained using ChatGPT Sol 5.6, and formally verified in Lean by Claude Opus 5.' arXiv comment: 'main result obtained via LLM (see methodology section)'

#### What happened
Two logicians, one of whom wrote his PhD on undecidability, suspected a targeted search could settle Medvedev's logic. They prompted ChatGPT Sol 5.6, which returned a tiling-based reduction on 4 September. Claude Opus 5 formalised the central argument in Lean, and the authors wrote the paper and published the whole prompt trail.

#### Why it matters
It answers a well-known question in non-classical logic, and other logicians already build on it: on 30 September Han Xiao derived the undecidability of the logic Cheq from it, citing "the AI-generated proof". It is also a model of transparent disclosure: full prompt and output archives are published alongside the paper and treated like a dataset.

Sources: [arXiv 2609.13359: Medvedev logic is undecidable (Almeida, Knudstorp)](https://arxiv.org/abs/2609.13359); [arXiv 2609.39577: Non-finite axiomatizability and undecidability of Cheq (Han Xiao)](https://arxiv.org/abs/2609.39577)

### 2026-09-14: The k-server conjecture, the 'holy grail' of online algorithms, is proved at Oxford; ChatGPT 6 Astra generalised the authors' k = 3 proof to all k

****; University of Oxford, Christian Coester, Elias Koutsoupias, Marek Zbysiński, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-09-14-k-server-conjecture-proved/

Christian Coester, Elias Koutsoupias and Marek Zbysiński (Oxford) posted a proof of the k-server conjecture (Manasse–McGeoch–Sleator, 1988): the work function algorithm is k-competitive on every metric space (arXiv 2609.15979, 14 Sep 2026). The authors designed the potential function and proved k = 3 without AI. ChatGPT 6 Astra then 'derived an algebraic proof of correctness for any k', which the authors adapted and revised.

- The k-server problem was introduced by Manasse, McGeoch and Sleator in 1988; the paper notes it has repeatedly been called the 'holy grail' of competitive analysis
- Previous best: the work function algorithm is (2k−1)-competitive (Koutsoupias and Papadimitriou, 1995); the conjecture asks for ratio k
- New proof: represent the work function algebraically as a matrix whose column determinants encode work-function values; the amortised analysis uses a potential defined on a larger matrix of coordinate pairs
- AI role (acknowledgments): the k = 3 potential and 'initial proof for three servers … was obtained without AI assistance'; discussions with ChatGPT 5.5 Pro and Gemini 3.1 Pro gave 'a deeper understanding of the potential'; 'ChatGPT 6 Astra subsequently derived an algebraic proof of correctness for any k'; the published proof adapts it, and Astra helped draft some sections
- Discussed on Hacker News (106 points, 15 Sep 2026)
- Status: preprint, not peer-reviewed

#### What happened
The Oxford team first found a new potential function that proved the open three-server case by hand. After refining it in conversations with ChatGPT and Gemini, they gave the reformulation to ChatGPT 6 Astra, which produced an algebraic proof for all k. The authors then recast that proof in a more natural matrix representation.

#### Why it matters
The k-server conjecture is one of the best-known open problems in algorithms, and Koutsoupias co-proved the previous best bound in 1995. Here the AI's contribution is the generalisation step that turned a special case into the full theorem.

Sources: [arXiv 2609.15979: The k-server conjecture is true (Coester, Koutsoupias, Zbysiński)](https://arxiv.org/abs/2609.15979); [Hacker News discussion](https://news.ycombinator.com/item?id=49709129)

### 2026-09-14: Sylvester's 1879 conjecture on sums of two cubes is fully proved: Burungale and Tian settle the last case (p ≡ 8 mod 9) after Hongbo Yin's proof of the 4, 7 cases; both later papers disclose AI use

****; University of Texas at Austin, Chinese Academy of Sciences, Shandong University; Science & math; confidence medium; https://postcutoff.com/e/2026-09-14-sylvester-conjecture-proved-burungale-tian/

On Sept 14, 2026 Ashay Burungale (UT Austin) and Ye Tian (AMSS, Chinese Academy of Sciences) posted "A proof of Sylvester's conjecture" (arXiv 2609.14893). It proves the remaining case p ≡ 8 (mod 9) of Sylvester's 1879 conjecture that every prime p ≡ 4, 7, 8 (mod 9) is a sum of two rational cubes. Hongbo Yin had posted the 4 and 7 cases in May 2026. The paper says ChatGPT and Claude were used for exploratory computations and editing. On Oct 6 Yin posted a much shorter proof of the 8 case, whose details he says were worked out by AI.

- Conjecture (Sylvester, 1879): every prime p ≡ 4, 7, 8 (mod 9) is a sum of two rational cubes. Primes p ≡ 2, 5 (mod 9) are not (Pépin, Lucas, Sylvester). BSD predicts the conjecture because the curve E_p: y² = x³ + p²/4 has root number −1 exactly for these classes
- History: Elkies announced a proof for p ≡ 4, 7 in 1994 but never published details; Dasgupta–Voight proved the 4, 7 cases under an extra cubic-residue condition (2009, 2018); Yin proved a weaker 8-case result (Trans. AMS 2022)
- May 25, 2026: Hongbo Yin (Shandong University), 'A proof of the 4,7 cases of Sylvester's conjecture on cube sums', arXiv 2605.25917. It uses recent progress on full BSD for rank-0 curves and the solution of the Unbounded Denominators Conjecture
- Sept 14, 2026: Ashay Burungale and Ye Tian, 'A proof of Sylvester's conjecture', arXiv 2609.14893 (v2 Sept 15). For p ≡ 8 (mod 9) they prove E_p has analytic rank one, adapting an auxiliary Rankin–Selberg construction from their work on the rank-one converse for CM elliptic curves
- Burungale–Tian AI disclosure: 'Use of artificial intelligence. ChatGPT and Claude tools were used for exploratory computations, editing and reorganization.'
- Oct 6, 2026: Hongbo Yin, 'A concise proof of the 8 case of Sylvester's conjecture', arXiv 2610.08015 (10 pages). It avoids the Gross–Zagier formula and calls Burungale–Tian's proof 'complicated and hard to follow'
- Yin's AI disclosure: 'The ideas are due to the author while the details realization is due to AI. But the output of AI is ugly and not readily readable by human, so the final simplification, reorganization and writing are due to the author. The example in Section 6 is computed by AI.' The AI tool is not named

#### What happened
Sylvester's conjecture goes back to his 1879 work on ternary cubic equations. It says that every prime p ≡ 4, 7 or 8 (mod 9)
can be written as a sum of two rational cubes. In modern terms, the elliptic curve E_p: y² = x³ + p²/4 should have positive
rank, which the Birch and Swinnerton-Dyer conjecture predicts from its root number.

Hongbo Yin (Shandong University) posted a proof of the 4 and 7 cases on arXiv on May 25, 2026 (revised in June), before the
June 2026 cutoff. On Sept 14, 2026 Ashay Burungale and Ye Tian posted "A proof of Sylvester's conjecture", which handles the
last class, p ≡ 8 (mod 9), by proving that E_p has analytic rank one. Together the two papers prove the full conjecture.
Burungale and Tian state that ChatGPT and Claude were used for exploratory computations, editing and reorganization.

On Oct 6 Yin posted a 10-page "concise proof" of the 8 case built on his May paper. He writes that the ideas are his, the
detailed working-out was done by an unnamed AI whose output was "ugly and not readily readable by human", and he rewrote it.

We found this through the arXiv sweep of Oct 10 (Yin's Oct 6 paper). We found no press coverage or expert reactions. Affiliations
from the papers: Burungale, University of Texas at Austin; Tian, Morningside Center / AMSS, Chinese Academy of Sciences.

#### Why it matters
This is one of the oldest named problems in Diophantine number theory, and the main proofs are human-led. AI shows up as a
tool for exploratory computation and, in the shorter follow-up proof, for writing out the details. It shows that by autumn 2026
even researchers proving major results routinely disclose AI help.

Sources: [arXiv 2609.14893: Burungale & Tian, A proof of Sylvester's conjecture](https://arxiv.org/abs/2609.14893); [arXiv 2605.25917: H. Yin, A proof of the 4,7 cases of Sylvester's conjecture on cube sums](https://arxiv.org/abs/2605.25917); [arXiv 2610.08015: H. Yin, A concise proof of the 8 case of Sylvester's conjecture](https://arxiv.org/abs/2610.08015); [Dasgupta & Voight, Sylvester's problem and mock Heegner points (background)](https://ar5iv.labs.arxiv.org/html/1707.05874)

### 2026-09-14: Friedgut's 2004 influential-coalitions conjecture resolved (Chattopadhyay–Gurumukhani, ChatGPT Pro used extensively); Friedgut then has GPT-6 Astra simplify the proof

***; Cornell University, Weizmann Institute of Science, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-09-14-friedgut-coalition-conjecture-resolved/

On Sept 14, 2026 Eshan Chattopadhyay and Mohit Gurumukhani (Cornell) posted a proof (arXiv 2609.16401) of Ehud Friedgut's 2004 conjecture: every monotone Boolean function on the continuous cube [0,1]^n has a coalition of O(n/√log n) coordinates that can force a fixed output with probability 1 − ε. It is the first sublinear bound independent of alphabet size for collective coin flipping. Their AI-use statement says they "used ChatGPT Pro extensively for exploring ideas, and proving several technical parts". On Oct 2 Friedgut posted a note (arXiv 2610.03086) with a simpler proof that he says ChatGPT-6 Astra wrote after two short prompts from him. He credits the breakthrough to Chattopadhyay and Gurumukhani.

- arXiv 2609.16401 (cs.CC, Sept 14, 2026), 'A Resolution of Friedgut's Conjecture on Influential Coalitions', 15 pages; key tools: an encoding relating influence on product spaces to p-biased influence, and Hatami's pseudo-junta theorem (Annals 2012)
- Previously only the Boolean-cube case was known (Kahn–Kalai–Linial 1988, O(n/log n)); no sublinear alphabet-independent bound existed
- C–G AI use statement: 'The authors used ChatGPT Pro extensively for exploring ideas, and proving several technical parts. The authors take full responsibility for the correctness of all results in the paper.'
- arXiv 2610.03086 (Oct 2; revised Sept 30), Friedgut, 'Small Influential Coalitions in [0,1]^n via the Junta Theorem': gets O(n/(ε log n)) using Friedgut's junta theorem instead of Hatami's, plus a direct discretization from [0,1]^n to {0,1}^n
- Friedgut's account: he fed the C–G manuscript to ChatGPT-6 Astra with the prompt 'see whether it can be simplified by using Friedgut's junta theorem instead of Hatami's pseudo-junta theorem'; 'After a few minutes the bot supplied me with a simpler proof'; a second prompt produced the discretization shortcut. The proof section is ChatGPT's output, 'lightly edited'
- Friedgut: 'my mathematical contribution… was limited to my two suggestions to the bot… I did not apply any creativity to the project'; he had earlier failed to solve the conjecture with a weaker ChatGPT model; he calls the AI's perturbed-measure argument 'really a beautiful idea'

#### What happened
Friedgut had carried the conjecture for over two decades and had tried it with AI before, without success. Chattopadhyay and Gurumukhani settled it in a 15-page paper that leaned heavily on ChatGPT Pro. After reading their proof, Friedgut suspected Hatami's deep pseudo-junta theorem was more than the problem needed. He asked GPT-6 Astra to swap in his own weaker junta theorem, then to reuse the discretization from his 2004 paper. The result is a shorter proof that passes through a family of perturbed product measures, which he likens to Dinur–Safra's "Importance of Being Biased". He posted it at the Cornell authors' suggestion.

#### Why it matters
It follows the Chvátal pattern from the same weeks (2026-09-16-chvatal-conjecture-proved): a human-led proof with heavy model use is followed within days by an AI-written simplification, here prompted by the conjecture's author himself.

Sources: [arXiv 2609.16401: A Resolution of Friedgut's Conjecture on Influential Coalitions (Chattopadhyay, Gurumukhani)](https://arxiv.org/abs/2609.16401); [arXiv 2610.03086: Small Influential Coalitions in [0,1]^n via the Junta Theorem (Friedgut)](https://arxiv.org/abs/2610.03086)

### 2026-09-14: FDA grants priority review to Takeda's zasocitinib, a computationally designed TYK2 inhibitor, with a decision due Q1 2027

***; Takeda, Nimbus Therapeutics, Schrödinger; Science & math; confidence high; https://postcutoff.com/e/2026-09-14-takeda-zasocitinib-fda-priority-review/

Takeda said on 14 Sept 2026 that the FDA had accepted, with priority review, its new drug application for zasocitinib (TAK-279), an oral TYK2 inhibitor for moderate-to-severe plaque psoriasis. The target action date is in Q1 2027. The molecule came from Nimbus Therapeutics and Schrödinger's physics-based (free energy perturbation) and machine-learning design. If approved, it may be called the first approved "AI-designed" drug, a label that Nimbus's own R&D head rejects.

- NDA accepted under priority review; PDUFA target action date in the first quarter of calendar 2027
- Phase 3 LATITUDE PsO 3001 (693 patients) and 3002 (1,108 patients): all primary endpoints and all 44 ranked secondary endpoints met; nearly 3,000 patients across the programme
- Head-to-head: statistically superior to BMS's Sotyktu (deucravacitinib); >35% of patients reached PASI 100 at week 16 (per press)
- Identified in 2020 by Nimbus with Schrödinger's FEP + ML; ~13,000 compounds assessed computationally (PharmaVoice)
- Takeda bought it from Nimbus in 2022 for $4B upfront plus up to $2B in sales milestones
- Nimbus R&D president Peter Tummino: 'I have heard people say it's going to be the first AI-approved drug and that's not the term I would use.'

#### What happened
Takeda's TYK2 inhibitor finished a Phase 3 programme of nearly 3,000 patients and was accepted for FDA priority review, with a decision expected in Q1 2027. The compound was found in 2020 when Nimbus and Schrödinger used free-energy-perturbation physics simulations and machine learning to evaluate about 13,000 designs computationally.

#### Why it matters
It could become the first FDA-approved drug widely described as computationally or AI-designed, just ahead of Insilico's rentosertib. The label is disputed. The design relied mainly on physics-based modelling and was not generative AI, and the drug was identified in 2020.

Sources: [Takeda: FDA accepts zasocitinib NDA with priority review](https://www.takeda.com/newsroom/newsreleases/2026/fda-priority-review-zasocitinib-psoriasis/); [PharmaVoice: Nimbus used AI to help develop Takeda's $4B psoriasis bet](https://www.pharmavoice.com/news/nimbus-takeda-zasocitinib-ai-drug-discovery/831289/); [BioSpace: Takeda's $4B Nimbus bet pays off with best-in-class Phase III data](https://www.biospace.com/drug-development/takedas-4b-nimbus-bet-pays-off-with-best-in-class-phase-iii-plaque-psoriasis-data); [IntuitionLabs: AI drug discovery FDA approvals, 2026 reality check](https://intuitionlabs.ai/articles/ai-drug-discovery-fda-approvals)

### 2026-09-15: Erdős–Hajnal high-girth problem (Erdős #108) disproved with ChatGPT/Codex help and a Lean proof, via the Conjectures.io bounty; experts sharpen it with GPT-6 Astra

****; Conjectures.io, Purdue University, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-09-15-erdos-108-high-girth-problem-disproved-conjectures-io/

On Sept 15, 2026 Purdue's Jensen Kohlmeyer and Liam Kruer (as "JenW1N") submitted a Lean 4 proof to the Conjectures.io bounty platform that disproves Erdős Problem #108, the Erdős–Hajnal question from the late 1960s of whether huge chromatic number forces a subgraph with large girth and large chromatic number. They built triangle-free graphs of arbitrarily large chromatic number whose four-cycle-free subgraphs are all 6-colourable. Their manuscript says the work "was developed with substantial assistance from OpenAI ChatGPT/Codex". On Oct 1 graph theorists Tung Nguyen and Bartosz Walczak published an exposition that lowers 6 to 3. They credit ChatGPT-6 Astra with discovering the key new proof.

- Problem: for every r ≥ 4 and k ≥ 2, is there f(k,r) such that every graph with chromatic number ≥ f(k,r) has a subgraph of girth ≥ r and chromatic number ≥ k? (Erdős–Hajnal; Rödl proved r = 4). The answer is no already at girth 5 and k = 7
- Construction: arc graphs over an ordered multipartite random base graph (the Janzer–Steiner–Sudakov model); every C4-free subgraph splits into a bounded-degree-controlled part (4-colourable) and a forest (2 colours)
- Verification: the main theorem and the negation of Google DeepMind's pinned Formal Conjectures statement (ErdosProblems/108.lean) were proved in Lean 4 (v4.33.1) using only standard axioms; Conjectures.io lists it as Lean-verified and approved in human review, with a $3,833 reward paid
- AI disclosure (Kohlmeyer & Kruer): 'This work was developed with substantial assistance from OpenAI ChatGPT/Codex, including mathematical exploration, the preparation and checking of estimates, construction and debugging of the Lean proofs, and drafting and typesetting of the manuscript.' Conjectures.io's own write-up was 'prepared by Conjectures with Codex assistance from the accepted Lean submission'
- Nguyen & Walczak (arXiv 2609.40192, Oct 1): improved 6-colourable to 3-colourable. 'ChatGPT-6 Astra discovered a proof of Theorem 2.1, in particular proposing Lemma 5.1 as a suitable extension of Lemma 4.3.' The step from 6 to 4 was done 'entirely by the authors … with no AI tools involved'
- Nguyen & Walczak note that both earlier manuscripts 'lack any references to the literature except for' Janzer–Steiner–Sudakov, and that the Lean-derived write-up 'may not be readily accessible to researchers in graph theory'
- Platform: Conjectures.io (Bittensor subnet 66) pays for Lean-verified proofs of formalised open problems. By Oct 5 its results page listed 26 problems solved and $127,713 paid; JenW1N also has accepted solves of Erdős #859, #18(b), #1062(ii) and Green's open problems 24 and 40 (Sept 15–25)
- erdosproblems.com still showed #108 as OPEN with one proof claim on Oct 5, 2026

#### What happened
Erdős and Hajnal asked whether a graph with enormous chromatic number must contain a subgraph that has both large girth and large chromatic
number. Rödl settled girth 4 (triangle-free subgraphs), and the general question stayed open as Erdős Problem #108. Google DeepMind's
Formal Conjectures project had pinned it as a Lean statement. Conjectures.io, a Bittensor subnet that pays for Lean-checked proofs, offered
a bounty for it.

On Sept 15, 2026 the solver "JenW1N", identified in the manuscripts as Purdue's Jensen Kohlmeyer with Liam Kruer, submitted a Lean proof of
the negation. They take arc graphs of a carefully sized random multipartite graph. In these graphs any subgraph without a four-cycle is
6-colourable, yet the whole graph has arbitrarily large chromatic number. The platform verified the proof and approved it in human review.
It paid the bounty and published an explanatory PDF that Codex prepared from the Lean code.

Graph theorists Tung Nguyen and Bartosz Walczak learned of the result from the Conjectures.io announcement. On Oct 1 they posted an
exposition on arXiv that ties each step to the literature. Working without AI, they observed that the construction already gives 4 colours.
They then asked ChatGPT-6 Astra for a 2-degenerate version of the key lemma. Astra found the proof, which brings the bound down to 3 colours.

#### Why it matters
This is a well-known problem from Erdős's list, and the counterexample comes with a machine-checked proof. Independent experts have
checked it and improved on it. It also shows a new route for AI-assisted mathematics: paid, Lean-gated bounties outside arXiv, where the
first write-up was itself machine-generated. Nguyen and Walczak's note criticises the lack of literature context in those write-ups.
We found no Hacker News thread about the result, and we did not locate the original Conjectures.io announcement post on X.

Sources: [Kohlmeyer & Kruer: A counterexample to Erdős problem 108 via arc graphs (dated Sept 15, 2026)](https://jenwin.io/papers/erdos108-arc-graphs.pdf); [Conjectures.io write-up: Erdős Problem 108, high chromatic number with six-colour C4-free subgraphs (Sept 17)](https://conjectures.io/papers/erdos108.pdf); [Conjectures.io problem page: Erdős 108](https://conjectures.io/problems/erdos108-erdos-108); [Conjectures.io results (bounties paid)](https://conjectures.io/results); [arXiv 2609.40192: Nguyen & Walczak, On the solution to the Erdős–Hajnal problem on high-girth high-chromatic subgraphs](https://arxiv.org/abs/2609.40192); [Erdős Problems #108](https://www.erdosproblems.com/108); [Google DeepMind Formal Conjectures repository](https://github.com/google-deepmind/formal-conjectures)

### 2026-09-15: OpenAI Foundation launches Public Data for Health with $125M+ in grants for open biomedical datasets

***; OpenAI Foundation; Science & math; confidence high; https://postcutoff.com/e/2026-09-15-openai-foundation-public-data-for-health/

On Sept 15, 2026 the OpenAI Foundation, the nonprofit that controls OpenAI, launched Public Data for Health. It is its second science program, after AI for Alzheimer's in April, and starts with more than $125M in grants to build and preserve open scientific datasets that AI can learn from. Initial grantees are OpenADMET (UCSF), CTD Commons (1Day Sooner) and UNC's Initiative for Generative Immunotherapy. Nature described the Foundation, which has committed to give away at least $25B, as on track to become one of the world's richest charities.

- Announced Sept 15, 2026 on openaifoundation.org; more than $125M in initial grants; the Foundation's second science program after AI for Alzheimer's (April 2026)
- OpenADMET (led by UCSF's James Fraser): datasets and competitions for predicting how small molecules move through the body (ADMET). Nature reports $15M
- CTD Commons (led by Josh Morrison, 1Day Sooner): preserves and publishes regulatory knowledge (FDA submission documents) from failed and bankrupt drug programs. Nature reports $500,000
- UNC Initiative for Generative Immunotherapy (Alex Rubinsteyn): public data for personalized cancer vaccines. Nature reports $40M
- Nature (Sept 2026): the Foundation was created in Oct 2025, holds about a 26% stake in OpenAI and has committed to give away at least $25B. Earlier it put about $100M into Alzheimer's research. Life-sciences lead Jacob Trefethen came from Coefficient Giving

#### What happened
The OpenAI Foundation announced Public Data for Health, a grant program that helps nonprofits and universities create, preserve and share
scientific datasets. The first three grantees cover the whole drug-development stack: molecular ADMET properties, regulatory filings from
failed programs, and immunology data for personalized cancer vaccines.

#### Why it matters
This is one of the first large disbursements from what may become the richest charity in the world, funded by its stake in OpenAI. The
Foundation is betting that data, not only models, limits AI-driven biomedicine. The per-grantee amounts come from Nature, not the
announcement itself.

Sources: [OpenAI Foundation: Public Data for Health](https://openaifoundation.org/news/public-data-for-health); [Nature: How to spend $25 billion on science: OpenAI's staggeringly rich research charity gets under way](https://www.nature.com/articles/d41586-026-03015-0); [Inside Precision Medicine: OpenAI Foundation gives $125M to Public Data for AI-driven health program](https://www.insideprecisionmedicine.com/topics/informatics/openai-foundation-gives-125m-to-public-data-for-ai-driven-health-program/); [Inside Philanthropy: The OpenAI Foundation pledged $1 billion in a year. How's that going six months in?](https://insidephilanthropy.com/home/the-openai-foundation-pledged-1-billion-in-a-year-hows-that-going-six-months-in)

### 2026-09-15: Periodic Labs introduces Periodic Neon, a 1T-parameter lab-trained model that beats GPT-6 Astra at X-ray diffraction analysis

***; Periodic Labs; Science & math; confidence high; https://postcutoff.com/e/2026-09-15-periodic-labs-neon/

On Sept 15, 2026 Periodic Labs, the materials-science startup of Liam Fedus and Ekin Dogus Cubuk, published its first research. It introduced Periodic Neon, a 1-trillion-parameter model built by mid-training and RL-tuning the open-weight Kimi K2.6 on the company's own laboratory data, using 1,300 H200 GPUs. On the internal FrontierXRD benchmark Neon reached 55.3%, up from the base model's 2.7%. The company says it beats GPT-6 Astra and Claude Fable 5.1 at diffraction analysis, and Neon is deployed in its Menlo Park labs.

- 1T parameters; post-trained from open-weight Kimi K2.6 with scientific midtraining + RL on proprietary XRD lab data
- Final training run on 1,300 H200 GPUs
- FrontierXRD (internal eval): Neon 55.3% vs Kimi K2.6 2.7%; company says it surpasses GPT-6 Astra and Claude Fable 5.1 'at a lower cost per analysis' (their scores not given in the post)
- Deployed in Periodic's physical labs to analyse experiments in the search for superconductors and magnets; no discoveries claimed yet
- Three posts the same day: 'Building Labs that Learn', 'Nature Is Our Learning Environment', 'AI Infrastructure at Periodic'
- No announcement of releasing Neon's weights
- Liam Fedus on The Frontier podcast with Mercor's Brendan Foody (reported by Synced via 36Kr, Oct 12): 'Machine learning systems are not magic', AlphaFold worked because decades of protein-structure data existed; published materials results are a biased sample because 'everything you see on the Internet is a success'; Periodic runs its own Menlo Park lab because 'without the ability to put ideas into practice, you can only stay at the thinking stage' (quotes as translated by 36Kr)

#### What happened
Periodic Labs came out of stealth in 2025 with a $300M seed round. Its first public results show the "lab as RL environment" idea:
experimental data from its own autonomous labs is used to specialise a large open-weight model for one scientific task, analysing XRD patterns.

#### Why it matters
It shows a startup taking a Chinese open-weight frontier model and, with modest compute and proprietary experimental data, beating
the best closed models on a narrow scientific task. That is an argument that experimental data, not general capability, is the bottleneck in AI-for-science.

Sources: [Periodic Labs: Building Labs that Learn](https://periodic.com/news/building-labs-that-learn); [Periodic Labs: Nature Is Our Learning Environment](https://periodic.com/news/nature-is-our-learning-environment); [Periodic Labs: AI Infrastructure at Periodic](https://periodic.com/news/ai-infrastructure-at-periodic); [Menlo Times: Periodic Labs introduces Periodic Neon](https://www.menlotimes.com/post/periodic-labs-introduces-periodic-neon); [36Kr (Synced): ChatGPT co-creator Liam Fedus: after mastering mathematics, it's time for AI to head to the lab](https://eu.36kr.com/en/p/4020983712272515)

### 2026-09-16: Chvátal's 1972 conjecture proved (Chang–Liu–Liu, ChatGPT-assisted), then a GPT-6 Astra 'proof from The Book' and a Codex-built Lean formalization

****; Institute for Basic Science, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-09-16-chvatal-conjecture-proved/

On Sept 16, 2026 Fan Chang, Hong Liu and Miao Liu posted a proof of Chvátal's conjecture (1972): every hereditary family of sets has a largest intersecting subfamily that is a star. It went through a sharp correlation inequality for increasing Boolean functions. ChatGPT tested candidate inequalities and proved a special case that pointed the authors to the general theorem. Within days the proof was formalised in Lean with GPT-6 via Codex. On Sept 23 Ellis, Filmus and Friedgut posted a 7-page spectral "proof from The Book" that GPT-6 Astra produced after reading the Chang–Liu–Liu manuscript.

- arXiv 2609.19123 (Sept 16, 2026), Chang, Liu & Liu: proves Chvátal's conjecture and the Friedgut–Kahn–Kalai–Keller correlation form Cov(f,g) ≥ ¼ min_i Inf_i[f] for antipodal g
- AI disclosure (Chang–Liu–Liu): 'We used ChatGPT to test candidate inequalities… ChatGPT proved the special case of the inequality in Theorem 1.2 when f is a symmetric threshold function, inspiring the authors to pursue and ultimately prove the stronger inequality'; all proofs written and checked by the authors
- Lean: boonsuan/chvatal formalizes the Chang–Liu–Liu proof, 'generated with GPT-6 through Codex'; registered with Palomar as PALOMAR-2026-09-17-000004; no sorry or extra axioms in the library; also imported into Lean Pool (PR #443)
- arXiv 2609.28404 (Sept 23), Ellis, Filmus & Friedgut, 'Chvátal's conjecture: a proof from The Book': a short direct spectral proof plus a projection-packing-number strengthening
- Their AI chronology: weeks of failed attempts with ChatGPT-6 Astra and Claude Fable 5.1 on their own conjectures (H, I); after they fed in the Chang–Liu–Liu paper, Astra announced a proof that the projection packing number is tight for every downset, then 'stripped [it] down to the spectral bare bones'; ChatGPT used only for proofreading of the final paper

#### What happened
Chvátal's conjecture, posed in 1972, is one of the best-known open problems in extremal set theory. Chang, Liu and Liu (IBS) proved it through
a sharp correlation inequality for increasing Boolean functions, and they credit ChatGPT with proving a special case that showed them the way. A
Lean formalization made with GPT-6 in Codex appeared the next day in the Palomar registry. A week later, Ellis, Filmus and Friedgut
described how GPT-6 Astra had failed for weeks on their related conjectures. Once given the new paper, it produced a short spectral proof, which they
present as "a proof from The Book".

#### Why it matters
It is a clear example of the late-2026 pattern in mathematics. A human-led breakthrough with model assistance is followed within days by AI
simplification and AI formalization. Disclosures are now detailed enough to separate the contribution of each model.

Sources: [arXiv 2609.19123: A proof of Chvátal's conjecture via a sharp correlation inequality](https://arxiv.org/abs/2609.19123); [arXiv 2609.28404: Chvátal's conjecture, a proof from The Book (Ellis, Filmus, Friedgut)](https://arxiv.org/abs/2609.28404); [GitHub boonsuan/chvatal: Lean formalization generated with GPT-6 through Codex](https://github.com/boonsuan/chvatal); [Lean Pool PR #443: import Chvátal's conjecture and sharp Boolean correlation](https://github.com/Vilin97/lean-pool/pull/443)

### 2026-09-16: Lean-verified 'Liouville Goldbach' theorem goes viral as a GPT-6 Astra 'Goldbach breakthrough'; the AI role is unconfirmed and it is not the Goldbach conjecture

**; Captain Sude (pseudonymous); Science & math; confidence medium; https://postcutoff.com/e/2026-09-16-liouville-goldbach-lean-proof/

On 16 Sep 2026 the pseudonymous X account @captain_sude released a Lean-verified proof that every even N > 2 is a sum of two positive integers that each have an odd number of prime factors (Liouville λ = −1). This answers a 2018 MathOverflow question that Mangerel had settled only under GRH and for large N. Chinese and crypto media reported it as 'GPT-6 Astra makes a major breakthrough on the Goldbach conjecture'. The classical conjecture is untouched, and the repository itself does not mention Astra.

- Statement (Lean theorem LiouvilleGoldbach.liouville_goldbach): for every even N > 2 there are a, b ≥ 1 with a + b = N and λ(a) = λ(b) = −1; a and b may be composite
- The question was asked on MathOverflow in August 2018; Mangerel (arXiv 2412.17199) attributes it to Shusterman and proved it assuming GRH for sufficiently large N
- Announcement (x.com/captain_sude, 16 Sep 2026, ~47k views at check): 'The Liouville version of the Goldbach conjecture is now fully proven and Lean verified!'
- Independent technical replay (GitHub sunnyspot114514/Liouville-Goldbach-audit): the build passes, and an axiom audit shows only the standard axioms; the auditor says this is not mathematical peer review
- AI provenance: the paper is authorless and the repository has 'no explicit mention of Astra / GPT-6' (ai4math-chronicle review, issue #116); the Astra attribution comes from secondary coverage (36kr, KuCoin, Lookonchain)
- Hacker News thread 'GPT-6 Astra has made a major breakthrough in the Goldbach Conjecture' (23 Sep 2026)

#### What happened
A pseudonymous account posted a paper and a complete Lean formalisation of a Goldbach-type statement for the Liouville function. Secondary coverage, especially in Chinese and crypto media, credited GPT-6 Astra and described it as progress on Goldbach's conjecture.

#### Why it matters
The mathematics is formally verified and does answer a small open question. The episode is mainly a caution: an AI model is attributed without any first-party evidence, and the headline claims a famous conjecture that the result does not touch. Models reading this should not repeat "Astra proved Goldbach".

Sources: [GitHub: CaptainSude/Liouville-Goldbach (paper and Lean proof)](https://github.com/CaptainSude/Liouville-Goldbach); [X: @captain_sude announcement (16 Sep 2026)](https://x.com/captain_sude/status/2100277626120163664); [GitHub: independent replay audit](https://github.com/sunnyspot114514/Liouville-Goldbach-audit); [ai4math-chronicle issue #116: provenance and review watch](https://github.com/Charlie-Wang-03/ai4math-chronicle/issues/116); [MathOverflow: Goldbach's conjecture for the Liouville function (2018)](https://mathoverflow.net/questions/307479/goldbachs-conjecture-for-the-liouville-function); [36kr: GPT-6 Astra Unveils Major Breakthrough in Proving the Goldbach Conjecture](https://eu.36kr.com/en/p/3992525310802946); [Hacker News discussion](https://news.ycombinator.com/item?id=49814019)

### 2026-09-17: ζ(5) proved irrational: Aabir Fauzan's Zenodo preprint, the first such result since Apéry's ζ(3) in 1978, is formally verified in Lean within a week, one formalization written by Claude

*****; Aalto University, Google DeepMind, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-09-17-zeta-5-irrational-fauzan-lean-verified/

On Sept 17, 2026 Aabir Fauzan (Aalto University) posted "ζ(5) is irrational" on Zenodo. It proves that the value ζ(5) is irrational, the first irrationality proof for a specific odd zeta value since Apéry's ζ(3) (1978). By Sept 23 a complete, sorry-free Lean 4 formalization by Google DeepMind's Moritz Firsching was checked by Comparator against DeepMind's Formal Conjectures statement. A second, independent formalization was written by Claude Opus 5/5.5 agents in Claude Code under Dan Romik. How much AI was used in the paper itself is disputed: the author discloses only supporting use, while Frank Calegari says it looks "almost entirely AI generated".

- Preprint: A. Fauzan, 'ζ(5) is irrational', Zenodo 10.5281/zenodo.22826419, Sept 17, 2026; posted on Zenodo, not arXiv (about 16.8k views and 2.4k downloads by Oct 5). The record calls it the initial presentation, with a full manuscript and arXiv version to follow
- Method (abstract): integer polynomials Q_n of degree 37n with 0 < Q_n(ζ(5)) < exp(−139n²/5), built as rationally normalized Hankel determinants with a positive moment representation and logarithmic-energy estimates. It also gives the irrationality measure bound |ζ(5) − a/b| > b^−260
- Before this: Ball–Rivoal (infinitely many odd zeta values are irrational) and Zudilin (at least one of ζ(5), ζ(7), ζ(9), ζ(11) is irrational), but no specific odd value beyond ζ(3)
- Paper's AI disclosure: 'A generative AI tool was used in a supporting role for editing the exposition, proofreading, LaTeX preparation, and consistency checks of the arguments, calculations, and references'
- Frank Calegari (Sept 24): 'The paper I saw gives me the impression of being almost entirely AI generated (if so, it then comes with a very dishonest disclosure), but never mind, thanks for the compute!' He traces the key ideas to Prévost and Prévost–Rivoal
- Lean formalization 1 (github.com/mo271/zeta5, Moritz Firsching, Google DeepMind; first commit Sept 23): complete with no sorry, axioms only propext, Classical.choice and Quot.sound. CI runs Comparator against Formal Conjectures' RiemannZetaValues.irrational_five. It deviates from the paper in three local estimates (rank loss r, cutoff M = 400) but still proves irrationality. How it was produced is not stated
- Lean formalization 2 (github.com/danromik/zeta5-irrationality): 70 files, about 29,400 lines, no sorry; the prime number theorem is assumed as an axiom. README: 'The Lean code and the documentation were written by Claude (Anthropic), using the models Claude Opus 5 and Claude Opus 5.5, working as multi-agent workflows in Claude Code directed by Dan Romik'. Its pre-formalization audit found one inequality (p. 10) stated without proof and supplied a proof
- Other efforts: an 'autonomous' formalization project (long-mathematics/zeta5-irrationality, Christopher D. Long) and a copy in Vilin97/lean-pool. One write-up reports the method handles ζ(k) for k = 2, 3, 4, 5 with changed parameters
- Reactions: Elliot Glazer (Sept 22, ~81k views): 'zeta(5) has very likely been proven irrational … I had Astra study the argument, and it fully vouches for its correctness'. Alex Kontorovich (Sept 23, ~74k views): 'WOW!! Zeta(5) is irrational! … Amazing what we'll learn (with AI help)'

#### What happened
Apéry's 1978 proof that ζ(3) is irrational was the last time a specific odd zeta value was shown to be irrational. Later work
(Rivoal, Ball–Rivoal, Zudilin) proved only statements about families, such as "at least one of ζ(5), ζ(7), ζ(9), ζ(11) is
irrational". On Sept 17, 2026 Aabir Fauzan, giving an Aalto University address and with no earlier research papers, posted a
31-page proof for ζ(5) on Zenodo. Commentators guessed he had no arXiv endorsement.

Verification was unusually fast. Elliot Glazer publicised the paper on Sept 22. On Sept 23 Moritz Firsching (Google DeepMind,
a maintainer of the Formal Conjectures benchmark) pushed a complete Lean 4 formalization, and Alex Kontorovich announced it.
Comparator CI then confirmed that it proves exactly the benchmark statement `irrational_five` with only the standard axioms.
A separate formalization by Dan Romik was, by its README, written entirely by Claude Opus 5 and Opus 5.5 agents in Claude
Code. It was done without consulting the other formalization, and its audit filled one gap in the paper.

The AI role in finding the proof is unclear. The paper's disclosure is limited to editing, LaTeX and consistency checks.
Frank Calegari wrote that the paper looks "almost entirely AI generated" and, if so, the disclosure is "very dishonest". He
then used ChatGPT to trace where its ideas came from. Andreas Holmstrom classifies it as "Human + AI (supporting role)". No AI
system has been named as the author of the argument, and Fauzan has made no public statement that we found.

#### Why it matters
It is the most famous number-theory result of the AI-assisted 2026 wave: a problem experts had worked on for about 48 years.
It was settled in a non-traditional venue and machine-verified in Lean within a week, with AI agents doing at least one
formalization. That pattern (preprint, then Lean, then expert reading) reverses the usual peer-review order. The paper has not
been refereed, and the ZFC-level guarantee rests on the Lean checks, one of which assumes the prime number theorem.

Sources: [Zenodo: A. Fauzan, ζ(5) is irrational (Sept 17, 2026)](https://zenodo.org/records/22826419); [Lean formalization (Moritz Firsching): mo271/zeta5](https://github.com/mo271/zeta5); [Lean formalization written by Claude (Dan Romik): danromik/zeta5-irrationality](https://github.com/danromik/zeta5-irrationality); [Formal Conjectures: RiemannZetaValues statement](https://github.com/google-deepmind/formal-conjectures/blob/2db01438fda960163f58ff2c2385b52e53f56dd2/FormalConjectures/Wikipedia/RiemannZetaValues.lean); [Frank Calegari (Persiflage): zeta(5) is irrational (Sept 24)](https://galoisrepresentations.org/2026/09/24/zeta5-is-irrational/); [Andreas Holmstrom: Number theory breakthroughs at the dawn of a new era](https://www.peakmath.org/number-theory-breakthroughs-at-the-dawn-of-a-new-era/); [Bogdan Grechuk: Zeta(5) is irrational (exposition, Sept 29)](https://functor.network/user/3333/entry/1997/pdf); [Elliot Glazer on X (Sept 22)](https://x.com/ElliotGlazer/status/2102435821542691202); [Alex Kontorovich on X (Sept 23)](https://x.com/AlexKontorovich/status/2102870642555269554); [long-mathematics/zeta5-irrationality (autonomous formalization attempt)](https://github.com/long-mathematics/zeta5-irrationality)

### 2026-09-17: Claude speeds up 30+ open-source biology models about 4x and folds 10,000+ token complexes on one GPU node

***; Anthropic, Adaptyv Bio; Science & math; confidence high; https://postcutoff.com/e/2026-09-17-claude-biomolecular-modeling-speedups/

On Sept 17, 2026 Anthropic reported that Claude, working in Claude Science, optimized more than 30 open-source biomolecular models (structure prediction, protein design, genomics, protein language models) in under four weeks. The optimized models run about 4x faster on average, and a new low-memory "Big" mode accurately folds systems over 10,000 tokens on a single GPU node. Anthropic open-sourced the code and launched a protein design competition with Adaptyv Bio backed by up to $1M in Claude credits.

- 30+ models optimized in just under four weeks by an internal general-purpose research model, supervised by two staff with no prior kernel-engineering experience
- Average speedup ~4x with minimal loss of precision, ~2x with identical outputs (~1.6x identical-output speedup for structure prediction models)
- FlashPairformer kernels: 2.7–2.9x faster triangle attention and 1.7–3.2x faster triangle multiplication than the field standard (NVIDIA cuEquivariance / BioNeMo-IR)
- 'Big' mode accurately folded human mitochondrial complex I, the TRiC chaperone, a proteasome and a bacterial 70S ribosome (>10,000 tokens) on one GPU node; for comparison, AlphaFold3's accurate 40S ribosome prediction was 7,663 tokens
- Inference ran on 31,000–70,000+ token viral capsids on one 8-GPU B300 node, but those predictions collapsed (no accurate structure)
- Binder design with one H200, 24 h, a ~1,100-word prompt and no sub-agents matched the in silico ipSAE scores of the earlier campaign at about two orders of magnitude fewer GPU hours (~$150 of GPU plus tokens); tested with Mythos 5.1, Mythos 5 and Opus 5 on 16 targets
- Competition with Adaptyv Bio: five problems, wet-lab validation of 5,000+ designs, up to $1M in Claude credits, $250K in Modal compute credits, DNA from Twist Bioscience
- Same day: Life Sciences Verification Program opened in public beta

#### What happened
Anthropic's August binder campaign had let Claude spend up to $10,000 of Modal compute per target, about 2,500 H100 hours. To make the approach
affordable, Anthropic had Claude optimize the open-source models it relies on. Claude wrote general kernels (FlashPairformer) for the cubic-cost
triangle operations in Pairformer-style models such as AlphaFold3, OpenFold3 and Boltz-2. It also made model-specific changes, such as caching
recomputed work and collapsing dead branches. Anthropic checked that downstream accuracy did not change. A low-memory mode extended accurate
folding to molecular machines about 1.5 orders of magnitude beyond the models' training context.

#### Why it matters
Expert engineers normally need weeks per model to make such optimizations, and the work rarely transfers between models. Here an AI did it
across a whole ecosystem of scientific software and released the code. It lowers the compute barrier for academic protein design by roughly
100x. The results are Anthropic's own benchmarks, and faster models do not add new scientific validation.

Sources: [Anthropic: How Claude is uplifting biomolecular modeling](https://www.anthropic.com/research/claude-uplifts-biomolecular-modeling); [Technical report: Accelerating open-source biomolecular models with Claude (PDF)](https://www-cdn.anthropic.com/e96b5807039a88168733d9687afe41dfbbd5de13.pdf); [Code: anthropics/uplifting-biomolecular-modeling](https://github.com/anthropics/uplifting-biomolecular-modeling); [Anthropic × Adaptyv 2026 protein design competition (Proteinbase)](https://proteinbase.com/competitions/anthropic-adaptyv-2026); [Introducing the Life Sciences Verification Program](https://www.anthropic.com/news/life-sciences-verification-program); [Unite.AI: Anthropic reports Claude optimized 30+ open-source biomolecular models](https://www.unite.ai/anthropic-reports-claude-optimized-30-plus-open-source-biomolecular-models/); [remio: Claude uplifts biomolecular modeling, but speed is not scientific validation](https://www.remio.ai/post/claude-uplifts-biomolecular-modeling-but-speed-is-not-scientific-validation)

### 2026-09-17: Science: Stanford's 'Virtual Biotech' of 37,000 AI agents mines ~50,000 clinical trials and designs a lung-cancer ADC later matched by pharma

***; Stanford University; Science & math; confidence high; https://postcutoff.com/e/2026-09-17-stanford-virtual-biotech-37000-agents/

On Sept 17, 2026 James Zou's Stanford group (lead author Harrison Zhang) published the "Virtual Biotech" in Science (aeg6779): about 37,000 specialized AI agents organized like a biotech company under a chief-scientific-officer agent. It analysed about 50,000 clinical trials in under a week and found that drugs targeting "switch-like" genes are more likely to succeed, and it designed a B7-H3 (CD276) antibody-drug conjugate for lung cancer that matched a strategy a drugmaker had independently developed and that received FDA breakthrough designation.

- Paper: Science, Sept 17, 2026, doi 10.1126/science.aeg6779; senior author James Zou, lead author Harrison Zhang (Stanford Medicine)
- Structure: a CSO agent takes a human's question, delegates to specialised divisions and combines results; about 37,000 agents in total
- Clinical-trial analysis: ~50,000 trials in under a week
- Finding: drugs whose targets show high specificity and bimodal ('switch-like') expression were 40% more likely to advance from phase 1 to 2, 48% more likely to reach market, and had 32% fewer adverse events; the pattern held across cancer, brain, heart, kidney and lung conditions
- Drug design: an ADC against B7-H3 for lung cancer, using data from before January 2025; an independent company developed the same strategy (August 2025), which received FDA breakthrough-therapy designation; VentureBeat names Merck
- Grew out of Zou's 5–8-agent 'Virtual Lab' that designed nanobodies (Nature, 2025); the LLM used is not named in the press coverage

#### What happened
Zou's group scaled its earlier Virtual Lab, a handful of agents mirroring a university lab, to a company-scale organisation of tens of thousands of
agents. Its two headline outputs were a statistical rule for which drug targets succeed and a concrete therapeutic design.

#### Why it matters
It is one of the largest multi-agent scientific systems published in a top journal, and an early test of whether "AI companies of agents" can do
drug-discovery work. The drug design was matched by an independent programme, which is a form of external validation, but it was not new to the field.

Sources: [Stanford Medicine: Virtual biotech company puts thousands of AI scientist agents to work on drug discovery](https://med.stanford.edu/news/all-news/2026/09/virtual-biotech-company.html); [VentureBeat: Stanford is running 37,000 AI agents as a virtual biotech](https://venturebeat.com/orchestration/stanford-is-running-37-000-ai-agents-as-a-virtual-biotech-and-one-of-its-drug-designs-got-independently-confirmed-by-merck); [Singularity Hub: Virtual biotech company puts 37,000 AI agents to work on drug discovery](https://singularityhub.com/2026/09/18/virtual-biotech-company-puts-37000-ai-agents-to-work-on-drug-discovery/)

### 2026-09-19: FutureHouse and Edison Scientific publish twelve 'Millennium Problems for Biology', pitched as the 'last reasonable eval' for AI in biology

***; FutureHouse, Edison Scientific; Science & math; confidence high; https://postcutoff.com/e/2026-09-19-millennium-problems-for-biology/

On Sept 19, 2026, citing AI's progress in mathematics, FutureHouse and Edison Scientific (Sam Rodriques, Michaela Hinks) published twelve "Millennium Problems for Biology": very hard to solve but easy to validate in a simple lab, from synthesizing life from chemicals and reversibly cryopreserving whole mice to a reverse translatase and a better Rubisco. Rodriques called them "the 'last reasonable eval' for AI in biology"; the launch post had ~508k views on X, and cash prizes and a judging panel were promised.

- The 12 problems: origins of life; whole-mouse cryopreservation; reverse translatase (protein → nucleic acid); improved Rubisco; quadruplet-codon cell; somatic limb regeneration in mammals; bacterial production of AAV/lentivirus; programmable proteases; cell-penetrating intracellular protein binders; protein amplification chain reaction; 5′ polymerases; new nitrogenases
- Design rule: each has acceptance criteria checkable in a simple laboratory environment
- Sept 21 update (Rodriques): the specs will be revised after community feedback; FutureHouse will run a community process for a longer list of grand challenges, convene a panel to adjudicate success and announce cash prizes
- Sept 28: Andrew White (FutureHouse co-founder) joked that 'Millennium problems for bio #4 lasted 9 days?', pointing to a UC Davis/LBNL bioRxiv preprint (posted Sept 9) in which protein-language-model-designed de novo Rubiscos were active and one showed an apparent CO2/O2 specificity beyond the natural Rubiscos assayed; no adjudication has been announced
- Hacker News discussion reached 164 points (Sept 20)

#### What happened
Rodriques framed the list as biology's answer to the run of AI results on open math problems: concrete tasks that, if an AI scientist (or anyone) solved them, would be unambiguous to check at the bench.

#### Why it matters
Benchmarks for AI in science are saturating. A small set of hard, lab-checkable targets from an AI-scientist lab gives a public yardstick for claims that AI systems are doing transformative biology, though how wins are judged is still being worked out.

Sources: [Millennium Problems for Biology (site)](https://millenniumproblems.bio); [FutureHouse: Millennium Problems for Biology](https://www.futurehouse.org/news/millennium-problems-for-biology); [Sam Rodriques on X: launch post (~508k views)](https://x.com/SGRodriques/status/2101449009546273138); [Sam Rodriques on X: Sept 21 update (panel, prizes, expanded list)](https://x.com/SGRodriques/status/2102030245394337967); [Andrew White on X: '#4 lasted 9 days?'](https://x.com/andrewwhite01/status/2104614015951683902); [bioRxiv: De novo Rubisco design with protein language models (Kehl, ..., Siegel)](https://www.biorxiv.org/content/10.64898/2026.09.04.749267v1); [Hacker News discussion](https://news.ycombinator.com/item?id=49775082)

### 2026-09-21: Grad's 1967 conjecture on 3D plasma equilibria falls: two AI-assisted papers give three families of counterexamples

*****; University of Maryland, OpenAI, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-09-21-grad-conjecture-counterexamples/

Two independent arXiv preprints posted a day apart (21 and 22 Sep 2026) construct smooth magnetohydrostatic plasma equilibria with nested toroidal pressure surfaces and non-constant pressure that have none of the symmetries Harold Grad conjectured were necessary, answering a question central to stellarator fusion theory. Matt Landreman's two explicit families were "discovered using the artificial intelligence model GPT-6 Astra Pro"; Gómez-Serrano, Liehr and Taylor's family was worked out with GPT-5.6 Sol, Claude Fable 5 and Claude Opus 5 and verified in Lean 4.

- Grad's conjecture (1967): smooth 3D equilibria with nested flux surfaces and non-constant pressure require symmetry (plane-reflection, axial or helical)
- arXiv:2609.24739 (21 Sep 2026), Gómez-Serrano, Liehr, Taylor: for each large N, a smooth one-parameter family on embedded solid tori whose only Euclidean symmetry group is the cyclic group C_N; magnetic field vanishes exactly on a round magnetic axis
- Their LLM disclosure: authors set up the problem and a detailed roadmap in July 2026; GPT-5.6 Sol, Claude Fable 5 and Claude Opus 5 filled in technical details and produced the Lean code under guidance; GPT-6 Astra and Claude Fable 5.1 used in the late Lean stage; authors say the models did not contribute mathematical content of the article itself and they checked everything
- Main theorem formally verified in Lean 4 (github.com/lukasliehr/Grad-Conjecture; depends only on standard axioms)
- arXiv:2609.26742 (22 Sep 2026, rev. 28 Sep), Matt Landreman (U. Maryland): explicit analytic solutions in elementary functions with exact nested toroidal surfaces; one family with uniform rotational transform, one with sheared transform; also steady incompressible Euler flows
- Landreman's acknowledgement: 'These solutions were discovered using the artificial intelligence model GPT-6 Astra Pro, which was also used to draft parts of the manuscript. All equations were confirmed manually by the author.'
- An X post summarising both papers (@0x_impai, 24 Sep) reached ~744k views
- Reach in Japan: a Japanese explainer post by @yumamaeda1210 (Oct 4, 2026) quoting the GPT-6 Astra Pro acknowledgement reached ~240k views in a day (fxtwitter)

#### What happened
In 1967 Harold Grad argued that, apart from special cases, smooth three-dimensional plasma equilibria with nested toroidal magnetic surfaces and a pressure gradient cannot exist without a symmetry. The question matters for stellarators, fusion devices that are deliberately non-symmetric and whose design codes assume nested surfaces.

On 21 September 2026 Javier Gómez-Serrano, Lukas Liehr and Mitchell A. Taylor posted a construction of such equilibria for every sufficiently large N, with only the discrete rotation group C_N as symmetry, and a Lean 4 certification of the main theorem. A day later Matt Landreman posted explicit analytic families written with elementary functions, which he says were discovered with GPT-6 Astra Pro.

#### Why it matters
It removes a long-standing theoretical doubt about smooth non-symmetric equilibria, which is relevant to stellarator design and gives exact test cases for equilibrium codes. It is also a clear case of a frontier model producing a physics result outright, with the author disclosing it.

#### Follow-up: a Google team generalizes Landreman's solutions with Gemini 4 (Oct 5, 2026)
Opal Issan, Jonathan Citrin, Matt Landreman and Brendan Tracey posted arXiv:2610.07304 on 5 October 2026. It adds a symmetry-breaking
parameter to both of Landreman's families, giving non-axisymmetric solutions that also break stellarator symmetry and reduce to his when
the parameter is zero. The paper's "Use of AI tools" section says: "This work was carried out using Gemini 4 running as a small team of
agents in Google Antigravity, an agentic development environment. Given Landreman [1] and a list of the open questions it raised, the
agents proposed the generalized solutions that remove the stellarator-symmetry assumption. Promising candidates appeared within hours.
Gemini 4 Argon was also used to draft parts of the manuscript and write symbolic and numerical verification code. ... Every derivation
was independently re-derived and checked symbolically by the authors." Status: preprint, not peer-reviewed. Google cited the paper in its
Oct 13 Gemini 4 Argon release post (see `2026-10-13-gemini-4-argon-general-availability`).

Sources: [arXiv:2610.07304 — Analytic non-axisymmetric 3D MHD equilibria without stellarator symmetry (Issan, Citrin, Landreman, Tracey; Gemini 4)](https://arxiv.org/abs/2610.07304); [Japanese explainer on X (@yumamaeda1210, Oct 4)](https://x.com/yumamaeda1210/status/2106672447617646846); [arXiv:2609.24739 — Counterexamples to Grad's conjecture (Gómez-Serrano, Liehr, Taylor)](https://arxiv.org/abs/2609.24739); [arXiv:2609.26742 — Analytic toroidal 3D MHD equilibria and steady Euler flows with invariant surfaces (Landreman)](https://arxiv.org/abs/2609.26742); [Lean 4 formalization (GitHub: lukasliehr/Grad-Conjecture)](https://github.com/lukasliehr/Grad-Conjecture); [X: @0x_impai summary thread (~744k views)](https://x.com/0x_impai/status/2103161612677107942)

### 2026-09-21: Courtade–Kumar 'most informative Boolean function' conjecture (2013) proved three times in two days, all with AI: Ky & Tran (ChatGPT), Google + CUHK (Gemini, Lean-verified end-to-end), Mahdavifar & Beirami

****; Google, Chinese University of Hong Kong, FPT University, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-09-21-courtade-kumar-conjecture-proved/

The Courtade–Kumar conjecture (Kumar & Courtade 2013) says that among all Boolean functions f of n uniform bits, a dictator f(x) = x_i maximises the mutual information I(f(X); Y) when Y is X passed through a binary symmetric channel. On 21 Sep 2026 two independent proofs appeared on arXiv: Vu Khac Ky and Tuan Tran's "Dictators are most informative" (2609.24184, 36 pages; "used ChatGPT to develop the proofs, prepare the verification code, and revise the exposition") and a ~250-page computer-assisted proof by Google (Mirrokni, Woodruff, Javanmard, Lin) and CUHK (Chen, Gohari, Nair) (2609.24931), which states that "the overwhelming majority of the novel ideas and results in this paper were generated by AI" (Gemini in Google's Stellar Colosseum harness) and that the entire proof, including numerical certificates, is formally verified in Lean end-to-end. A third proof by Mahdavifar & Beirami (2609.26444, 22 Sep) says its single-bit proof "was generated by generative AI systems". OpenAI's 6 Oct catalogue (family 119) also claims the conjecture.

- Statement: for X uniform on {−1,1}^n and Y obtained by flipping each bit independently with probability p, every Boolean f satisfies I(f(X); Y) ≤ 1 − h(p), with equality for dictators (Courtade & Kumar, ISIT 2013; IEEE Trans. Inf. Theory 2014)
- Ky & Tran (FPT University; USTC), arXiv 2609.24184, posted 21 Sep 2026 06:56 UTC, 36 pages. Disclosure: 'The authors formulated the approach and used ChatGPT to develop the proofs, prepare the verification code, and revise the exposition.' Ky told VnExpress AI helped 'quickly test ideas and identify counterexamples'
- Chen, Gohari, Javanmard, Lin, Mirrokni, Nair & Woodruff (CUHK + Google), arXiv 2609.24931, v1 21 Sep 2026 17:26 UTC, v2 24 Sep (added the Lean link and concurrent work). Abstract: 'The entire proof, including all numerical certificates, has been formally verified in Lean end-to-end'
- Google paper, Part 4 'Human–AI collaboration': Gemini in the Stellar Colosseum framework first found counterexamples to a generalised version (Conjecture 5) of the CUHK reduction; CUHK then supplied the 'max-phi Bellman function', reducing the conjecture to a 14-variable optimisation; 'The AI then led the effort to partition the space into different regions, developing novel, non-trivial theoretical ideas and numerical interval arithmetic calculations'; the move from the balanced to the unbalanced case 'operated entirely without human input'; 'Additional ideas, verification, and critiques were contributed by human collaborators working with Anthropic and OpenAI models'
- Stellar Colosseum (Lin, Woodruff, Deng, Mao, Zuo, Mirrokni; arXiv 2609.15983, 14 Sep 2026) is a many-agent harness; Google says it is available externally as the 'Long Proof' pattern in Antigravity Teamwork
- Mahdavifar & Beirami, arXiv 2609.26444 (v1 22 Sep, v2 28 Sep), also prove multibit extensions. Disclosure: 'The proof of the single-bit result, including its detailed mathematical derivations, was generated by generative AI systems; the authors' role was high-level direction, feedback, and final review. The multibit extension was entirely human-directed'
- OpenAI's 6 Oct catalogue, family 119 (manuscripts dated 24 Sep), claims the Courtade–Kumar conjecture and the Hellinger conjecture with arbitrary output bias. On 7 Oct Ky posted a separate paper proving the balanced Hellinger conjecture for antipodal Boolean functions (arXiv 2610.10618), citing all of these
- Reception: Google Research VP Vahab Mirrokni announced the Google proof on X, called the conjecture a 'longstanding central open problem at the intersection of information theory and Boolean function analysis' and congratulated Ky and Tran (VietNamNet, VnExpress); the two teams coordinated a simultaneous release

#### What happened
The Courtade–Kumar conjecture asks how much a single bit can say about a noisy copy of its input. Courtade and Kumar
conjectured in 2013 that the best one-bit summary is simply one of the input coordinates. Partial results covered high
noise, balanced functions and weaker forms, and Google listed it in its February 2026 "Accelerating scientific research with
Gemini" paper as a problem it was trying to solve with AI.

On 21 September 2026 Vu Khac Ky (FPT University) and Tuan Tran (USTC) posted a 36-page proof. The same day a Google–CUHK team
posted a roughly 250-page computer-assisted proof. Its Part 4 describes the collaboration: Gemini, running in Google's
Stellar Colosseum many-agent harness, first found counterexamples to an over-general version of CUHK's earlier reduction.
CUHK then supplied the right Bellman function, after which "the AI then led the effort", partitioning a 14-variable domain
into regions and proving bounds with interval arithmetic. The authors write that "the overwhelming majority of the novel
ideas and results in this paper were generated by AI", and that the whole proof, numerical certificates included, is
checked by Lean's kernel. A day later Hessam Mahdavifar and Ahmad Beirami posted a third proof whose single-bit part was,
by their account, generated by AI.

The two first teams coordinated their release. OpenAI's 6 October catalogue later included the same conjecture (family 119,
manuscripts dated 24 September).

#### Why it matters
This is one of the clearest cases of AI-driven results arriving simultaneously from several directions: a lab team with
its own agent harness, a two-person academic team using ChatGPT, and a third team, all within about 30 hours. For a
model with an older cutoff: the Courtade–Kumar conjecture should now be treated as proved, with a machine-checked proof.

Sources: [Chen, Gohari, Javanmard, Lin, Mirrokni, Nair, Woodruff: A Proof of the Most Informative Boolean Function Conjecture (arXiv 2609.24931)](https://arxiv.org/abs/2609.24931); [Ky & Tran: Dictators are most informative (arXiv 2609.24184)](https://arxiv.org/abs/2609.24184); [Mahdavifar & Beirami: The Most Informative Bit and Beyond (arXiv 2609.26444)](https://arxiv.org/abs/2609.26444); [Lin et al.: Stellar Colosseum, a many-agent harness for long-horizon research (arXiv 2609.15983)](https://arxiv.org/abs/2609.15983); [VietNamNet: Vietnamese mathematicians find proof for decade-old information theory puzzle (29 Sep)](https://vietnamnet.vn/en/vietnamese-mathematicians-find-proof-for-decade-old-information-theory-puzzle-2559814.html); [VnExpress: Vietnamese mathematicians solve decade-old information theory problem (3 Oct)](https://e.vnexpress.net/news/tech/personalities/vietnamese-mathematicians-solve-decade-old-information-theory-problem-5125621.html); [Ky: The Hellinger Conjecture for Antipodal Boolean Functions (arXiv 2610.10618, 7 Oct)](https://arxiv.org/abs/2610.10618); [OpenAI math catalogue (CONTENTS.md, family 119)](https://github.com/openai/math/blob/main/CONTENTS.md)

### 2026-09-21: OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts)

***; OpenAI; Science & math; confidence low; https://postcutoff.com/e/2026-09-21-openai-100-open-problems-claim/

On 21 Sep 2026 OpenAI said an unnamed internal model had resolved more than 100 long-standing open problems during about 24 days of training (28 Aug – 21 Sep). It released no list and no proofs, and did not define 'resolved'. It also formed a 9-member Advisory Group on Mathematics and AI at IAS Princeton, including Timothy Gowers, Edward Witten and Martin Hairer. On 6 Oct 2026 OpenAI released the list: 722 manuscripts in 372 result families on GitHub, partly Lean-formalized (see 2026-10-06-openai-math-release-722-manuscripts).

- Claim: 100+ open problems resolved in ~24 days of training; no evidence released as of 29 Sep 2026
- Released 6 Oct 2026: github.com/openai/math, 722 manuscripts in 372 families from ~4,000 posed problems (~3 h ChatGPT Pro compute per result), with Lean formalizations for 162 manuscripts (review 'unchecked'); claims include the quasi-Riemann hypothesis, Unique Games, Hodge for CM abelian varieties and free group factor isomorphism
- Advisory Group on Mathematics and AI (9 members) at the Institute for Advanced Study; per its own announcement (Tao blog) it formed after OpenAI approached members, but it is independent of any AI company and unpaid
- Sober counterpoint: Epoch's 'FrontierMath Erdős' benchmark (68 open Erdős problems, Lean, $300/problem): GPT-6 Astra 3%, all others 0% (arXiv 2609.25050)
- OEIS Open benchmark: models resolved 147 of 492 formalised open OEIS conjectures (30%) at $50/attempt (arXiv 2608.11941)
- Side effect: on 22 Sep Francesco Deangelis posted an unchecked ChatGPT Astra proof of the Mumford–Shah conjecture to document priority, explicitly because of this announcement (arXiv 2609.26732)
- The Verge (28 Sep): AGMAI member Martin Hairer said 'We don't work for OpenAI, are not paid by them, and it's totally independent', and that the group had begun talks with other frontier labs. Asked about plans: 'we, well, you know, started two days ago'. Hairer also described OpenAI altering manuscripts 'sneakily' after criticism
- Critics quoted by The Verge: Álvaro Lozano-Robledo (UConn) on OpenAI's 'almost solved another Millennium problem' line, 'That sentence makes no sense in mathematics'; Colva Roney-Dougal (St Andrews) said it is 'somewhat agonizing' to know a large number of results may soon be announced

#### What happened
OpenAI made a sweeping claim about a model still in training while announcing an advisory body of leading mathematicians.

### Update, 6 Oct 2026: the list is out
OpenAI published the promised results as a GitHub repository with 722 manuscripts in 372 families, far more than "100+".
Many come with Lean proofs, and it released reasoning summaries and compute estimates, as AGMAI had recommended. See
2026-10-06-openai-math-release-722-manuscripts and the separate entries for the most famous claims.

#### Why it matters
If substantiated, it would mean open problems are being resolved at industrial scale. Until a list and proofs appear it is an unverified claim, and it contrasts with independent benchmarks where most open Erdős problems still resist all models.

Sources: [GitHub: openai/math (the released list, 6 Oct 2026)](https://github.com/openai/math); [OpenAI: Sharing AI progress in mathematics (6 Oct 2026)](https://openai.com/index/sharing-ai-progress-in-mathematics/); [TechCrunch: OpenAI forms math advisory group as its AI resolves more than 100 open problems](https://techcrunch.com/2026/09/21/openai-forms-math-advisory-group-as-its-ai-resolves-more-than-100-open-problems/); [The Decoder: OpenAI says internal model solved over 100 long-standing math problems](https://the-decoder.com/openai-says-its-internal-model-solved-over-100-long-standing-math-problems-after-just-a-month-of-training/); [FrontierMath Erdős benchmark (arXiv 2609.25050)](https://arxiv.org/abs/2609.25050); [OEIS Open benchmark (arXiv 2608.11941)](https://arxiv.org/abs/2608.11941); [OpenAI: Advisory Group on Mathematics and Artificial Intelligence](https://openai.com/index/advisory-group-on-mathematics-and-ai/); [Terence Tao blog: Announcing the Advisory Group on Mathematics and Artificial Intelligence](https://terrytao.wordpress.com/2026/09/21/advisory-group-on-mathematics-and-artificial-intelligence/); [Thomas Bloom on X: FrontierMath Erdős thread](https://x.com/thomasfbloom/status/2095630765035864260); [The Verge (23 Sep): OpenAI wants to consult elite mathematicians about how to not fumble again](https://www.theverge.com/ai-artificial-intelligence/999167/openai-elite-mathematicians-panel); [The Verge (28 Sep): OpenAI keeps bulldozing mathematicians (Hairer interview)](https://www.theverge.com/ai-artificial-intelligence/1001477/openai-math-advisory-group)

### 2026-09-22: Mathematician posts an unchecked ChatGPT Astra proof of the planar Mumford–Shah conjecture (1989), citing OpenAI's '100 open problems' claim

****; Francesco Deangelis, University of Münster, OpenAI; Science & math; confidence low; https://postcutoff.com/e/2026-09-22-mumford-shah-conjecture-astra-claim/

On 22 Sep 2026 Francesco Deangelis (University of Münster) posted a claimed proof of the planar Mumford–Shah conjecture (1989), a central open problem in the calculus of variations and image segmentation. He says ChatGPT Astra produced it on 9 Sep. He calls it an 'AI-generated draft' that he is still checking. He released it early to document priority after OpenAI's 21 Sep claim that an internal model had solved 100+ open problems.

- arXiv 2609.26732 (22 Sep 2026): 'On September 9, 2026, I obtained a proof of the Mumford–Shah conjecture, formulated in 1989, using ChatGPT Astra'
- Timeline in the author's note: 5 Sep, Astra in Work mode ran 6 hours without a complete proof; 6 Sep, it proved the author's Dirichlet-boundary conjecture in 1 h 20 min, building on his unpublished April 2026 work; 9 Sep, given that result, it produced a complete proof after 2 h 30 min (37 pages); 11 Sep, a shorter version, which is the one posted
- Author's caveat: 'This prompted me to make the current AI-generated draft publicly available now, to document the chronology of this work. I will continue checking the mathematical details'
- Claimed method: classify planar generalized global minimizers with a second-variation argument (two perpendicular translations of a compact crack portion, a Hodge identity and holomorphic rigidity). The known equivalence between that classification and interior regularity then gives the conjecture
- Status: unrefereed, not formalised, and not publicly checked by independent experts as of 30 Sep 2026

#### What happened
Deangelis, who worked on Mumford–Shah regularity in his PhD, gave ChatGPT Astra his thesis, reference books and unpublished notes. First it proved a boundary-regularity conjecture of his. Then, given that result, it produced a full claimed proof of the interior conjecture. He posted the model's shortened manuscript almost unchanged, with a note on its provenance.

#### Why it matters
If correct, it would settle one of the best-known open problems in the calculus of variations. The case also shows OpenAI's unverified "100 open problems" announcement leading independent researchers to rush their own AI-generated claims onto arXiv before checking them.

Sources: [arXiv 2609.26732: Solution of the Mumford–Shah conjecture (Deangelis)](https://arxiv.org/abs/2609.26732)

### 2026-09-22: Odlyzko–Poonen conjecture (1993) proved unconditionally: 'The proofs are due to GPT-6 Astra', which also wrote a 22,000-line Lean formalisation

****; Constantin Kogler, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-09-22-odlyzko-poonen-conjecture-proved/

Constantin Kogler posted an unconditional proof of the Odlyzko–Poonen conjecture: a random monic 0/1 polynomial with constant term 1 is irreducible over Q with probability tending to 1 (arXiv 2609.26771, 22 Sep 2026). Breuillard and Varjú had proved this in 2019 only under the Generalized Riemann Hypothesis. The paper says the proofs are due to GPT-6 Astra. The model also formalised everything in Lean (about 22,000 lines in about 30 hours), and Emmanuel Breuillard checked the proof.

- Conjecture: Odlyzko and Poonen (1993); Konyagin (1999) gave a c/log n lower bound; Breuillard–Varjú (2019) proved it under GRH; Bary-Soroker–Koukoulopoulos–Kozma (2023) handled coefficients from at least 35 consecutive integers
- New: P(P_n irreducible) → 1 unconditionally, with P(reducible) = sqrt(2/(πn)) + O(1/n)
- Machine Contribution Statement: 'The proofs are due to GPT-6 Astra. The author rewrote and checked the arguments'; first found on 14 Sep 2026, after the model had proved a related result on Bernoulli convolutions (first under GRH, then unconditionally)
- 'GPT-6 Astra formalized all results from this paper and all necessary results from previous work. The formalization took around 30 hours and around 22'000 lines of code were written' (repository ckkogler/odlyzko-poonen-lean)
- Kogler thanks Emmanuel Breuillard 'for checking the proof', and thanks Breuillard and Péter Varjú for help with the writing

#### What happened
While studying Bernoulli convolutions with GPT-6 Astra, Kogler saw the model remove a GRH assumption from a related result. He then asked it to try the Odlyzko–Poonen conjecture. The model produced a proof and formalised it together with all the prior results it needed.

#### Why it matters
The unconditional case had stayed open after Breuillard and Varjú's conditional proof in 2019. The paper assigns the mathematics fully to the model and includes both a machine-checked proof and a check by a leading expert, which is unusually complete verification for an AI result.

Sources: [arXiv 2609.26771: The Odlyzko–Poonen Conjecture on Irreducibility of Random Polynomials (Kogler)](https://arxiv.org/abs/2609.26771); [GitHub: ckkogler/odlyzko-poonen-lean](https://github.com/ckkogler/odlyzko-poonen-lean)

### 2026-09-23: Claude agents discover a novel CRISPR-like enzyme system; Anthropic reveals its own biology wet lab

****; Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-09-23-claude-discovers-novel-enzyme-system/

On September 23, 2026 Anthropic reported that about 950 Claude agents, running for 21 hours on 210 million tokens over a large DNA-sequence database, found array-associated reverse transcriptases (ARTs). These are a previously unknown enzyme system in bacteriophages with CRISPR-like repeat arrays. It is the first result from Anthropic's new molecular biology research group and Bay Area wet lab, which the company confirmed on Sept 18.

- Announced Sept 23, 2026; technical preprint released
- ~950 Claude agents, 21 hours, 210M tokens
- 200,000+ reverse transcriptases gathered, 3,500 candidate systems, top 20 analyzed
- CRISPR pioneer Feng Zhang (MIT): 'an exciting example of how AI agents can contribute to biological discovery'
- Anthropic's wet lab (BSL-1/BSL-2, no human pathogens, all bench work by human scientists) confirmed Sept 18 by head of life sciences Eric Kauderer-Abrams
- Disputed novelty/significance: biologist Lucas Harrington: 'finding a weird cluster of genes and repeats is often the easy part... the hard part is figuring out what the system actually does'
- Mario Rodríguez Mestre (Univ. of Copenhagen) says his team had already found the pattern and suspects it leaked from his own Claude conversations; Anthropic denies this (says Claude is not trained on user transcripts and its biology team had no access to them). Mestre's group calls the system "jumbotrons", first seen in jumbo phages in 2022, still unpublished (NYT 2026-09-27)
- CNN (Katie Hunt, Oct 8): Rodríguez Mestre told CNN 'It is essentially the same finding' and 'I am not claiming that Anthropic deliberately took our work. I cannot demonstrate that'; he thinks it more plausible that 'there was considerably more prior scientific knowledge and human direction behind the search than the phrase autonomous discovery suggests'. Anthropic did not respond to CNN
- Other experts in CNN: Aaron Engelhart (Univ. of Minnesota): 'very interesting preliminary results, but… they don't demonstrate gene editing or provide a mechanistic picture'; Ilya Finkelstein (UT Austin) noted lead author Peter Yoon is a former Doudna-lab scientist and 'four of six authors are senior molecular biologists', and that the case may make scientists wary of putting unpublished results into AI tools; Gustavo Sudre (King's College London) called the broad-direction search 'a real shift' but said 'the gap between interesting pattern and useful knowledge is where most of the work lives'. The work is not peer-reviewed

#### What happened
Anthropic formed the life-sciences research group in spring 2026 to test whether general-purpose models can speed up biological discovery. The announcement does not say which Claude model version the agents used.

#### Why it matters
It is an example of massively parallel agent search yielding a biologically novel finding endorsed by a leading domain expert. It also marks Anthropic's move into running its own physical experiments.

Sources: [The Atlantic (Hana Kiros): AI's Real Gift to Science, 'Advanced models can help researchers sort through mountains of biological data' (Oct 4, 2026; paywalled, only the dek was read)](https://www.theatlantic.com/science/2026/10/anthropic-artificial-intelligence-science-biology/688878/); [Claude discovers a novel enzyme system with CRISPR-like repeats (Anthropic)](https://www.anthropic.com/news/claude-discovers-novel-enzyme-system); [Technical preprint (PDF)](https://www-cdn.anthropic.com/22573675ada52a8ca8a97a1a4b4326b2f208a071.pdf); [TechCrunch: Anthropic says its biology lab has already found something big](https://techcrunch.com/2026/09/23/anthropic-says-its-biology-lab-has-already-found-something-big/); [TechCrunch: Anthropic is operating a lab that conducts biology experiments](https://techcrunch.com/2026/09/18/anthropic-is-operating-a-lab-that-conducts-biology-experiments/); [SiliconANGLE: Anthropic opens AI-powered biology research lab](https://siliconangle.com/2026/09/18/anthropic-opens-ai-powered-biology-research-lab/); [Phys.org: Anthropic touts AI-led biology discovery](https://phys.org/news/2026-09-anthropic-touts-ai-biology-discovery.html); [MIT Technology Review: When can we say AI made a scientific discovery?](https://www.technologyreview.com/2026/09/28/1145230/when-can-we-say-ai-made-a-scientific-discovery/); [Irish Times (NYT syndication): Did Anthropic's AI really make a scientific discovery on its own? (Rodríguez Mestre 'jumbotron' priority claim)](https://www.irishtimes.com/world/2026/09/28/did-anthropics-artificial-intelligence-really-make-a-scientific-discovery-on-its-own/); [Benzinga: scientist says he had already studied the enzymes for 4 years](https://www.benzinga.com/markets/private-markets/26/09/62022648/anthropic-claudes-claimed-breakthrough-in-biology-faces-a-major-question-scientist-says-he-had-already-studied-the-enzymes-for-4-years); [Inside Anthropic's molecular biology lab (video)](https://www.youtube.com/watch?v=DdCEmlAydcw); [Anthropic on X: Claude discovers an enzyme system](https://x.com/AnthropicAI/status/2102824959827742916); [Lucas Harrington on X: genome-mining critique thread](https://x.com/CRISPR_LuCas/status/2102878373160906938); [The Verge: Anthropic's biolab: Claude finds a CRISPR-like enzyme system](https://www.theverge.com/ai-artificial-intelligence/999470/anthropic-biolab-claude-crispr); [CNN: Anthropic's announcement of a 'Claude-led' CRISPR-like discovery is sparking controversy. Here's why](https://www.cnn.com/2026/10/08/science/ai-biology-anthropic-dna-discovery)

### 2026-09-24: DeepMind, EMBL-EBI, NVIDIA and CEPI add predicted protein-complex structures for 2,800+ viruses to the AlphaFold Database

***; Google DeepMind, EMBL-EBI, NVIDIA, CEPI; Science & math; confidence high; https://postcutoff.com/e/2026-09-24-viral-protein-complex-structures-alphafold-db/

On Sept 24, 2026 a consortium of Google DeepMind, EMBL-EBI, NVIDIA, CEPI and universities in Korea, Switzerland and the UK released predicted viral protein-complex structures for more than 2,800 viruses in the AlphaFold Database. About 30% of the interactions are new to science. The predictions were made with AlphaFold2 on NVIDIA's BioNeMo inference stack, and the database now holds 260M+ predictions.

- Complex structures for 2,800+ viruses; ~30% of the interactions are new to science
- Partners: Google DeepMind, EMBL-EBI, NVIDIA, CEPI, Seoul National University, Sungkyunkwan University, SIB, University of Glasgow
- Built with AlphaFold2 on the NVIDIA BioNeMo Inference Runtime; NVIDIA open-sourced its BioNeMo Structure Prediction Pipeline
- AlphaFold Database now holds 260M+ predictions

#### What happened
An open data release applying structure prediction to viral protein complexes at pandemic-preparedness scale, coordinated with CEPI.

#### Why it matters
It gives vaccine and antiviral researchers structural hypotheses for thousands of viruses at once. The interactions are predictions and still need experimental checks.

Sources: [NVIDIA blog: open protein dataset](https://blogs.nvidia.com/blog/open-protein-dataset/)

### 2026-09-25: Claude (Fable 5.1 in Claude Science) computes the nine-loop six-gluon amplitude in planar N=4 super-Yang-Mills, answering a physicist's public challenge

****; Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-09-25-claude-nine-loop-amplitude-n4-sym/

On Sept 25, 2026 Anthropic published a guest post by physicist-turned-writer Matt von Hippel. In August he had challenged AI companies to compute "N=8 supergravity to seven loops, or N=4 super Yang-Mills to nine loops". Anthropic physicists Liam Fitzpatrick and Siddharth Mishra-Sharma had Claude Fable 5.1, running in the Claude Science harness, compute the six-particle (hexagon) amplitude in planar N=4 SYM at nine loops, one loop past the previous record. Their prompting was little more than "keep going". Claude did it two independent ways (form-factor/antipodal-duality route and direct hexagon bootstrap), for about $1–2K of usage. SLAC's Lance Dixon verified the result. A Beijing group led by Song He had independently obtained most of it with GPT-6-based assistance.

- Challenge: von Hippel's 4gravitons post of Aug 7, 2026 ('It only counts when AI gets to my field') asked for N=8 supergravity at seven loops or N=4 SYM at nine loops on academic-scale compute; Anthropic reached out at the end of August
- Setup: Claude Fable 5.1 in Claude Science; initial prompt 'The problem is to compute the Six-particle (hexagon) amplitude in planar N=4 SYM at nine loops', then instructions like 'Keep working on this until I tell you to stop. Give me updates every 4-6 hours'
- Two routes: (1) bootstrap the nine-loop three-point form factor and map it to the amplitude via antipodal duality (the method Dixon used for eight loops, arXiv 2308.08199); (2) a direct bootstrap of the symbol in the space of hexagon functions. The two agree on every coefficient compared
- Cost: about $1,000–2,000 end-user price per approach, mostly Claude usage; the SymPy bootstrap cost ~$100 of compute (96 CPUs for about a week)
- Verification: checked with Lance Dixon (SLAC), who holds the six-, seven- and eight-loop results; the form factor is certified over the rationals at three 31-bit primes, the amplitude computed modulo two primes (99.62% of nonzero symbol coordinates reconstruct to certified rationals)
- Data released on Mishra-Sharma's 'Cosmic9' page and Zenodo (symbol as 424 quintuple coproducts over a 5,431-dimensional weight-13 hexagon symbol space); the programs are not distributed
- Priority: Song He's group (Chinese Academy of Sciences, Beijing) had independently obtained 'the majority of the result' using GPT-6-based AI assistance; the human groups will publish the physics analysis
- Von Hippel's verdict: Claude 'used known methods, with a bit more compute than people had tried to use before', not new methods; but 'this is a real frontier calculation' done essentially autonomously, and 'there is more low-hanging fruit out there than you'd expect'

#### What happened
Scattering amplitudes are usually computed to two or three loops. The planar N=4 super-Yang-Mills "toy model" has been pushed furthest with
the amplitude bootstrap. Dixon and collaborators reached eight loops indirectly through a form factor. Anthropic's physicists asked Claude
which of von Hippel's two challenges it was most likely to solve, then let the Claude Science harness run for days with minimal oversight.
Von Hippel says either route would have cost an end user about one or two thousand dollars.

#### Why it matters
It is a frontier-level computation in theoretical physics done almost autonomously by an AI agent on a modest budget. As the challenger
notes, it used known methods rather than new ones, and humans (Song He's group, with GPT-6 help) were close behind. It suggests that much
"compute-limited" science is really limited by labour and engineering.

Sources: [Zenodo 22800071: The Symbols of Six-Gluon MHV Amplitudes through Nine Loops (Song He, Jirong Jing, Xiang Li; ITP-CAS/UCAS; Sept 17, 2026)](https://zenodo.org/records/22800071); [Zenodo 22949278: Nine-loop six-gluon MHV amplitude in planar N=4 SYM, large data files (author: Claude (Anthropic), Sept 25, 2026, CC BY 4.0)](https://zenodo.org/records/22949278); [Anthropic (guest post by Matt von Hippel): Yes, Claude can do nine loops](https://www.anthropic.com/research/yes-claude-can-do-nine-loops); [Cosmic9: nine-loop six-point amplitudes computed by Claude (data page)](https://smsharma.io/cosmic-nine-loops/); [4gravitons: It only counts when AI gets to my field (the challenge, Aug 7, 2026)](https://4gravitons.com/2026/08/07/it-only-counts-when-ai-gets-to-my-field/); [arXiv 2308.08199: eight-loop amplitude via form factor and antipodal duality (Dixon et al.)](https://arxiv.org/abs/2308.08199)

### 2026-09-25: Conjectures.io bounty platform pays out for Lean proofs of Erdős #859, #18(b), #1062(ii) and Ben Green's problems 24, 39, 40 within ten days, mostly to Purdue's 'JenW1N'

***; Conjectures.io, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-09-25-conjectures-io-erdos-green-bounty-solves/

Between Sept 15 and Sept 25, 2026, Conjectures.io paid bounties for machine-checked Lean proofs of seven formalised open problems. The platform runs on Bittensor subnet 66, publishes open problems as pinned Lean statements and pays in its alpha token after a kernel check and human review. Six of the seven went to "JenW1N" (Purdue students Jensen Kohlmeyer and Liam Kruer): Erdős #108 (see its own entry), #18(b), #859, #1062(ii), and Green's open problems 24 and 40. Green's problem 39 went to "Jordan". Proofs run to 14k–74k lines of Lean, and the JenW1N submissions credit OpenAI ChatGPT/Codex. By Oct 5 the platform listed 26 problems solved and about $128–130k paid. erdosproblems.com still marked #18, #859 and #1062 as open, with pending proof claims, so expert acceptance is not yet established.

- Results page (Oct 5, 2026): 26 problems solved, 33 verified proofs, $129,963 (127,226.57 α) paid, 257 live bounties; two submissions rejected (Erdős #70, #252) as already solved/formalised; Erdős #579 (by 'Jordan', Oct 3) pending review. (The lead recorded $127,713 earlier the same day; the total moves with the token price)
- Erdős #18(b), JenW1N, Sept 16, $3,232: proves h(n!) < n^ε for every ε > 0 and large n (h(m) = number of divisors needed to represent every 1 ≤ k < m as a sum of distinct divisors of a practical m); Lean file 14,043 lines. Erdős had shown h(n!) < n
- Erdős #859, JenW1N, Sept 18, $3,820: the density d_t of n for which t is a sum of distinct divisors of n satisfies d_t ~ c₁/(log t)^c₂ (Erdős 1970 had only bounds of this shape); Lean file 36,197 lines
- Erdős #1062(ii), JenW1N, Sept 21, $4,007: on fork-free subsets of {1,…,n} (no a | b and a | c), whether lim f(n)/n is irrational (Guy's problem B24); Lean file 74,209 lines; the page itself notes that 'a complete ordinary-kernel check and target axiom audit are still required'
- Green's open problem 24, JenW1N, Sept 25, $4,247: a (1/3 + o(1))n² asymptotic from Ben Green's list of 100 open problems; 52,946 lines of Lean; disclosure: 'OpenAI ChatGPT/Codex assisted with the mathematics, certificate generation, Lean formalization, verification, and presentation'
- Green's open problem 40, JenW1N, Sept 15, $4,436; Green's open problem 39, 'Jordan', Sept 19, $2,793
- Platform model: 'checking a proof takes seconds where finding one can take months'; accepts proofs made with 'any AI, tool, or method'; payment only after Lean kernel check against the pinned statement plus human review for gaming or plagiarism
- erdosproblems.com on Oct 5, 2026: #18 OPEN with 3 proof claims, #859 OPEN with 1 proof claim, #1062 OPEN with 1 proof claim, #579 OPEN with none

#### What happened
After the Erdős #108 disproof (Sept 15), the same two-person Purdue team kept claiming Conjectures.io bounties. They took several number-theory problems on divisors and practical numbers that Erdős posed in the 1970s, plus three problems from Ben Green's list. Each solution is a single Lean file checked against the platform's pinned formal statement, and the platform describes it in a write-up. Rewards were paid in Bittensor alpha, worth about $2.8k–4.4k per problem.

We read the solution pages through a summarising fetcher. Exact statements, especially the precise formal form of #1062(ii) and Green 24, should be checked against the pinned Lean targets.

#### Why it matters
A crypto-funded bounty market is now paying for formal proofs of open problems, and AI-assisted solvers are collecting the payouts in days. The Lean check guarantees that the pinned statement was proved. It does not guarantee that the statement matches what Erdős or Green meant, or that mathematicians find the proof illuminating. Graph theorists raised the same complaint about the #108 write-up. Watch whether erdosproblems.com accepts the claims.

Sources: [Conjectures.io results](https://conjectures.io/results); [Conjectures.io: Erdős #18(b) solution](https://conjectures.io/results/e93a2766-4c70-4564-b565-d0c556f35929/solution); [Conjectures.io: Erdős #859 solution](https://conjectures.io/results/e060355f-a728-4b06-af11-6302b5780d19/solution); [Conjectures.io: Erdős #1062(ii) solution](https://conjectures.io/results/8d59a0af-6762-4606-93c9-72dd356a57bc/solution); [Conjectures.io: Green's open problem 24 solution](https://conjectures.io/results/4b04c9b9-323d-446c-84e1-65f9dd09231d/solution); [erdosproblems.com #18](https://www.erdosproblems.com/18); [erdosproblems.com #859](https://www.erdosproblems.com/859); [erdosproblems.com #1062](https://www.erdosproblems.com/1062)

### 2026-09-25: Lila Sciences' AI-run lab screens 2,942 catalysts and finds iridium- and ruthenium-free palladium oxides for green hydrogen

***; Lila Sciences; Science & math; confidence medium; https://postcutoff.com/e/2026-09-25-lila-ai-lab-palladium-oer-catalysts/

On 25 Sept 2026 Lila Sciences reported that its AI-directed autonomous lab proposed, synthesized and screened 2,942 oxide catalysts (53 material systems, 26 elements) for the acidic oxygen evolution reaction used in PEM water electrolysis. It identified six palladium-based families on or near the activity–stability Pareto front. The best performed comparably to ruthenium over 1,000+ hours of stability tests. The results are in a preprint (arXiv 2609.30133) and have not been peer-reviewed.

- 2,942 catalysts across 53 material systems and 26 elements; 6 Pd-based families on or near the Pareto front (e.g. InMnPdOx, NiTaPdOx)
- Lead composition performed comparably to ruthenium in activity after 1,000+ hours of stability testing (company claim)
- Palladium had been widely considered a dead end for acidic OER
- Bayesian models combined with language models chose experiments; humans handled safety review and some manual sample transfers; Lila claims ~17x faster screening and >90% less human time per sample
- Preprint: Jenewein et al., 21 authors, all Lila Sciences, submitted 24 Sept 2026
- Company context: Flagship Pioneering spin-out; $550M raised by Oct 2025 (incl. NVentures), valuation >$1.3B; Bloomberg (3 June 2026) reported talks to raise ~$2B at ~$8.5B pre-money

#### What happened
Lila's autonomous materials lab ran closed-loop campaigns in which AI models proposed oxide compositions. Robotic sputtering and electrochemical stations made and tested them, and the results fed back into the models. The AI pushed into palladium compositions that experts had largely written off and found stable, active catalysts without iridium or ruthenium.

#### Why it matters
It is one of the first concrete, data-backed discovery claims from the heavily funded "scientific superintelligence" startups. It addresses a real bottleneck for gigawatt-scale green hydrogen, where iridium supply is scarce. It is still a company preprint and needs peer review and industrial-scale testing.

Sources: [Lila: How an AI-run lab cracked open green hydrogen's catalyst problem](https://www.lila.ai/news/how-an-ai-run-lab-cracked-open-green-hydrogens-catalyst-problem); [arXiv 2609.30133: AI-guided high-throughput discovery of Ir- and Ru-free palladium-oxide catalysts](https://arxiv.org/abs/2609.30133); [Unite.AI: Lila Sciences' AI lab uncovers palladium catalysts for green hydrogen](https://www.unite.ai/lila-sciences-ai-lab-uncovers-palladium-catalysts-for-green-hydrogen/); [Bloomberg: Lila Sciences said in talks for funds at $8.5B valuation](https://www.bloomberg.com/news/articles/2026-06-03/lila-sciences-said-in-talks-for-funds-at-8-5-billion-valuation); [Lila: $350M Series A announcement](https://www.lila.ai/news/announcing-the-close-of-our-series-a); [MIT Technology Review: AI materials-discovery startups (Dec 2025)](https://www.technologyreview.com/2025/12/15/1129210/ai-materials-science-discovery-startups-investment/)

### 2026-09-25: New record bound on the irrationality measure of ζ(2), μ ≤ 5.0495243, released as a 92k-line Lean proof written by Claude; beaten a week later by a Claude-assisted paper

***; Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-09-25-zeta2-irrationality-measure-claude-lean/

On Sept 25, 2026 Jonathan Kleid published a machine-checked Lean 4 proof that the irrationality measure of ζ(2) = π²/6 is at most 5.0495243, improving Zudilin's 2014 record of 5.095412. Kleid found the parameters, (13, 11, 9, 15; 26) in Zudilin's construction, with his own computer-algebra system qedbook. Per the repository, "the Lean text was written by Claude (Anthropic) sessions directed by Jonathan Kleid". The development has 166 hand-structured modules (91,672 lines) plus 559 generated certificate modules, with no sorry and only Lean's three standard axioms. On Oct 2 David Niedbala Giraudin posted a 6-page paper (arXiv 2610.02912; "Technical assistance: Claude (Anthropic)") lowering the bound to 5.0193784; it cites Kleid's bound as the previous record.

- Theorem (Lean): ¬ LiouvilleWith 5.0495243 ζ(2), plus the explicit rational-approximation form C/n^5.0495243 ≤ |ζ(2) − m/n| for large n; certified value of the construction 5.04952429053…
- Previous records: 5.441243 (Rhin–Viola, 1996), 5.095412 (Zudilin, Ann. Math. Québec 2014)
- Provenance (README): 'The Lean text was written by Claude (Anthropic) sessions directed by Jonathan Kleid, who also designed the search program and the verification discipline (every gate proven able to go red before its green was accepted)'
- Size: 166 modules / 91,672 lines, plus 559 generated modules holding a creative-telescoping certificate as exact integer data; full build ~12.6 CPU-hours, ~13 GB disk, assembly modules peak near 18 GB RAM
- Fills Mathlib gaps: explicit complex Binet bound for log Γ, a linear-forms criterion for ¬LiouvilleWith, Poincaré-type growth bounds for three-term recurrences, p-adic analysis of the common factor; the PNT comes from the vendored PrimeNumberTheoremAnd project (Kontorovich, Tao et al.)
- Repository qedbook/zeta2-irrationality-measure created 2026-09-25 (Apache-2.0); no accompanying paper
- Superseded Oct 2, 2026: Niedbala Giraudin, arXiv 2610.02912, μ(ζ(2)) < 5.0193784, using Zudilin's two hypergeometric constructions at new parameters without creative telescoping, constants checked by ball arithmetic; acknowledgements: 'Technical assistance: Claude (Anthropic)'. The same author's arXiv 2609.26980 (Sept 22) improved the 25-year-old ζ(3) record to μ(ζ(3)) < 5.5138800 (from 5.513891), 'carried out with the assistance of an AI system (Claude, Anthropic)'; both follow Bai's GPT-assisted π bound (2026-09-10-pi-irrationality-measure-bai-gpt)

#### What happened
Kleid searched the parameter space of Zudilin's Mellin–Barnes construction with qedbook, his own computer-algebra system, using Marcovecchio's pairing theorem to make each candidate tractable. He found a member whose growth rates give the exponent 1 + (42.034 + 15.019)/(29.108 − 15.019) ≈ 5.0495. He then had Claude write a complete Lean 4 proof. Every value computed in qedbook enters the proof as data that Lean's kernel re-checks. The repository says what is and is not proved: the sharper figure 5.04952429 printed by the search is *not* claimed.

A week later a preprint by David Niedbala Giraudin (Claude credited for technical assistance) beat the bound, citing Kleid's "machine-verified bound … published in September 2026" as the intermediate record.

#### Why it matters
It is a new way to publish: a record in analytic number theory released as a fully kernel-checked, AI-written formal proof rather than a paper. It also shows the pace of autumn 2026, when an AI-formalized record stood for only a week before a short paper, itself written with Claude as technical assistant, improved it. The open question is whether qedbook-style search plus AI formalization can now reach the new bound.

Sources: [GitHub qedbook/zeta2-irrationality-measure: μ(ζ(2)) ≤ 5.0495243, machine-checked in Lean 4](https://github.com/qedbook/zeta2-irrationality-measure); [qedbook](https://qedbook.com); [arXiv 2610.02912: A new upper bound for the irrationality exponent of ζ(2) (Niedbala Giraudin)](https://arxiv.org/abs/2610.02912); [arXiv 2609.26980: A new upper bound for the irrationality exponent of ζ(3) (Niedbala Giraudin)](https://arxiv.org/abs/2609.26980); [Zudilin (2014): Two hypergeometric tales and a new irrationality measure of ζ(2)](https://arxiv.org/abs/1310.1526)

### 2026-09-26: GPT-6 Astra finds a proof improving the Kővári–Sós–Turán bound: diagonal bipartite Ramsey numbers b(t,t) = O(2^t) (Mubayi)

***; University of Illinois Chicago, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-09-26-mubayi-zarankiewicz-astra/

On Sept 26, 2026 Dhruv Mubayi posted a proof that every bipartite graph with N vertices per side and at least N²/2 edges contains K_{t,t} once N ≥ c·2^t. That improves the classical Kővári–Sós–Turán requirement of order t·2^t and cuts the diagonal bipartite Ramsey upper bound from Conlon's O(2^t log t) to O(2^t). The abstract says "The proof was found by GPT-6 Astra": an 8-page proof the author first found incomprehensible and then rewrote with the model.

- arXiv 2609.32937 (math.CO), Dhruv Mubayi, 'Sharper Zarankiewicz and Diagonal Bipartite Ramsey Bounds', Sept 26, 2026
- Result: density ≥ 1/2 bipartite graphs contain K_{t,t} when N ≥ c·2^t; hence b(t,t) = O(2^t), improving Conlon's O(2^t log t)
- AI declaration: 'The proof of the main result was found by GPT-6 Astra after several guided prompts by the author. Astra first produced a proof that was only 8 pages long and was incomprehensible to the author'; later write-ups are largely the author's
- Mubayi had earlier prompted Astra to extensions of the Erdős–Sós work (see 2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra)

#### What happened
Mubayi, a leading extremal combinatorialist, reports that GPT-6 Astra found a proof removing the log factor from the best bound on diagonal
bipartite Ramsey numbers. The improvement over the 1954 Kővári–Sós–Turán counting bound comes from a new method that he says "will probably have further applications".

#### Why it matters
It is a quantitative improvement on a classical bound in a central area, credited outright to a model in the abstract. That kind of
disclosure was rare before September 2026.

Sources: [arXiv 2609.32937: Sharper Zarankiewicz and Diagonal Bipartite Ramsey Bounds](https://arxiv.org/abs/2609.32937)

### 2026-09-27: Erdős Problem #993 (1987): independent-set sequences of all forests claimed unimodal; a second proof and its Lean 4 formalization are attributed entirely to AI agents (Zhang, Li, Vallier)

****; Northwestern Polytechnical University, University of Toledo, OpenAI, Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-09-27-erdos-993-forest-independence-polynomials-unimodal/

On 27 Sept 2026 Tong Zhang and Wei Li (Northwestern Polytechnical University, Xi'an) deposited on Zenodo a computer-assisted proof that the independence sequence of every finite forest is unimodal, the 1987 question of Alavi, Malde, Schwenk and Erdős listed as Erdős Problem #993. Their record says "OpenAI's GPT 6.0 Astra assisted with the computational work; the authors provided the overall proof strategy." On 6 Oct 2026 Wei Li, Kevin Vallier (University of Toledo) and Tong Zhang posted arXiv 2610.07943 with a second proof and a Lean 4 formalization. It states: "AI systems produced the new mathematical arguments and formal proofs reported here and wrote most of the text", naming Claude Opus 5.5, Claude Fable 5.1 and several OpenAI models. Both are unrefereed; the arXiv paper calls itself a provisional manuscript shared "for examination by the community".

- Problem: for a graph G let i_k(G) be the number of independent vertex sets of size k. Alavi, Malde, Schwenk and Erdős (1987) showed the sequence need not be unimodal for general graphs and asked: 'For trees (or perhaps forests), is the vertex independence sequence unimodal?' It is Problem #993 on erdosproblems.com
- Why it was hard (paper): the stronger property, log-concavity, fails. Levit and Mandrescu conjectured it in 2004; Kadrawi, Levit, Yosef and Mizrachi found two trees on 26 vertices that are not log-concave
- 17 Sept 2026: Fang, Lu, Nevo, Yao and Zheng (arXiv 2609.20961) proved unimodality for all forests with at least N_0 vertices, for an unspecified constant N_0. Their proof claim on erdosproblems.com says 'using Odin Automatic AI Research Agent'
- 27 Sept 2026: Zhang and Li, 'Unimodality of Forest Independence Polynomials', Zenodo doi 10.5281/zenodo.22999166. Abstract: 'We reduce the large-order threshold to 59 vertices and extend the finite-order coverage through 60 vertices; the overlapping ranges complete the proof. OpenAI's GPT 6.0 Astra assisted with the computational work; the authors provided the overall proof strategy.'
- Zenodo update of 30 Sept: 'The arXiv submission was withdrawn on 28 September 2026, before it was publicly announced. We are revising the manuscript to make its exposition clearer and easier to understand.'
- 6 Oct 2026: Li, Vallier and Zhang, arXiv 2610.07943 [math.CO], 81 pages, 'Draft, version 2.2': a second proof. Forests with at most 24 vertices by exact counting; forests with at least 25 vertices by one moment argument on the hard-core model. 'No forest is enumerated.'
- AI disclosure (abstract): 'AI systems produced the new mathematical arguments and formal proofs reported here and wrote most of the text, under the direction described in Section 9. The paper has not undergone external peer review'
- Section 9: 'The project account attributes all of the new mathematics of this paper and all of its formal proofs to AI systems in a project directed by K. Vallier. The systems were Claude agents, running Anthropic's models Claude Opus 5.5 and Claude Fable 5.1; two OpenAI Codex agents, called Sol and Astra in the project, which over the period ran the OpenAI models GPT-5.6 Sol, GPT-6 Astra in Codex mode, GPT-6 Sol and GPT-6.1 Sol; and ChatGPT conversations with the OpenAI model GPT-6 Astra in Pro mode'
- Human roles per Section 9: Zhang and Li gave the earlier proof whose decomposition, binomial-mixture reduction and several inequalities the new proof reuses; Zhang wrote the introduction, Li reviewed it, Vallier directed the AI project. The section adds: 'This attribution does not assert that every author has reviewed the complete integrated manuscript.'
- Timeline per Section 9: the moment argument was developed on 27–29 Sept 2026, the proofs by hand on 28–30 Sept, the final route fixed on 1 Oct and the last inequality produced on 1–2 Oct
- Lean 4 (Mathlib): final theorem erdos993_v22_final : Erdos993Statement, with 'no premise'; code at github.com/selfreferencing/erdos993-lean. The paper notes the use of compiled evaluations; Zhang and Li's Zenodo note on the earlier release v1.0-claim says the large-order part 'uses native_decide and therefore additionally relies on Lean's compiler and runtime'
- erdosproblems.com (read 13 Oct 2026) lists two proof claims for #993 (Fang et al., 21 Sept; Tong Zhang 'using GPT6.0 astra', 27 Sept) under the notice 'Appearing on this page is no guarantee of proof correctness'. The site froze new proof claims and removed status labels on 6 Oct
- AI credit in arXiv 2609.20961 (Fang, Lu, Nevo, Yao, Zheng; PDF read 2026-10-16): 'The Odin Automatic AI Research Agent was used to find the initial proof'. The same agent is credited with the S³×S³ positive-curvature metric (2026-09-06-odin-ai-agent-positive-curvature-s3xs3-hopf-sign-conjecture)

#### What happened
An independent set in a graph is a set of vertices with no edge between any two of them. Count the independent sets of size 0, 1,
2, … and you get the graph's independence sequence. In 1987 Alavi, Malde, Schwenk and Erdős showed that for general graphs this
sequence can go up and down in any pattern, and asked whether for trees and forests it always rises to a peak and then falls
(unimodality). The question became one of the best-known open problems about graph polynomials. A natural route, proving the
stronger log-concavity, was closed when non-log-concave trees on 26 vertices were found.

In September 2026 the problem moved quickly. On 17 Sept, Fang, Lu, Nevo, Yao and Zheng proved the statement for all sufficiently
large forests. On 27 Sept, Tong Zhang and Wei Li claimed the full result: large forests (59 vertices and up) by a probabilistic
reduction, small ones (up to 60) by structural inequalities with exact computer certificates. They credit GPT-6 Astra with the
computational work and themselves with the strategy. They withdrew the arXiv submission the next day to rewrite the exposition
and left the manuscript on Zenodo.

Kevin Vallier (University of Toledo) then ran a project in which Claude and OpenAI Codex agents built a different argument on top
of Zhang and Li's reductions and formalized it in Lean 4. The resulting paper, posted on 6 Oct with all three as authors, is
unusually explicit: it attributes "all of the new mathematics of this paper and all of its formal proofs" to AI systems, lists the
models, dates each step between 27 Sept and 2 Oct, and says that most of the prose was written by Claude agents. It also says the
authors are rewriting the manuscript before journal submission and that not every author has necessarily reviewed the whole text.

#### Why it matters
It is an Erdős problem with a long literature, not an obscure one, and the paper describes a mode of work that goes beyond
"AI-assisted": agents of two companies' models produced the arguments, checked one another's certificates and wrote the formal
proof, with a human directing. The Lean theorem gives the claim a firm anchor that most AI-written proofs of 2026 lack, with the
caveat that parts of it rely on compiled evaluation. The human-readable proof is 81 pages and by the authors' own account still
provisional.

#### Not verified here
The Lean development was not re-run for this entry, and the Zenodo PDF of the first proof was not read (its abstract and update
note were). No expert assessment, press coverage or Hacker News discussion was found on 13 Oct 2026; erdosproblems.com no longer
shows open/solved labels. Vallier's field and role beyond "directing the project" are not described in the paper.

Sources: [arXiv 2610.07943: Unimodality of Forest Independence Polynomials (Li, Vallier, Zhang; second proof)](https://arxiv.org/abs/2610.07943); [Zenodo: Zhang & Li, Unimodality of Forest Independence Polynomials (first proof, 27 Sept 2026)](https://doi.org/10.5281/zenodo.22999166); [Erdős Problem #993 (erdosproblems.com)](https://www.erdosproblems.com/993); [Erdős Problem #993: proof claims thread](https://www.erdosproblems.com/forum/thread/993/proof-claims); [Lean 4 formalization (selfreferencing/erdos993-lean)](https://github.com/selfreferencing/erdos993-lean); [Fang, Lu, Nevo, Yao & Zheng, Unimodality of independence polynomials for sufficiently large forests (arXiv 2609.20961)](https://arxiv.org/abs/2609.20961)

### 2026-09-27: Perelman's proof of the Poincaré conjecture formalised in Lean, with AI-generated code (Chow, Qin, Liao, Khaitan)

****; UC San Diego, Cornell University, Princeton University; Science & math; confidence medium; https://postcutoff.com/e/2026-09-27-poincare-conjecture-lean-formalization/

On Sept 27, 2026 Ziyang Qin, Yuan Liao, Ayush Khaitan and Ricci-flow expert Bennett Chow posted a Lean 4 formalization of the Hamilton–Perelman proof of the 3D Poincaré conjecture. Via the Moise smoothing theorem it covers both the smooth and the topological versions, and it proves the statement Mathlib had marked `proof_wanted`. The axiom check shows no `sorry` and no project axioms. The paper says "automated assistance was used for proofs and prose". Secondary reports say about 2.7M of 4.7M Lean lines were produced in the last two weeks with ChatGPT/Codex and Claude. It is the first machine-checked proof of a solved Millennium Prize problem.

- arXiv 2609.33842 (math.DG), submitted Sept 27, 2026; code: github.com/qinz1yang/differential-geometry, release v0.1.3 (Lean/Mathlib v4.33.1)
- Lean endpoints: exists_isManifold_three (Moise), smoothPoincareConjecture_holds (smooth), DifferentialGeometry.Topology.poincare_conjecture (topological); all four queried closures use only propext, Classical.choice, Quot.sound, with no sorryAx or project-specific axiom (paper)
- Proves the topological 3D Poincaré statement marked proof_wanted in Mathlib v4.33.1, checked by applying the endpoints directly (paper)
- Route: Ricci flow with surgery and finite-time extinction (Hamilton–Perelman), following Kleiner–Lott and Morgan–Tian
- AI disclosure in the paper: 'Automated assistance was used for proofs and prose; the authors retain responsibility'; it reuses parts of Tau Ceti, a Lean library credited to 'J. Tooby-Smith, Codex, and The Tau Ceti contributors'
- Secondary report (KuCoin flash citing Chinese outlet Xinzhiyuan): ~4.7M lines in total, ~2.7M in the final two weeks with ChatGPT (Astra), Codex and Claude (Fable 5.1); Qin made 7,477 of 11,000+ commits; ~7 months of work. Unverified against a primary source
- Repo README 'AI Disclaimer' (checked Oct 1): 'AI agents assisted with formalizing individual proofs and writing boilerplate'; names 'Generative AI (ChatGPT, Claude, Deepseek, Gemini, GLM, etc.)' without saying which models; 'The high-level architecture is human-designed'; humans checked definitions and core theorem statements; the repo shows 11,175 commits. Line counts and the Astra/Fable 5.1 attribution are still only in secondary reports
- Preceded by the same group's 'A Lean Formalization of Hamilton's Three-Manifold Theorem' (arXiv 2608.21502, Aug 2026)
- Second, independent formalization (6 Oct 2026): Zhang, Delaval, Chen, Chen, Xu, Liao, Jiang, Liu and Bin Dong (Peking University/BICMR and others), arXiv 2610.08329, code github.com/frenzymath/PoincareConjecture. It follows Morgan–Tian, proves the topological and smooth theorems, passes a full Lean build and Comparator checks against Mathlib-only target statements (also re-checked with the independent Nanoda kernel)
- Frenzymath numbers (paper): preparation from May 2026 with three AI engineers and a group of mathematicians; ~90 milestones; main formalization restarted in September and finished in 'slightly more than two weeks' with publicly available commercial models; ~3.2M Lean lines, 2.8M after dependency extraction; cost ~$25,000 (~$20,000 of AI subscriptions, mainly GPT Pro 20× plans, plus ~$5,000 of servers)
- Frenzymath models (paper): 'GPT-6-Astra formalized the milestone statements in Lean, and both GPT-6-Astra and Fable5.1 performed multiple rounds of statement review. Proof construction relied primarily on GPT-6-Astra, with occasional consultation of Fable-5.1'; interactive sessions used Codex and Claude Code plus a distributed 'Archon Horizon' mission workflow
- Follow-up (9 Oct 2026): the same group, now ten authors, formalized the full geometrization conjecture (arXiv 2610.13897, release v0.1.5) and named its models for the first time: the Humanize builder–reviewer workflow with 'GPT-6-Astra as the reviewer' and builders chosen among GPT-6 Luna, GPT-6 Sol, GPT-6 Astra, Claude Opus 5.5 and DeepSeek V4.1 Flash. That statement is about the geometrization work; the Poincaré paper itself still gives no model names or line counts (see 2026-10-09-geometrization-conjecture-lean-formalization)

#### What happened
Bennett Chow (UCSD), a leading Ricci-flow researcher, and three younger collaborators released a Lean 4 library that formalizes the whole
Hamilton–Perelman proof: short-time existence via DeTurck, singularity analysis, surgery and finite-time extinction. With the Moise smoothing
theorem it yields the topological Poincaré conjecture. The paper is short (it points to the code) and includes axiom-check commands that anyone can rerun.

On 6 October a second, independent team (Frenzymath, led from Peking University by Bin Dong and others) posted its own complete formalization of both the smooth and topological Poincaré theorems, following Morgan–Tian (arXiv 2610.08329). Unlike Chow's group, it documents the AI side in detail: GPT-6 Astra wrote most of the proofs, Claude Fable 5.1 helped review statements, the main phase took just over two weeks, the code is about 2.8 million lines and the project cost about $25,000. Two independent machine-checked proofs now exist.

Chow's group's paper discloses AI help in one sentence. The scale (millions of lines in months) and the model names come from secondary reports, and
confidence is medium until the authors or the labs confirm them.

#### Why it matters
After Fermat's Last Theorem (Claude, Sept 2026), this is the second landmark formalization of a famous proof in a month. Perelman's proof
needed years of expert refereeing in 2003–06. A kernel-checked version suggests that AI-generated Lean at scale now reaches the hardest
parts of geometric analysis. Experts still have to check that the Lean definitions match the mathematics.

Sources: [arXiv: A Lean Formalization of the Hamilton–Perelman Proof of the Three-Dimensional Poincaré Conjecture](https://arxiv.org/abs/2609.33842); [GitHub: qinz1yang/differential-geometry](https://github.com/qinz1yang/differential-geometry); [arXiv: A Lean Formalization of Hamilton's Three-Manifold Theorem (2608.21502)](https://arxiv.org/abs/2608.21502); [KuCoin News: Poincaré Conjecture proof fully verified by AI in 4.7 million lines of code](https://www.kucoin.com/news/flash/poincar-conjecture-proof-fully-verified-by-ai-in-4-7-million-lines-of-code); [Bennett Chow: Ricci Flow as a Test for AI (CMSA talk slides, Sept 2025)](https://mathweb.ucsd.edu/~bechow/GML-Talk-CMSA-091625.pdf); [Frenzymath: Formalizing the Poincaré Conjecture in Lean 4 (separate project)](https://github.com/frenzymath/Poincare-Conjecture); [arXiv 2610.08329: An AI-Assisted Formalization of the Poincaré Conjecture (Frenzymath/PKU)](https://arxiv.org/abs/2610.08329); [GitHub: frenzymath/PoincareConjecture (second, independent formalization)](https://github.com/frenzymath/PoincareConjecture)

### 2026-09-28: Four-body problem: uniqueness of convex central configurations for all masses claimed (Simó–Yoccoz conjecture), AI-found and formalized in Lean (Tomar)

***; independent; Science & math; confidence medium; https://postcutoff.com/e/2026-09-28-four-body-convex-central-configurations-unique-tomar/

On 28 Sept 2026 Tejasvi Singh Tomar (independent researcher) posted arXiv 2609.35632: for every four positive masses and every cyclic ordering there is exactly one strictly convex planar central configuration up to similarity. This answers Problem 10 of Albouy, Cabral and Santos's 2012 list of open problems on the n-body problem, conjectured by Albouy, Fu and Sun in 2008. The proof uses interval arithmetic and is formalized in Lean; the paper states that "Generative AI tools were used to find and prove the results". Unrefereed, no model named, no expert reaction found. It preceded the same author's claims on Smale's sixth problem and on 3×3 matrix multiplication by ten days.

- Paper: 'Uniqueness of four-body convex central configurations for all masses', arXiv 2609.35632 [math.DS], 28 Sept 2026, 34 pages, 1 figure; Tejasvi Singh Tomar
- Claim (abstract): 'for every choice of four positive masses and every cyclic ordering of the bodies there is exactly one strictly convex planar central configuration with that ordering, up to similarity. This answers Problem 10 in the list of Albouy, Cabral and Santos, which Santoprete calls the Simó-Yoccoz conjecture.'
- History per the paper: MacMillan and Bartky proved existence in 1932; Albouy, Fu and Sun conjectured uniqueness in 2008; before this paper it was known for equal masses (Albouy 1995–96), equal masses at the ends of a diagonal, two pairs of equal adjacent masses (Fernandes, Llibre and Mello 2017), restricted classes (Santoprete), three small masses, and a region of masses by interval arithmetic (Sun, Xie and You 2023)
- Method: a uniform lower bound for the Hessian at convex central configurations ('at least one quarter of its radial part'); Dziobek's relations and a Cauchy–Schwarz argument reduce it to showing that the trace of an explicit 2×2 matrix is below a fixed bound, proved by interval arithmetic on the three-dimensional set of normalized convex configurations; uniqueness follows by a covering argument from the equal-mass case
- Checks: 'The computation has been repeated with a second, independently written program. The Hessian bound and the uniqueness theorem, including the computation, have been formalized in the Lean proof assistant and checked by its kernel, using only the standard axioms.' The paper says the proof was also checked by nanoda, a second type checker for Lean
- Consequences stated: every degenerate four-body central configuration is concave; the convex configuration depends analytically on the masses; the known symmetry theorems for kites, isosceles trapezoids and rhombi follow in a few lines
- AI disclosure ('Use of AI tools'): 'Generative AI tools were used to find and prove the results, to write both programs of Section 5 and the Lean formalization, and to draft the text.' No model is named
- Code, certificates and Lean project: github.com/tejasvi/lean4body, archived at doi.org/10.5281/zenodo.23004297
- Status on 13 Oct 2026: unrefereed; no public comment found from specialists in central configurations (web and Hacker News searched; X not searched)

#### What happened
A central configuration is an arrangement of point masses that can rotate rigidly under Newtonian gravity. For four bodies in the
plane, a non-collinear one is either convex (the bodies are the corners of a convex quadrilateral) or concave (one body inside
the triangle of the others). It has been known since 1932 that a convex one exists for every choice of masses and every cyclic
order of the bodies. Whether it is the only one was an open question that specialists had settled only for special families of
masses.

Tomar's paper claims the general answer: it is unique. The argument bounds the Hessian of the relevant function at every convex
central configuration, which makes each one non-degenerate, and then continues from the equal-mass square, where uniqueness is
classical. One inequality is checked by interval arithmetic over a three-dimensional region, and the whole chain, including that
computation, is reported as formalized in Lean.

The author states that generative AI tools found and proved the results and wrote the programs and the formalization. This was the
first of three such papers by the same author in under two weeks.

#### Why it matters
The problem is less famous than Smale's sixth, but it sat on the standard list of open problems in the field for 14 years and had
attracted steady partial results. If the proof holds, it is another case where an outsider with AI tools and a proof assistant
closed a problem that specialists had worked on case by case.

#### Not verified here
Nobody else is known to have re-run the Lean project or compared its formal statement with the theorem. The paper does not name
the AI tools. We found no response from the authors of the earlier partial results.

Sources: [arXiv 2609.35632: Uniqueness of four-body convex central configurations for all masses](https://arxiv.org/abs/2609.35632); [GitHub: tejasvi/lean4body (programs, certificates, Lean formalization)](https://github.com/tejasvi/lean4body); [Zenodo archive of the repository](https://doi.org/10.5281/zenodo.23004297)

### 2026-09-28: Convex counterexamples to Schiffer and Pompeiu in dimensions 3, 4, 6, 8, 10 and 14, made with Claude Opus 5.5 and GPT-6 Astra/Sol

***; Jizhou Guo; Science & math; confidence medium; https://postcutoff.com/e/2026-09-28-schiffer-pompeiu-convex-higher-dimensions/

On 28 Sep 2026 Jizhou Guo posted what he says are the first counterexamples to the Schiffer and Pompeiu conjectures in dimension three and higher. They are also the first convex counterexamples in any dimension: bounded convex non-ball domains in R^3, R^4, R^6, R^8, R^10 and R^14, with computer-assisted existence proofs. The paper says the work 'was carried out with the assistance of AI models (Claude Opus 5.5, GPT-6 Astra and GPT-6 Sol)'.

- arXiv 2609.35419 (87 pages, v1 28 Sep 2026); supersedes the author's Zenodo preprint on dimensions four and six (doi 10.5281/zenodo.22999505)
- Dimensions 4, 6, 8, 10 and 14: adjoint-invariant domains in the rank-two compact Lie algebras u(2), so(4), su(3), so(5) and g2. Harish-Chandra's radial-part formula reduces each case to a planar Helmholtz problem, and Kostant's theorem reduces convexity to the Cartan section
- Dimension 3: an axisymmetric convex domain, plus a separate strictly star-shaped non-convex example
- Existence by Newton–Kantorovich contractions in weighted coefficient spaces: finite blocks checked in Arb ball arithmetic, infinite tails bounded analytically; verification code and certificates are in the arXiv ancillary files and on GitHub
- Acknowledgement, verbatim: 'This work was carried out with the assistance of AI models (Claude Opus 5.5, GPT-6 Astra and GPT-6 Sol). The author takes full responsibility for the content of this paper.' The paper does not break down which parts each model did
- Builds on the August 2026 planar counterexamples of Colbrook–Stepaniants and Cao-Labora–de Dios Pont
- Status: single-author preprint, not peer-reviewed

#### What happened
Eight weeks after the planar disproofs, a single author extended the counterexamples to higher dimensions and to convex domains. He used Lie-theoretic symmetry to reduce each case to a planar problem and certified the solutions with interval arithmetic.

#### Why it matters
If it holds up, the Schiffer and Pompeiu conjectures fail even for convex domains, which was one of the natural fallback forms of the conjectures. It is also an example of a September 2026 paper that names the newest frontier models (Claude Opus 5.5 was released on 22 Sep) without saying what each one did.

Sources: [arXiv 2609.35419: Convex counterexamples to the Schiffer and Pompeiu conjectures in dimensions three, four, six, eight, ten and fourteen](https://arxiv.org/abs/2609.35419); [GitHub: aster2024/schiffer-pompeiu-counterexamples (paper and verification)](https://github.com/aster2024/schiffer-pompeiu-counterexamples); [GitHub: aster2024/schiffer-pompeiu-r4 (earlier dimension-four version)](https://github.com/aster2024/schiffer-pompeiu-r4)

### 2026-09-28: Caltech 'Mathathon' is reworked into 'Old Problems, New Proofs' after an open letter from mathematicians

**; Caltech, XTX Markets, SAIR; Science & math; confidence high; https://postcutoff.com/e/2026-09-28-mathathon-joint-statement/

On Sept 28, 2026 Caltech's organizers and the signatories of an open letter against the AI-sponsored 'Mathathon' published a joint statement on Terence Tao's blog. The event is renamed 'Old Problems, New Proofs' (Nov 13–15, 2026). Participants learn existing hard proofs, then write new expositions, and cash grants usable for any tool (LLM credits, HPC or pen and paper) replace proprietary-AI sponsorship.

- Joint statement posted Sept 28, 2026 on terrytao.wordpress.com
- The original open letter (proofsandprompts.com, Sept 10) gathered 2,000+ signatures
- New format: 40 hours learning existing hard proofs, then two months producing new expositions
- Tool-neutral cash grants replace proprietary-AI sponsorship; XTX Markets is lead donor, SAIR co-organizes
- Also that week on Tao's blog: recommendations of the Harvard/CMSA Summit on PhD Math Education in the Age of AI (Sept 25)

#### What happened
A public dispute over an AI-lab-sponsored math competition ended with the event redesigned around understanding and exposition rather than AI problem-solving.

#### Why it matters
It is part of the September 2026 push by research mathematicians to set norms for AI in their field, after the Fields medalists' letter, the ICIAM statement and OpenAI's advisory group.

Sources: [Terence Tao: Joint statement about Mathathon](https://terrytao.wordpress.com/2026/09/28/joint-statement-about-mathathon/); [Recommendations of the Summit on PhD Math Education in the Age of AI](https://terrytao.wordpress.com/2026/09/25/recommendations-of-the-summit-on-phd-math-education-in-the-age-of-ai/); [Open letter about the Mathathon](https://proofsandprompts.com/2026/09/10/open-letter-about-the-mathathon)

### 2026-09-29: List Total Colouring Conjecture (late 1990s) disproved; counterexample found by ChatGPT 6 Astra Ultra 'with little input'

****; OpenAI, University of Victoria; Science & math; confidence high; https://postcutoff.com/e/2026-09-29-list-total-colouring-conjecture-false-astra/

On Sept 29, 2026 Jonathan A. Noel posted a 20-vertex cubic graph whose total chromatic number is 4 but whose list total chromatic number is 5. This disproves the List Total Colouring Conjecture of Borodin–Kostochka–Woodall, Juvan–Mohar–Škrekovski and Hilton–Johnson from the late 1990s. The paper's AI declaration says Noel prompted ChatGPT 6 Astra Ultra on Sept 24 to disprove the conjecture and it produced the counterexample; Noel checked the arguments and rewrote the exposition. It is his second list-colouring conjecture disproof by Astra Ultra in September, after the Partial List Colouring Conjecture.

- Result: a cubic simple graph G on 20 vertices (four copies of K_{2,3} joined by one cross edge between each pair) with χ''(G) = 4 and list total chromatic number χ''_ℓ(G) = 5 (arXiv 2609.38417 abstract and Section 2)
- Conjecture disproved: the List Total Colouring Conjecture (list total chromatic number = total chromatic number for every multigraph), posed independently by Borodin–Kostochka–Woodall, Juvan–Mohar–Škrekovski and Hilton–Johnson in the late 1990s
- AI declaration (verbatim): 'On September 24, 2026, the author prompted ChatGPT 6 Astra Ultra to disprove the List Total Colouring Conjecture and it produced the counterexample in this paper. The author checked the arguments and rewrote the exposition based on drafts generated by ChatGPT ... The figures were also generated by ChatGPT. The author takes full responsibility for correctness.'
- Abstract: 'ChatGPT 6 Astra Ultra discovered the counterexample with little input from the author.'
- Noel asks whether χ''_ℓ(G) ≤ χ''(G) + 1 might still hold for all multigraphs
- Same author, same model, same month: Noel's arXiv 2609.23291 disproved the Partial List Colouring Conjecture (Albertson–Grossman–Haas) with a counterexample 'discovered and fully verified by ChatGPT 6 Astra Ultra after some persistent prompting'
- Verification status: unrefereed arXiv preprint (math.CO); the proof is short and hand-checkable; no Lean formalisation mentioned. No HN thread or high-reach X post found as of Oct 5

#### What happened
Jonathan A. Noel posted "The List Total Colouring Conjecture is False" to arXiv (math.CO) on Sept 29, 2026. The total chromatic number χ''(G) is the fewest colours needed to colour both the vertices and the edges of a graph so that adjacent or incident elements differ. The list version gives every vertex and edge its own list of allowed colours. The List Total Colouring Conjecture, posed independently by three groups in the late 1990s, says the list version never needs more colours than the ordinary one.

The counterexample is a cubic graph on 20 vertices. It is built from four disjoint copies of K_{2,3}, with exactly one edge between each pair of copies. It has a total colouring with 4 colours, but Noel gives a list assignment from {1,…,5} with 4 colours per element that admits no proper colouring. So the list total chromatic number is 5.

The paper's AI declaration says Noel prompted ChatGPT 6 Astra Ultra on Sept 24, 2026 to disprove the conjecture, and the model produced the counterexample. The author checked the arguments, rewrote the exposition from ChatGPT drafts and used ChatGPT to proofread, suggest references and generate the figures.

#### Why it matters
This is a named conjecture from the late 1990s, listed on Open Problem Garden, falling to a prompted AI search, with a counterexample small enough to verify by hand. Together with Noel's Partial List Colouring disproof earlier in September, it is a clear case of a frontier model finding counterexamples to long-standing graph-colouring conjectures with little human guidance. It remains an unrefereed preprint.

Sources: [arXiv 2609.38417: The List Total Colouring Conjecture is False (Noel)](https://arxiv.org/abs/2609.38417); [arXiv 2609.23291: The Partial List Colouring Conjecture is False (Noel)](https://arxiv.org/abs/2609.23291); [Open Problem Garden: List Total Colouring Conjecture](http://www.openproblemgarden.org/op/list_total_colouring_conjecture)

### 2026-09-29: EPFL's LinCodeEvolve (Viazovska, Abbe) finds seven record-breaking binary linear codes with LLM-guided program search

***; EPFL; Science & math; confidence high; https://postcutoff.com/e/2026-09-29-lincodeevolve-record-binary-codes/

On Sept 29, 2026 EPFL researchers Amal Seddas, Vladyslav Shashkov, Maryna Viazovska (Fields Medal 2022) and Emmanuel Abbe posted LinCodeEvolve, an LLM-driven evolutionary program search (built on ShinkaEvolve and EvoTune) that found seven binary linear codes beating the best-known minimum distances in Grassl's CodeTables, e.g. [172,21,66] and [200,21,77]. With standard modifications they improve 22 table entries. Every code was verified by exhaustive enumeration, and the table maintainer checked them. The work applies the FunSearch/AlphaEvolve approach to a classic coding-theory benchmark.

- arXiv 2609.37056 [cs.IT], 29 Sep 2026; authors Seddas, Shashkov, Viazovska, Abbe (EPFL)
- Seven new record codes: [172,21,66], [173,20,68], [176,21,68], [181,21,70], [184,21,72], [189,22,72], [200,21,77]; six have concise quasi-cyclic descriptions
- Standard code modifications extend these to 22 improved entries of the binary tables at codetables.de; 'the results have been checked by the maintainer of the code tables and will be incorporated into them'
- Method: an LLM proposes code-construction programs (a chain of oracle, strategist, contract-repair and implementer calls), scored by an exact minimum-distance evaluator; a strategy loop with expert supervision redirects the search when progress plateaus
- Comparison: directly prompting GPT-6 Astra at extra-high reasoning effort gave distance 65 for (172,21) vs 66 from LinCodeEvolve; at (200,21) both reached 77, but LinCodeEvolve's code has 25 minimum-weight codewords vs 75
- Compute: a single 80 GB H100; each experiment took 3–4 hours on average

#### What happened
Finding binary linear codes with large minimum distance is a central coding-theory problem, and certifying minimum distance is NP-hard.
LinCodeEvolve keeps ShinkaEvolve's island-based archive, novelty judge and model selection. Candidates enter the archive only with an exact
certificate (full weight distribution). Among codes with equal distance it prefers fewer minimum-weight codewords.

#### Why it matters
Viazovska, who solved sphere packing in dimensions 8 and 24, is now co-authoring LLM-search papers. It is another case of FunSearch-style
search improving a long-maintained table of records with modest compute (one H100). The authors note that careful prompting of a frontier
model (GPT-6 Astra) came close on some parameters, but systematic search did better across many parameter pairs.

Sources: [arXiv 2609.37056: Evolving Towards Better Codes: LLM-Guided Search for High-Distance Binary Linear Codes](https://arxiv.org/abs/2609.37056); [CodeTables.de (Grassl): bounds on linear codes](http://www.codetables.de)

### 2026-09-29: Lovász conjecture (1969): long paths in vertex-transitive graphs pushed to n^(1−o(1)), with GPT-5.6 Sol supplying key steps

***; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-09-29-lovasz-conjecture-nearly-linear-bound-gpt-5-6-sol/

On Sept 29, 2026 Bowen Li and Abhishek Methuku posted a proof that every connected vertex-transitive graph on n vertices contains a cycle of length n^(1−o(1)). The previous bound was n^(2/3−o(1)), set earlier in 2026. This is the strongest general progress so far towards Lovász's 1969 conjecture that every such graph has a Hamiltonian path. The paper's AI statement says GPT-5.6 Sol pointed the authors to two key tools (the Tessera–Tointon structure theorem and Babai's contraction lemma) and "supplied the required group-theoretic arguments" for the last missing lemma.

- arXiv 2609.38135 (math.CO), 'A nearly linear bound for the Lovász conjecture', Bowen Li and Abhishek Methuku, Sept 29, 2026
- Result: every connected vertex-transitive graph on n vertices has a cycle (and path) of length n^(1−o(1)); the previous record n^(2/3−o(1)) was by Bucić, Christoph, Pokrovskiy and Steiner (arXiv 2606.09742, June 2026); Babai's 1979 bound was about √n
- AI statement: 'In July 2026, when the authors asked OpenAI's GPT-5.6 Sol whether a decomposition into vertex-transitive parts is possible, it directed them to the structure theorem of Tessera and Tointon'
- AI statement: for the small-orbit case 'GPT pointed them to Babai's contraction lemma', and 'GPT supplied the required group-theoretic arguments for controlling both the number and the word lengths of the required generators (Lemma 6.1) … This provided the key remaining step'
- Human part (per the paper): the 2025 spanning-tree traversal plus Lovász local lemma strategy and the path-lifting bound (Lemma 5.7) were the authors' own
- Context: Christoph, Hunter and Sudakov (arXiv 2609.30165, Sept 24, no AI disclosure) separately showed (1−o(1))n cycles when the degree is polylogarithmically large

#### What happened
Lovász asked in 1969 whether every connected vertex-transitive graph has a Hamiltonian path. For decades the best general guarantee was a cycle of length about √n (Babai, 1979). In 2026 the bound moved quickly:
first n^(2/3−o(1)) by Bucić, Christoph, Pokrovskiy and Steiner (June), and now n^(1−o(1)) by Li and Methuku.

The paper has a detailed "Statement of AI use". The authors had a 2025 strategy: traverse a spanning tree of the quotient graph repeatedly
and use the Lovász local lemma to join random short paths. It worked only if the vertex set could be split into large vertex-transitive
parts. In July 2026 GPT-5.6 Sol directed them to the Tessera–Tointon structure theorem, which supplies such a partition, and the
large-orbit cases followed "quickly". For the small-orbit case GPT pointed them to Babai's contraction lemma, which reduces the problem
to Cayley graphs of nilpotent groups. When the authors could not bound the word lengths of the generators they needed, "GPT supplied
the required group-theoretic arguments" (Lemma 6.1), "the key remaining step".

#### Why it matters
This is a large quantitative step on a famous 57-year-old problem in graph theory. The disclosure follows a pattern seen across 2026: the
model did not produce the whole proof, but it supplied the right literature tools and one hard lemma. The conjecture itself (a full
Hamiltonian path) remains open. The result is an unrefereed preprint, and no expert reactions were found as of Oct 5.

Sources: [arXiv 2609.38135: A nearly linear bound for the Lovász conjecture](https://arxiv.org/abs/2609.38135); [arXiv 2606.09742: Towards the Lovász conjecture via sublinear expanders (previous n^(2/3) bound)](https://arxiv.org/abs/2606.09742); [arXiv 2609.30165: Thinning and sprinkling, from robust sampling to almost Hamiltonicity (Christoph, Hunter, Sudakov)](https://arxiv.org/abs/2609.30165)

### 2026-09-29: Microsoft Research unveils Quine, a multimodal 'world model of biology' validated in pancreatic-cancer screens with the Broad Institute

**; Microsoft Research, Broad Institute; Science & math; confidence medium; https://postcutoff.com/e/2026-09-29-microsoft-quine-biology-world-model/

On Sept 29, 2026 Microsoft Research introduced Quine, an early-stage AI research system that learns shared representations across genomics, proteins, chemistry, cell state and bioimaging and is wired into a tool/literature harness. With the Broad Institute it prioritized thousands of compounds to shift pancreatic-cancer (PDAC) cell states in about a weekend, and Microsoft says the top-ranked candidates produced the largest intended transcriptional shifts in the lab. Access is limited to a Quine Fellows program.

- Announced Sept 29, 2026 by Nicolo Fusi (VP, Distinguished Scientist) and Jonathan M. Carlson (VP), Microsoft Research
- Multimodal model across genomics, proteins, chemistry, cellular state and bioimaging, plus a harness connecting tools, literature and researchers in a closed loop with wet-lab experiments
- PDAC case study with the Broad Institute: thousands of compounds predicted and prioritized for cell-state transitions in about one weekend; top candidates gave the largest intended transcriptional shifts; predictions pointed to a third phenotype beyond the classical/basal axis and to repurposing candidates (company claims, no paper linked)
- Access: Quine Fellows program (applications open) and selected collaborations; planned to reach customers through Microsoft Discovery; research use only

#### What happened
Microsoft Research presented Quine as a step toward a "world model of biology" that lets scientists search far more interventions
computationally than intuition allows, then test the best in the lab. The only result shown is a PDAC compound-prioritization study with
the Broad Institute; no paper, benchmark or weights were released.

#### Why it matters
It joins a crowded field of lab-in-the-loop biology agents (FutureHouse/Edison Robin, Lila, OpenAI GPT-Rosalind) and is Microsoft's
first named biology foundation-model effort tied to its Discovery platform.

Sources: [Microsoft Research: Introducing Quine, an AI research system designed for the complexity of biology](https://www.microsoft.com/en-us/research/blog/introducing-quine-an-ai-research-system-designed-for-the-complexity-of-biology/)

### 2026-09-30: Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help (July–September catalogue)

****; OpenAI, Anthropic, Google DeepMind, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/

Between July and September 2026 arXiv saw a steady stream of papers that resolve a named conjecture or open question and disclose that a frontier model (mostly GPT-5.6 Sol/Pro and GPT-6 Astra, also Claude Fable 5/5.1, Opus 5/5.5, Gemini) found the key idea, the counterexample or the whole proof. This entry catalogues about 50 of them, with the AI role as the authors state it. Most are preprints without peer review. The most important ones have their own entries.

- Scale: the 'Gold Rush in AI4Math' survey counted 1,712 arXiv math papers with substantive AI contributions from 1 Mar to 20 Aug 2026 (14% of submissions by August). The VibeMathed tracker listed 748 AI-involved problems (511 marked resolved, 158 Lean-verified) when checked on 30 Sep 2026
- Typical disclosure pattern: GPT-5.6 Sol/Pro dominates July–August; GPT-6 Astra dominates September after its early-September release; Anthropic models appear mostly as Claude Code agents, for Lean formalisation or for review
- Fully AI-generated, per the authors: Talagrand's Conjecture 9.1 (Park & Talagrand, 'generated entirely by the AI model GPT-6 Astra'); the dominating Hadwiger conjecture disproof (found by ChatGPT 6 Astra Ultra); the Partial List Colouring Conjecture disproof ('discovered and fully verified by ChatGPT 6 Astra Ultra'); Daykin–Frankl ('LLM-generated proof'); Yau's scalar-curvature integral bound refuted ('ChatGPT generated the main theorems and their proofs'); optimal shallow circuits for Majority ('originally found by GPT-6 Astra')
- Key idea from AI, human completion: the stable forking conjecture (Hart–Kim–Pillay 1996) refuted with GPT-5.6 Sol; a smooth random fast dynamo on T³ ('central proof idea was generated autonomously by ChatGPT 5.6 Sol Ultra'); Kahn–Saks conjecture on linear extensions (ChatGPT 6 Astra 'used primarily to aid in the discovery'); Talagrand's operator cotype problem ('discovered by ChatGPT (GPT-5.6)'); Khachiyan's ellipsoid conjecture ('An AI language model discovered the proof'; Lean-checked)
- Counterexamples found by chatbots on the first or second prompt: Schubitopes are not Ehrhart positive (GPT-5.6 Sol Pro, first prompt, 38 min); claw-free graphs' chromatic symmetric functions are not Schur positive (ChatGPT-5.6 Sol Pro); a very ample polytope with non-unimodal h*-vector (ChatGPT 5.6 Sol); the Hinrichs–Vybíral conjecture ('found in a single prompt' with ChatGPT 6)
- Agent systems: Kazhdan–Lusztig polynomials of matroids need not be unimodal (Rethlas agent on GPT-5.6 Sol, 4 h 59 min, where the GPT-5.6 Sol Ultra web interface failed); Ehrhart volume conjecture equality case (GPT-5.6 Sol, Fable 5 and Danus); the Fröberg conjecture for quintics and septics in four variables (GPT-5.6 Sol, Claude Fable 5 and Grok 4.6)
- Contrasting disclosures: Reed & Stein's dense-case Erdős–Sós proof was found 'without any use of AI'; Kielak et al. solved the uniform Turán tetrahedron problem with no AI-derived arguments, while a competing author completed an alternative proof with ChatGPT-6; Cairo's Ehlers–Kundt counterexample states 'All ideas in this paper are of human origin'
- Priority and credibility problems: the inhomogeneous Duffin–Schaeffer counterexample (GPT-5.6 Sol) had been announced by Pollington a year earlier; Kumar & Volk 'were unable to follow the details and verify' Sheshadri's AI-assisted determinantal-complexity proof and gave their own short quadratic lower bound; the 'Liouville Goldbach' proof was misreported as a Goldbach breakthrough
- A specialist's own count: Alexandre Eremenko (Purdue, retired) wrote on MathOverflow that of about 40 unsolved problems he had collected on his web page, 'In August 2026, at least 7 of them were solved using AI. Some of these solutions are truly exciting, and contain completely new ideas.'

#### What happened
This catalogue comes from a systematic pass over arXiv. It combined API queries for math, physics and theoretical-CS papers from 1 Jul to 30 Sep 2026 that mention a model name, "conjecture" or "open problem", with the AI-disclosure section of the project sweep and the public trackers. For each paper below, the PDF's AI statement was read. Quotes are from the papers, and nothing here was independently checked unless noted.

### Combinatorics and discrete geometry
- **Kahn–Saks conjecture** (balance in posets) proved. Aires, [arXiv 2609.30895](https://arxiv.org/abs/2609.30895). ChatGPT 6 Astra "used primarily to aid in the discovery".
- **Talagrand's Conjecture 9.1**. Park & Talagrand, [arXiv 2609.33644](https://arxiv.org/abs/2609.33644). "The proofs presented in this document were generated entirely by the AI model GPT-6 Astra, and none of the authors claim any credit for them."
- **Dominating Hadwiger conjecture** disproved. Illingworth & Steiner, [arXiv 2609.35361](https://arxiv.org/abs/2609.35361). Found by ChatGPT 6 Astra Ultra; the humans wrote the exposition.
- **Partial List Colouring Conjecture** (Albertson–Grossman–Haas) false. Noel, [arXiv 2609.23291](https://arxiv.org/abs/2609.23291). Counterexample "discovered and fully verified by ChatGPT 6 Astra Ultra after some persistent prompting".
- **Daykin–Frankl conjecture** confirmed. Williams, [arXiv 2609.03087](https://arxiv.org/abs/2609.03087). "We verify and communicate an LLM-generated proof" (ChatGPT 5.6 Sol Pro).
- **Teschner's bondage-number conjecture** counterexample. Yavari, [arXiv 2609.04257](https://arxiv.org/abs/2609.04257). Generated by GPT-5.6 Sol Max.
- **Anstee–Sali conjecture** counterexample. Wu, [arXiv 2608.07646](https://arxiv.org/abs/2608.07646). GPT-5.6 Sol helped identify the example.
- **Stanley's rankwise lower-bound conjecture for differential posets** (1988) disproved. Dai, [arXiv 2607.22988](https://arxiv.org/abs/2607.22988) (25 Jul). Counterexample (for r=3, rank sequence 1,3,9,22,50 instead of 1,3,9,22,51) found by the TARS agent system in an autonomous search; Xinan Dai reconstructed and checked the proof by hand.
- **Bernhart–Kainen dispersability conjecture** (1979): a new smallest counterexample, a 5-regular two-orbit graph on 16 vertices (the previous smallest was the 20-vertex Folkman graph), given an explicit algebraic construction by an AI agent. [arXiv 2608.08118](https://arxiv.org/abs/2608.08118) (NeSy 2026); the Gold Rush survey's Table 3 lists GPT-5.5, Claude Opus 4.7, Gemini 3 Flash, Gemini 3.1 Pro and Claude Sonnet 4.6 as the models used.
- **Chromatic symmetric functions of claw-free graphs are not Schur positive**. Matherne & Morales, [arXiv 2607.21508](https://arxiv.org/abs/2607.21508). Found with ChatGPT-5.6 Sol Pro; independently found by Prajapati.
- **Schubitopes are not Ehrhart positive**. Li & St. Dizier, [arXiv 2608.00377](https://arxiv.org/abs/2608.00377). GPT-5.6 Sol Pro, first prompt, 38 minutes; verified in SageMath.
- **Very ample lattice polytope with non-unimodal h\*-vector**. Hofscheier, Kurylenko & Nill, [arXiv 2608.21507](https://arxiv.org/abs/2608.21507). Found with ChatGPT 5.6 Sol; answers a question of Ferroni–Higashitani.
- **Equality case of Ehrhart's volume conjecture**. J. Liu, [arXiv 2608.01040](https://arxiv.org/abs/2608.01040). Complements OpenAI's inequality proof; "obtained by generative AI" (GPT-5.6 Sol, Fable 5, Danus).
- **Kazhdan–Lusztig polynomials of matroids need not be unimodal**. Cheng et al., [arXiv 2607.24186](https://arxiv.org/abs/2607.24186). The Rethlas agent on GPT-5.6 Sol found it in 4 h 59 min.
- **Laplacian S_{n,n} conjecture**. Johnston, [arXiv 2609.26895](https://arxiv.org/abs/2609.26895). "Significant amount of help" from ChatGPT 5.5, 5.6 Sol and 6 Astra.
- **Comon's conjecture**, 27×27×27 counterexample. Lovitz, [arXiv 2609.28292](https://arxiv.org/abs/2609.28292). GPT-6 Astra gave "an initial proof of the main result".
- **Topological Bárány–Larman conjecture** proved for prime r; the optimal colorful Tverberg theorem of Blagojević–Matschke–Ziegler cannot extend to prime powers. Soberón, [arXiv 2609.37876](https://arxiv.org/abs/2609.37876). "An LLM was used to simplify the proof of the main result and to construct the counterexample" (a 13-point example in R³, "found with the use of an LLM"; model not named).
- **Nagamochi's scoring lemma** (unit-square packing, 2005) counterexample, showing the published proof of his rectangle bound is incomplete. Karakuş, [arXiv 2609.37410](https://arxiv.org/abs/2609.37410). "AI tools were used during exploratory work, for symbolic and numerical checks, and for language and typesetting assistance."
- **Quantum percolation on regular trees**: infinite clusters appear strictly before absolutely continuous spectrum (p_c < p_ac < 1). Becker & Oltman, [arXiv 2609.38017](https://arxiv.org/abs/2609.38017). ChatGPT used "as a research and writing aid".
- **Kaplansky's conjecture** (semifields) counterexamples. Nagy & Zhou, [arXiv 2609.32651](https://arxiv.org/abs/2609.32651). Developed with ChatGPT 6 Pro; formalised by Harmonic's Aristotle in Lean 4.
- **Generalized packing–covering conjecture**. Alfarano, Marino, Neri & Trombetti, [arXiv 2609.34910](https://arxiv.org/abs/2609.34910). ChatGPT 6 Astra turned the authors' strategy into a complete argument.
- **Conway's subprime closure grows by the golden ratio** (conjecture of Caragiu–Vicol–Zaki). Popescu, [arXiv 2609.14188](https://arxiv.org/abs/2609.14188). GPT-6 Astra assisted; Lean-verified.
- **Kalai's conjecture for tight trees** and **Erdős–Sós for digraphs**: see the Erdős–Sós entry.
- **Strongly aperiodic monotile in 3D** ("Chair44"). Tsiokos, [arXiv 2609.19214](https://arxiv.org/abs/2609.19214). Found by "an OpenAI reasoning model (ChatGPT, Astra)"; the text was largely written by Claude Fable 5.1 and reviewed by agents. Follow-ups: [arXiv 2609.24779](https://arxiv.org/abs/2609.24779) (notes) and [arXiv 2609.23783](https://arxiv.org/abs/2609.23783) (matching rules). A guest post on Terence Tao's blog (Kate Stange and Edmund Harriss, [11 Oct 2026](https://terrytao.wordpress.com/2026/10/11/illustration-beyond-proof/)) says Tsiokos shared a 61-page paper and "understood the intuitive geometric argument but not the mathematics", and that Chaim Goodman-Strauss then distilled it into eight pages built around a diagram of matching arrows, "giving a classical hierarchical proof of aperiodicity".

- **Chvátal's conjecture** (1972) proved. Chang, Liu & Liu, [arXiv 2609.19123](https://arxiv.org/abs/2609.19123): ChatGPT proved a guiding special case. Short spectral "Book proof" by GPT-6 Astra in Ellis–Filmus–Friedgut, [arXiv 2609.28404](https://arxiv.org/abs/2609.28404). Lean formalization via Codex. Own entry: `2026-09-16-chvatal-conjecture-proved`.
- **Diagonal bipartite Ramsey b(t,t) = O(2^t)**. Mubayi, [arXiv 2609.32937](https://arxiv.org/abs/2609.32937). "The proof was found by GPT-6 Astra." Own entry: `2026-09-26-mubayi-zarankiewicz-astra`.
- **1/3–2/3 conjecture**: first general improvement of the Brightwell–Felsner–Trotter balance constant in 30 years. Aires, Chan, Pak & Panova, [arXiv 2609.30888](https://arxiv.org/abs/2609.30888). Minor AI role: "we directed an AI assistant to explore the underlying algebraic optimization problem" for one lemma's inequalities.
- **List Total Colouring Conjecture** (Borodin–Kostochka–Woodall 1997; Juvan–Mohar–Škrekovski; Hilton–Johnson) false: a 20-vertex cubic graph with total chromatic number 4 and list total chromatic number 5. Noel, [arXiv 2609.38417](https://arxiv.org/abs/2609.38417) (29 Sep). "ChatGPT 6 Astra Ultra discovered the counterexample with little input from the author" (prompted on Sept 24). Own entry: `2026-09-29-list-total-colouring-conjecture-false-astra`.
- **Fishburn's latent-subset conjecture** (1987) proved. Dong & Mao, [arXiv 2609.35920](https://arxiv.org/abs/2609.35920) (28 Sep). "This proof was found by GPT-6 Astra, following an approach suggested by the author and using the weighted star inequality of Chang, Liu, and Liu" (the inequality from the Chvátal proof).
- **Burr–Erdős–Graham–Sós conjecture for the seven-cycle**, the remaining case k=3 after Bucić–Chen–Ma: f(n, ⌊n²/4⌋+1, C₇) = (1/8+o(1))n². Shahab, [arXiv 2609.38286](https://arxiv.org/abs/2609.38286) (29 Sep). "Made extensive use of AI systems, principally agents built on OpenAI's Codex and Anthropic's Claude" for the proof search, the exact rational certificate, the Lean formalisation and the manuscript; one counting lemma proved with Harmonic's Aristotle; main results Lean-checked.
- **Aigner's majorization conjecture** (adaptive group testing on star forests) false: an explicit six-test counterexample and a family for every k ≥ 6; the converse holds for k ≤ 5. Karpelevitch, [arXiv 2609.38284](https://arxiv.org/abs/2609.38284) (29 Sep). "Generative-AI assistants, primarily OpenAI Codex and, to a lesser extent, Anthropic Claude" were used for arguments, software, experiments and writing.
- **Erdős–Ulam conjectures on monochromatic union-closed families**, both claimed: every 2-colouring of subsets of [n] has a monochromatic union-closed family of super-polynomial size, but not necessarily of exponential size; Lean-verified. Bhattacharjee, Mandal & Bhattacharya, [arXiv 2610.02833](https://arxiv.org/abs/2610.02833) (2 Oct). Minor AI role: "Claude (Anthropic) assisted with the coding" of the Lean formalisation.

### Analysis, PDE and geometry
- **Landis conjecture** fails in dimensions ≥ 3 (real potentials). Frank & Ivanisvili, [arXiv 2608.00802](https://arxiv.org/abs/2608.00802). "The authors acknowledge the use of AI tools"; no detail.
- **Pólya's conjecture for higher-dimensional Neumann balls**. Filonov, Levitin, Polterovich & Sher, [arXiv 2607.29305](https://arxiv.org/abs/2607.29305). ChatGPT and Claude "contributed to the development of several technical lemmas" and the rigorous computer-assisted algorithm.
- **Yau's conjectured scalar-curvature integral bound** refuted. Hao & Zhu, [arXiv 2609.06533](https://arxiv.org/abs/2609.06533). "ChatGPT generated the main theorems and their proofs" (GPT-5.6 Pro).
- **AI-discovered smooth random fast dynamo on T³**. Rowan, [arXiv 2608.20105](https://arxiv.org/abs/2608.20105). Central idea "generated essentially autonomously by ChatGPT 5.6 Sol Ultra"; the original AI manuscript is in the arXiv source.
- **Nevanlinna's century-old half-plane problem** counterexample. He & Zhang, [arXiv 2608.24829](https://arxiv.org/abs/2608.24829). "AI-assisted exploration" (model not named).
- **Fuchs's conjecture** counterexample. Eremenko & Zhang, [arXiv 2609.28443](https://arxiv.org/abs/2609.28443). ChatGPT as an "exploratory tool" and for editing.
- **Forsythe's conjecture for restarted conjugate gradients**. Colbrook, Stepaniants & Townsend, [arXiv 2609.04659](https://arxiv.org/abs/2609.04659). Framed as an experiment in "how far a frontier language model could be pushed" (GPT-5.6, GPT-6).
- **Rockafellar's sum conjecture** fails. Boţ, [arXiv 2609.13906](https://arxiv.org/abs/2609.13906). GPT-6 Astra assisted with the development.
- **Hinrichs–Vybíral conjecture** counterexample. Vybíral, [arXiv 2609.21733](https://arxiv.org/abs/2609.21733). Found by a colleague "in a single prompt try" with ChatGPT 6.
- **Talagrand's operator cotype problem**. Wu, [arXiv 2609.19731](https://arxiv.org/abs/2609.19731). "The counterexample was discovered by ChatGPT (GPT-5.6)."
- **Khachiyan's ellipsoid conjecture**. Zhou, Zou & Liu, [arXiv 2609.28447](https://arxiv.org/abs/2609.28447). "An AI language model discovered the proof"; main theorem Lean-verified.
- **Smooth Hamiltonian diffeomorphism with two fixed points on S²×S²**. Jiao, [arXiv 2609.33626](https://arxiv.org/abs/2609.33626). "GPT suggested a key idea … most of the computation is done by GPT."
- **Explicit mono-monostatic polyhedron** (a certified polyhedral Gömböc). Schettini Gherardini, [arXiv 2609.07827](https://arxiv.org/abs/2609.07827). A Claude Opus 4.8 / Fable 5 agent designed and ran the experiments; exact certificates.
- **Lukic conjecture** counterexample. Yan, [arXiv 2607.26419](https://arxiv.org/abs/2607.26419). "This example was generated by GPT-5.6."
- **Feige's conjecture**. Nie & Wei, [arXiv 2607.24528](https://arxiv.org/abs/2607.24528). Proof "obtained with the assistance of GPT-5.6 Sol".
- **Pólya–Szegő logarithmic vs Newtonian capacity conjecture** (1945): only a conditional reduction, the conjecture "remains open". Clark & Laugesen, [arXiv 2609.35438](https://arxiv.org/abs/2609.35438) (28 Sep). Review role only: ChatGPT/Codex with GPT-5.6 Sol "assisted with a detailed critical review" of calculations, logic and citations.

### Algebra, number theory, topology and logic
- **Stable forking conjecture** (Hart–Kim–Pillay 1996) refuted. Freitag & Mutchnik, [arXiv 2609.00436](https://arxiv.org/abs/2609.00436). "This is an AI-generated result proven with the help of GPT-5.6 Sol."
- **Huneke–Wiegand conjecture** counterexample. Pham, [arXiv 2609.07615](https://arxiv.org/abs/2609.07615). AI-assisted search with GPT-5.6 Pro; Craig Huneke independently recomputed the data.
- **Sato's weak F-equivalence conjecture** counterexamples. Chakravarty, Choi & Xu, [arXiv 2608.18054](https://arxiv.org/abs/2608.18054). GPT-5.6 Sol "produced the key construction".
- **Qin's quasimodularity conjecture** for Hilbert schemes of points. Alekseev et al., [arXiv 2609.33884](https://arxiv.org/abs/2609.33884). "Most of the formal arguments … were initially generated by GPT-5.6 Sol."
- **Fröberg's conjecture** for quintics and septics in four variables. Wang & Zhang, [arXiv 2608.24797](https://arxiv.org/abs/2608.24797). GPT-5.6 Sol, Claude Fable 5 and Grok 4.6 workflow.
- **Mod 4 Kawauchi conjecture**. Conant, [arXiv 2607.18655](https://arxiv.org/abs/2607.18655). Claude Fable 5 "proposed the quotient-tower strategy" and drafted the first version.
- **HZ/4 is not an E2-Thom spectrum** (the remaining case). Ji, [arXiv 2609.19446](https://arxiv.org/abs/2609.19446). GPT-5.6 Sol and GPT-6 Astra; reviewed with Claude Fable 5.1.
- **Yang's conjecture** (tempered xi function) disproved. Kazin & Kadyrov, [arXiv 2609.29898](https://arxiv.org/abs/2609.29898). GPT-6 Astra identified the key sine-transform formulation.
- **Inhomogeneous Duffin–Schaeffer conjecture** counterexamples. He & Liao, [arXiv 2609.30870](https://arxiv.org/abs/2609.30870). Found with GPT-5.6 Sol; the authors later learned that Pollington had announced a counterexample more than a year earlier.
- **Fraenkel's conjecture** (Beatty sequences). Tan & Zhang, [arXiv 2609.01570](https://arxiv.org/abs/2609.01570). GPT-5.6 Sol resolved the cases m = 8–11, which inspired the proof strategy; Codex helped with the finite-case verification code.
- **Separable Jacobian conjecture in characteristic two**, dimension-2 counterexample. Mondello, [arXiv 2608.02634](https://arxiv.org/abs/2608.02634). Lean-checked; ChatGPT/Codex used for organisation; Aristotle replay.
- **Colombo's determinant problem**. [arXiv 2609.00101](https://arxiv.org/abs/2609.00101). The WuJie agent, DeepSeek, Qwen, Kimi and GPT played "a substantial role in identifying the proof strategy"; Lean 4.
- **Hirose's duality conjecture** (q-discretised iterated integrals on the four-punctured line). Seki, [arXiv 2609.40213](https://arxiv.org/abs/2609.40213). "ChatGPT, powered by GPT-6 Astra Pro, was used for mathematical discussions and to refine the English prose" (minor role).
- **Stuck–Zimmer conjecture**, two new cases (ergodic actions of irreducible higher-rank lattices; irreducible actions with an SL₂(ℝ) factor). Machado & Yifrach, [arXiv 2609.39424](https://arxiv.org/abs/2609.39424). "GPT6 Astra was used to help with the literature review, explore research leads, carry out some computations and proofread" (supporting role).
- **De Bruijn–Erdős consecutive-gap problem**. Korsky, [arXiv 2609.07196](https://arxiv.org/abs/2609.07196). GPT Astra used "for completing the mathematical argument".

- **Lehmer's permutation conjecture** (1965; a research problem in Knuth's TAOCP). Verhoeff, [arXiv 2610.01240](https://arxiv.org/abs/2610.01240). "The hypercube proof at the heart of this article was found and developed by Claude Opus 5.5 in a multi-agent proof effort"; Lean formalization by Aristotle (own entry: 2026-10-01-lehmer-permutation-conjecture-opus-5-5).
- **Amenable actions of free groups on unital simple AF-algebras** (the last open case, for free groups, of the existence problem for amenable actions of non-amenable groups on classifiable simple C*-algebras). Suzuki, [arXiv 2610.01460](https://arxiv.org/abs/2610.01460). "GPT-6 Astra (OpenAI) proposed a proof, including the idea of using [5], a key reference of which the author had previously been unaware."
- **Erdős–Graham square products of factorials** (Erdős Problem #374): order of growth of D_k(X) settled for every k, including D_5 ≍ D_6 ≍ X. Yudin, [arXiv 2610.01899](https://arxiv.org/abs/2610.01899). "Large language models (ChatGPT and Claude) were used in developing and checking arguments, locating references, and editing the exposition."
- **Explicit polynomial counterexample to Connes' embedding conjecture** (degree 12 in 65 variables; the conjecture was already known to be false via MIP* = RE, but no explicit polynomial was known). Wang & Zhi, [arXiv 2610.01536](https://arxiv.org/abs/2610.01536). "Generative AI tools assisted with exploring proof strategies, checking calculations, and revising the exposition."
- **Pappas's conjecture on ramified unitary local models**, case r>s at self-dual level for p ≥ 2s−1 (flatness of the wedge local model). Luo, [arXiv 2610.02602](https://arxiv.org/abs/2610.02602) (1 Oct). Minor AI role: "The author used ChatGPT and Claude for general research assistance".
- **Cyclic steepest descent is not universally R-superlinear** (disproves the form stated in Dai's ICM 2022 survey). Li & Wang, [arXiv 2610.00939](https://arxiv.org/abs/2610.00939). "The main results of this paper were obtained through a generative-AI workflow using OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, and OpenAI GPT-6 Astra."
- **Borg's conjecture on intersecting integer partitions**: counterexamples at every scale. Person & Schweser, [arXiv 2610.01747](https://arxiv.org/abs/2610.01747). "ChatGPT (5.6 Sol and later GPT-6 Astra) also performed the asymptotic calculations based on Szekeres' formula and assisted with the exposition."
- Also on Oct 1, with AI disclosures but smaller or unstated AI roles: Borcea's 2-variance conjecture and Baernstein's quasi-norm monotonicity conjecture (Zhang, [2610.02035](https://arxiv.org/abs/2610.02035), [2610.02009](https://arxiv.org/abs/2610.02009); Lean 4 for Borcea), the Bryant conjecture for minimal hypersurfaces in S⁴ (Qian & Tao, [2610.00886](https://arxiv.org/abs/2610.00886); "ChatGPT … was used in the preparation"), and higher-dimensional Morley simplices (Tran, [2610.01216](https://arxiv.org/abs/2610.01216)).
### Theoretical CS, information theory, quantum
- **Optimal shallow circuits for Majority**. Lecomte & Ramakrishnan, [arXiv 2609.34029](https://arxiv.org/abs/2609.34029). Constructions "originally found by GPT-6 Astra"; the authors rebuilt the proofs from high-level ideas.
- **Quadratic lower bound on determinantal complexity**. Kumar & Volk, [arXiv 2609.34462](https://arxiv.org/abs/2609.34462). Came out of using ChatGPT Astra to parse Sheshadri's AI-assisted proof, whose details they were 'unable to follow' and verify.
- **Strong secretary conjecture for linear matroids**. Bérczi, Dughmi, Livanos & Soto, [arXiv 2609.20797](https://arxiv.org/abs/2609.20797). Astra "identified the supermodularity … and proposed the uncrossing argument" on 15 Sep; concurrent-discovery note.
- **Nelson–Nguyen conjecture**. Mai & Rao, [arXiv 2609.22548](https://arxiv.org/abs/2609.22548). ChatGPT-5.6 Pro "used in proving and writing".
- **Markovity conjecture for two-receiver broadcast channels** refuted. Liu & Huang, [arXiv 2608.13170](https://arxiv.org/abs/2608.13170). GPT-5.6 Sol; Chandra Nair suggested the AI search and checked the result.
- **Shor's orthogonal-measurement conjecture**. [arXiv 2609.27992](https://arxiv.org/abs/2609.27992). Codex and GPT-5.6 Sol for exploration and error checking only.
- **Distributional variants of the Aaronson–Ambainis conjecture**. [arXiv 2609.35327](https://arxiv.org/abs/2609.35327). "Google Gemini suggested the core idea underlying the inner-gadget construction."

- **Haah's 3D cubic code thermalizes rapidly** (settles whether it is a self-correcting quantum memory: it is not, at any positive temperature; first analytic rapid-mixing proof for a fracton code). Stengele, Caha, Capel & Warzel, [arXiv 2609.39959](https://arxiv.org/abs/2609.39959). "We acknowledge the use of LLMs for a proof of Lemma 4.1" and for testing proof ideas (model not named).
- **Generalized semi-Clifford conjecture** (Zeng et al. 2007) at level k ≤ 4, any prime dimension. Marcus, Subramanian & Dall'Agnol, [arXiv 2609.38751](https://arxiv.org/abs/2609.38751). OpenAI Codex helped develop proof strategies and check algebra; Claude Code for literature and drafting.
- **Edit-distance codes over a four-letter alphabet** (DNA barcodes): E₄(6,3) ≥ 120 (was 114) plus 12 further improved lower bounds at lengths 6–9. Yeung, [arXiv 2609.39081](https://arxiv.org/abs/2609.39081). An LLM coding agent wrote the search and verifier code over five weeks; the paper documents failures in which unchecked intermediate results stopped the search early.
- **Cogentic** (Google Research multi-agent prover on Gemini): novel results on five open problems in online learning, auction theory and mechanism design, each checked by domain experts. Cai, Gupta, Jiang, Liaw, Mehta, Velegkas & Wang, [arXiv 2609.40324](https://arxiv.org/abs/2609.40324); results listed at [sites.google.com/view/cogentic](https://sites.google.com/view/cogentic).

### Sept 30 – Oct 2 additions (found Oct 5)
- **Erdős #108 (Erdős–Hajnal high-girth problem)** disproved through a Lean bounty (Kohlmeyer–Kruer, ChatGPT/Codex). The 3-colour sharpening by Nguyen & Walczak, [arXiv 2609.40192](https://arxiv.org/abs/2609.40192), has "ChatGPT-6 Astra discovered a proof of Theorem 2.1". Own entry: `2026-09-15-erdos-108-high-girth-problem-disproved-conjectures-io`.
- **ε-Dvoretzky conjecture** proved. Klartag & Moshe, [arXiv 2610.03204](https://arxiv.org/abs/2610.03204). The key idea "stemmed from a discussion between ChatGPT and the second named author". Own entry.
- **Barvinok's log-concavity question (line version)**. Leake & Mohammadi Yekta, [arXiv 2609.39917](https://arxiv.org/abs/2609.39917): "proven using ChatGPT 6 Astra". Own entry.
- **Triangle-free box graphs beating Burling (1965)**. Tomon, [arXiv 2610.03517](https://arxiv.org/abs/2610.03517): "found by ChatGPT Astra after guidance from the author". Own entry.
- **Friedgut's continuous-cube coalition conjecture**: a simplified proof of Chattopadhyay–Gurumukhani's result. Friedgut, [arXiv 2610.03086](https://arxiv.org/abs/2610.03086): "discovered with the aid of ChatGPT-6 Astra … my mathematical contribution to this current project was limited to my two suggestions to the bot".
- **Mond's conjecture** for corank-one germs C³→C⁴. Rimányi, [arXiv 2610.03331](https://arxiv.org/abs/2610.03331). Experiments, first drafts of several proofs and audits came "with substantial help from AI (Anthropic's Claude; OpenAI's ChatGPT)".
- **Simpson's closedness conjecture** in arbitrary rank. Hu, [arXiv 2610.03542](https://arxiv.org/abs/2610.03542). "ChatGPT was used for calculations, proving intermediate results, and polishing the writing."
- **Lam–Schilling–Shimozono conjecture**: K-k-Schur functions are k-Schur positive. Bai & Guo, [arXiv 2610.01708](https://arxiv.org/abs/2610.01708). The proof is human; "the counterexample given in Section 5 was found by ChatGPT-6 Astra".
- **Strong Roberson conjecture** (homomorphism distinguishing power) refuted. Kristjánsson, [arXiv 2610.03550](https://arxiv.org/abs/2610.03550): "developed through substantial interaction with an AI agent".
- **Exact eight-point angular-energy conjecture** (Hiriart-Urruty's list) counterexample. Zhao, [arXiv 2610.03207](https://arxiv.org/abs/2610.03207): "produced by OpenAI's GPT-5.6 Sol Ultra through Codex".
- **Hegselmann–Krause models**: the proximity digraph need not freeze. Hegarty, Ognissanti & Wedin, [arXiv 2610.03229](https://arxiv.org/abs/2610.03229): "All of the examples above were found by Claude Opus 5.5".
- **Delooping level is not additive under tensor products**. Lam, [arXiv 2609.39745](https://arxiv.org/abs/2609.39745). The counterexample was found by "the newest publicly available (Pro Subscription) model of ChatGPT" (Oct 1).
- **Quantum random self-reduction for linear problems**. Asadi, Hirahara & Shimizu, [arXiv 2609.39823](https://arxiv.org/abs/2609.39823): "The algorithm and proof strategy underlying the main theorem were suggested by ChatGPT 5.5."
- **Nearest stabilizer product state hardness**. Grier, Pashayan & Schaeffer, [arXiv 2610.02037](https://arxiv.org/abs/2610.02037): "Significant parts of the hardness proofs … were found by an interactive prompting of Gemini 3.1 Pro."
- **Disjoint-data inverse problem** sharp spectral threshold. Ylinen, [arXiv 2610.00871](https://arxiv.org/abs/2610.00871): "GPT-5.6-sol produced the initial nonuniqueness construction".
- **Ramanujan–Sato series of level 11** (open since Cooper et al. 2015). Campbell, [arXiv 2610.00551](https://arxiv.org/abs/2610.00551). Built on "a construction described by GPT-5.6 Sol Pro".
- **Prime labelings of trees**. Ho, [arXiv 2610.02271](https://arxiv.org/abs/2610.02271): "Most of the proof development, computational verification, and drafting … were carried out by generative AI systems, principally GPT Astra and Claude Opus".
- **Standard conjecture of Hodge type**, explicit form for Hermitian varieties. Harashita, [arXiv 2610.01731](https://arxiv.org/abs/2610.01731), with "extensive use of Claude Opus 5 and Claude Opus 5.5 for formulating assertions, proving results, drafting".
- Smaller or mostly human: Quillen's finiteness conjecture for the cotangent complex in characteristic 2 (Schürg, [2610.03208](https://arxiv.org/abs/2610.03208); ChatGPT as "somebody to talk to"); irrationality exponent of ζ(2) below 5.0193784 (Niedbala Giraudin, [2610.02912](https://arxiv.org/abs/2610.02912); "Technical assistance: Claude"); typical intersecting families at n=2k+1, 2k+2 (Li, [2610.02119](https://arxiv.org/abs/2610.02119); ChatGPT 5.6 Sol helped with technical lemmas); Matsumura's extension problem, smooth case, confirming Siu's plurigenera conjecture in that setting (Chen, Rao & Wang, [2609.40040](https://arxiv.org/abs/2609.40040); ChatGPT 5.6 and 6 Pro as auxiliary tools).

### Late-September disclosures the sweep missed (PDF scan, Oct 5 evening)
- **Lovász conjecture (1969)**, long cycles in vertex-transitive graphs: n^(1−o(1)), up from n^(2/3−o(1)). Li & Methuku, [arXiv 2609.38135](https://arxiv.org/abs/2609.38135) (29 Sep). GPT-5.6 Sol pointed the authors to the Tessera–Tointon structure theorem and Babai's contraction lemma, and "supplied the required group-theoretic arguments" for the key remaining Lemma 6.1. Own entry: `2026-09-29-lovasz-conjecture-nearly-linear-bound-gpt-5-6-sol`.
- **Tight convergence of Nesterov's fast gradient method**: Peppy, an AI workflow built on performance-estimation problems (PEP), closes Conjectures 4–5 of Taylor et al. (2017) for FGM and OGM and the FGM/FISTA rational-coefficient conjectures. The authors call these the first analytic proofs of the tight FGM bounds, open "roughly a decade". Suh, Yoon, Nguyen, Ying & Ma, [arXiv 2609.35762](https://arxiv.org/abs/2609.35762) (28 Sep). Proofs generated by Codex with GPT-6 Astra (Ultra reasoning) and checked symbolically in SymPy; "the Codex-generated drafts required human editing". Code: [PEPFlow peppy-v1](https://github.com/pepflow-lib/PEPFlow/tree/peppy-v1).
- **Watkins's conjecture holds for all infinite groups** (graphical regular representations; the Cayley index of every infinite group is 1, 2 or 8). Sutherland, [arXiv 2610.01049](https://arxiv.org/abs/2610.01049) (1 Oct). "OpenAI's ChatGPT provided substantial assistance with developing arguments, locating references, and preparing the checking programs"; the checks "do not constitute formal verification".
- **Rational points near the moment curve**: counterexamples showing the Hickman–Srivastava range is nearly sharp, against the belief that an O(n⁻¹) improvement was possible. Chen, Srivastava & Technau, [arXiv 2610.02443](https://arxiv.org/abs/2610.02443) (1 Oct). A self-contained argument (the recursive identity (2.5)) "was later suggested to us by Chat G.P.T. 6 Astra"; written and verified by the authors.
- **Chen's conjecture for flat n-tori** confirmed (sharp Willmore lower bound, Clifford torus the unique minimiser), but it fails for the total mean curvature of general n-tori when n ≥ 3. Ni, Wang & Xie, [arXiv 2609.36491](https://arxiv.org/abs/2609.36491) (29 Sep). "ChatGPT 5.6 Sol was used to generate preliminary versions of certain arguments in Lemma 3.2, Remark 4.6, and Proposition 5.5"; the authors substantially revised them.
- **Flat normal bundles are not preserved by mean curvature flow** (an embedded torus and an entire graph in R⁴). Kunikawa, [arXiv 2609.34291](https://arxiv.org/abs/2609.34291) (28 Sep). ChatGPT 5.6 "was used to suggest suitable ansatzes and to explore examples".
- Smaller AI roles: union-closed sets conjecture bound raised past Liu's conjectured 0.382709 to 0.38288 (Costa & Sadhu, [2610.02295](https://arxiv.org/abs/2610.02295); ChatGPT for "exploratory symbolic calculations, proof checking" and code); Fang–Lin spectral Turán conjecture counterexample (Wu & Lu, [2609.37208](https://arxiv.org/abs/2609.37208); ChatGPT "to discuss proof strategies, check intermediate arguments"); Boston's unramified conjecture for odd GL₂ over Q and McLeman's (3,3)-conjecture for p > 3 (Luo, [2609.37252](https://arxiv.org/abs/2609.37252); ChatGPT 6 for proofreading and "verifying the logical consistency"); Cordero-Erausquin–Eskenazis question on Talagrand type of S₁ (Luo, Wang & Zhou, [2610.03407](https://arxiv.org/abs/2610.03407); Codex for literature and editing); FPT-size resolution refutations (Cai & Li, [2610.02047](https://arxiv.org/abs/2610.02047); one construction from a ChatGPT-found gap).

### October 2026 listings (moved)
The blocks that used to follow here (the Oct 2–5, Oct 5–6, early-October, 6–8 Oct, 8–9 Oct, 5–8 Oct and 6–9 Oct arXiv listings) were moved unchanged on 13 Oct 2026 to a catalogue of their own: [AI-assisted results on arXiv, October 2026](/e/2026-10-13-ai-assisted-results-october-2026/). The smaller 5–9 Oct batch is in [a separate entry](/e/2026-10-09-arxiv-ai-disclosed-results-early-october-2026/).

### Outside arXiv (claims with weaker provenance)
- **Conway's refinement conjecture for omnific integers** (1976): Dan Abramov's "I vibed a proof of Conway's conjecture" (18 Sep; Claude, ChatGPT/Sol and Codex agents; Lean proof passing Palomar checks; not reviewed by mathematicians). [overreacted.io](https://overreacted.io/how-i-vibed-a-proof-of-conways-conjecture/)
- **Theo-Conjecture** (FirstPrinciples) on a 35-year-old question about common-divisor graphs ([arXiv 2608.04040](https://arxiv.org/abs/2608.04040); "not yet completed peer review").
- **Star Fleet Math**: 20 parallel Codex/GPT "starships" claiming 13 Lean-verified Erdős-problem solutions (July 2026). [starfleetmath.com](https://www.starfleetmath.com/)

### Community response: AIM problem list (Oct 2, 2026)
In a guest post on Terence Tao's blog, Matthew Colbrook announced **AIM**, organised with Steve Brunton, Maarten V. de Hoop, George Stepaniants, Alex Townsend and Rachel Ward. It is a community-curated GitHub list of precise open problems in applied mathematics, spectral theory and related fields, each with a dated literature-status review. On Oct 2 the repo listed 642 open targets (531 open, 111 partial), 23 solved entries, 0 Lean-verified. Contributions with or without AI are welcome. The organisers say the list is "rather than using a set of problems to benchmark AI systems" and is meant to give mathematicians, especially early-career researchers, credit for understanding and explaining proofs. A companion site, [AIM, explained](https://mathematics-explained.com/), runs a competition for explanatory papers and videos of up to 10 minutes (entries close 2027-01-01). [Tao's blog](https://terrytao.wordpress.com/2026/10/02/aim-an-invitation-to-explore-mathematics-together/) · [GitHub](https://github.com/MColbrook/AIM)

#### Why it matters
By September 2026, "settled with AI help" had stopped being news in itself: several named conjectures a week, across most of mathematics, with disclosures ranging from "fully generated by the model" to "used for proofreading". For a model with an older cutoff, the practical lesson is to check the current status of any conjecture before calling it open. Many that were open in early 2026 have claimed resolutions, but most are still unrefereed.

Sources: [arXiv 2608.24961: The Gold Rush in AI4Math: Where Are We Now?](https://arxiv.org/abs/2608.24961); [arXiv 2609.40324: Cogentic, multi-agent orchestration for automated proof discovery (Google, Gemini)](https://arxiv.org/abs/2609.40324); [arXiv 2609.39959: Haah's 3D cubic code thermalizes rapidly](https://arxiv.org/abs/2609.39959); [VibeMathed: tracker of AI-involved math problems](https://vibemathed.com/); [Wikipedia: List of mathematical discoveries by artificial intelligence](https://en.wikipedia.org/wiki/List_of_mathematical_discoveries_by_artificial_intelligence); [Kingy AI: Mathematics & Science Breakthrough Tracker](https://kingy.ai/mathematics-science-breakthrough-tracker/); [GitHub: ai4math-chronicle (provenance tracker)](https://github.com/Charlie-Wang-03/ai4math-chronicle); [Terence Tao's blog (guest post by Matthew Colbrook): AIM, an invitation to explore mathematics together (Oct 2, 2026)](https://terrytao.wordpress.com/2026/10/02/aim-an-invitation-to-explore-mathematics-together/); [GitHub: MColbrook/AIM (open applied problems, statuses and reviewed solutions)](https://github.com/MColbrook/AIM); [MathOverflow: Alexandre Eremenko's answer, 'In August 2026, at least 7 of them were solved using AI'](https://mathoverflow.net/a/515515); [arXiv 2607.23307: Chen & Klartag, sharp thin-shell inequality (proof found by GPT-5.6 Pro; own entry)](https://arxiv.org/abs/2607.23307)

### 2026-09-30: Anonymous 'AI-generated research' account mt0-svg posts Lean-checked solutions to the Tammes problem for 15 points, Robbins' V > 2 and the Hoffman–Johnson–Junge frog-model conjecture

****; mt0-svg (pseudonymous); Science & math; confidence medium; https://postcutoff.com/e/2026-09-30-mt0-svg-ai-generated-tammes-15-robbins-frog-model-lean/

The GitHub account mt0-svg (created Aug 1, 2026, no name) published, with Zenodo DOIs, six results labelled "AI-generated research: the results, proofs and code were found and written by AI". (1) Tammes problem for N = 15: the largest minimal angular distance of 15 points on the sphere is arccos u ≈ 53.65785°, attained by exactly two configurations (Buddenhagen–Kottwitz) whose contact graphs differ, which answers Musin–Tarasov's uniqueness question negatively for N = 15 (v1.0.0 Sept 30, v1.4.0 Oct 18). (2) Robbins' problem: V(n) > 2.0038 for n ≥ 300, so V > 2, answering Bruss and Ferguson (1996). (3) The frog model on rooted d-ary trees is recurrent for d = 3 and transient for d = 4, the open cases of Hoffman–Johnson–Junge's conjecture (Ann. Probab. 2017). Also: Keith–Zanello Conjecture B is false (p = 73), the Furio–Lombardo quartic has exactly four rational points, and the minimax regret constant of 1D online linear regression is 3. Each has a Lean 4 proof with only standard axioms, comparator-checked; Tammes and the frog model also rely on certified computations entered as hypotheses or replayed by Lean checkers. Not peer reviewed; no expert reaction found by Oct 18.

- Label on every repo and paper: 'This is AI-generated research: the results, proofs and code were found and written by AI. Credit goes to all the humans whose work it builds on.' No AI model or company is named
- Tammes N = 15: optimum arccos u ≈ 53.65785°, u the root in (1/2, 7/10) of 13u⁵ − u⁴ + 6u³ + 2u² − 3u − 1; method follows Musin–Tarasov (N = 13 in 2012, N = 14 in 2015): written reduction to plane graphs with ≤ 15 vertices, interval computation with replayed search-tree certificates, local rigidity near the two optima
- Tammes Lean scope (quoted): 'Lean 4 proves every step except the enumeration of the plane graphs and their refutation by the program, which enter the formal proof as hypotheses' (plantri enumeration and a few thousand lines of Rust)
- Tammes Theorem 1.3 (exactly two optimal configurations) is credited to GitHub user Satvik-patil, who posted it as issue #1 on Oct 6–7, 'Produced with an AI system (Claude)', 'not reviewed by a specialist'
- Robbins' problem: Lean theorem Robbins.main_so gives V(n) ≥ 137704521264/2^36 (≈ 2.0038) for all n ≥ 300, no sorry, no native_decide; the kernel checks 300 tables of a certificate by evaluation; question from Bruss & Ferguson (1996) and Bruss, 'What is known about Robbins' problem?' (J. Appl. Probab. 2005)
- Frog model: Lean theorems FrogModel.recurrent_three and FrogModel.transient_four, mean visits to the root ≤ 5.756 for d = 4; Lean checkers proved sound run on certificate data; 'Conjecture 2 of Hoffman, Johnson and Junge (Ann. Probab. 2017; arXiv:1404.6238)'
- Others: Keith–Zanello Conjecture B (Ann. Comb. 2026) false, exactly twenty odd t work at p = 73, largest t = 203; Furio–Lombardo Conjecture 1.6 (Proc. LMS 2026) proved via descent over a degree-21 field and Selmer group Chabauty; regret constant κ = 3 for doubly uniform 1D online linear regression (question of Gaillard et al., ALT 2019)
- Zenodo first versions: Keith–Zanello and Furio–Lombardo Oct 2, regret Oct 2–3, frog model and Robbins Oct 4, Tammes Sept 30 (v1.0.0) to Oct 18 (v1.4.0); all repos have 0 stars on Oct 18

#### What happened
On Oct 18 the sweep's Zenodo check turned up "The Tammes problem for fifteen points" (v1.4.0). Its author is listed only as
mt0-svg, and it opens with: "This is AI-generated research: the results, proofs and code were found and written by AI." The same
GitHub account had released five more such papers since Sept 28. Each comes with a Lean 4 development, a comparator challenge,
CI that rebuilds everything, and computation certificates.

- **Tammes, N = 15.** The paper extends the Musin–Tarasov method used for 13 and 14 points: a written reduction to irreducible
  contact graphs, an interval-arithmetic search with recorded and replayed trees, and a local rigidity estimate near the two known
  optima. Lean proves everything except that the plantri enumeration is complete and that the Rust program matches its formal model;
  those enter as stated hypotheses. Exactly two non-isomorphic optimal configurations exist. That answers Musin and Tarasov's
  question whether the maximal contact graph is unique (it is not, for N = 15). The two-configuration classification (Theorem 1.3)
  was contributed by GitHub user Satvik-patil in an issue that says it was "produced with an AI system (Claude)".
- **Robbins' problem.** It asks for the least expected rank of a value chosen by a stopping rule from n i.i.d. uniform values with full
  information. The paper relaxes it to a memory of the five smallest values seen and certifies V(n) > 2.0038 for n ≥ 300. That
  answers Bruss and Ferguson's 1996 question whether V > 2. The Lean proof has no hypotheses beyond the definitions.
- **Frog model.** Recurrence on the rooted 3-ary tree and transience on the 4-ary tree, the cases Hoffman, Johnson and Junge left as a
  conjecture, proved through exact recursions on subtree response laws, with certificate checkers verified in Lean.
- Smaller items: a counterexample to Keith–Zanello's Conjecture B at p = 73, a proof of Furio–Lombardo's conjecture that a twist of
  the Klein quartic has exactly four rational points, and κ = 3 for doubly uniform regret in 1D online linear regression.

Web searches on Oct 18 found no news coverage, no posts by experts and no Hacker News thread. The repos had no stars.

#### Why it matters
If they hold up, these are real advances on named problems. The Tammes problem is a classical problem in discrete geometry, and each new case so far has
needed a major computer-assisted effort. Robbins' problem is one of the best-known open questions in optimal stopping. The account
names no model and no person, so we cannot tell which system made them or how much human direction there was. The Lean proofs and
certificates are public and can be checked, which is not true of many AI maths claims. The Tammes proof's trust boundary
(enumeration and program correctness) is stated explicitly.

Sources: [GitHub mt0-svg/tammes-15 (paper, Lean proofs, certificates)](https://github.com/mt0-svg/tammes-15); [Zenodo: The Tammes problem for fifteen points (v1.4.0)](https://zenodo.org/records/23450670); [tammes-15 issue #1 by Satvik-patil: the two optimal configurations are the only ones (Claude-assisted)](https://github.com/mt0-svg/tammes-15/issues/1); [GitHub mt0-svg/robbins-problem: The value of Robbins' problem is larger than 2](https://github.com/mt0-svg/robbins-problem); [Zenodo: The value of Robbins' problem is larger than 2](https://zenodo.org/records/23124501); [GitHub mt0-svg/frog-model-tree: frog model recurrent on the 3-ary tree, transient on the 4-ary tree](https://github.com/mt0-svg/frog-model-tree); [GitHub mt0-svg/keith-zanello-73: Keith and Zanello's Conjecture B is false](https://github.com/mt0-svg/keith-zanello-73); [GitHub mt0-svg/furio-lombardo-quartic](https://github.com/mt0-svg/furio-lombardo-quartic); [GitHub mt0-svg/regret-kappa](https://github.com/mt0-svg/regret-kappa); [Musin & Tarasov: The Tammes problem for N = 14 (arXiv 1410.2536)](https://arxiv.org/abs/1410.2536)

### 2026-09-30: GPT-6 Astra proves Barvinok's log-concavity question for contingency tables on lines, giving lattice-point bounds for all totally unimodular polytopes

***; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-09-30-barvinok-log-concavity-totally-unimodular-astra/

Jonathan Leake and Maryam Mohammadi Yekta's Sept 30, 2026 preprint gives a new lower bound on the number of lattice points of every totally unimodular polytope, and with it a deterministic approximate-counting algorithm. The key ingredient, resolving Barvinok's log-concavity conjecture for contingency tables on lines, "was proven using ChatGPT 6 Astra". The result also implies a conjecture of Ferroni and Higashitani on Ehrhart polynomials of unimodular polytopes.

- arXiv 2609.39917 (math.CO), 'Log-concavity and Approximate Counting for Totally Unimodular Polytopes', Sept 30, 2026
- New class of 'variable-wise log-concave' (VLC) polynomials; bounds based on Gurvits capacity, so counting up to an explicit exponential factor by convex optimisation
- Barvinok's 2007 question was resolved in its 'line' version for all (totally) unimodular polytopes; the full-strength question for TU polytopes remains open
- AI declaration: 'The resolution of Barvinok's conjecture/question on the log-concavity on lines of the contingency tables counts (Theorem 1.4) was the main AI input to this paper. We generalized this result to all TU polytopes … and in fact the AI suggested in a remark that this should be possible.'

#### What happened
Barvinok asked in 2007 whether counts of contingency tables are log-concave as the margins vary. Leake and Mohammadi Yekta report that
ChatGPT 6 Astra proved the "on lines" version, which they call the paper's main AI input. They extended it to all totally unimodular
polytopes with entrywise bounds. That gives lattice-point lower bounds generalising earlier work on contingency tables and integer flows, and
a deterministic approximate-counting algorithm. They also state a generalised conjecture.

#### Why it matters
It is a named question in a central area of combinatorial counting, with the core proof credited to the model. It is an unrefereed preprint,
and we found no reactions as of Oct 5.

Sources: [arXiv 2609.39917: Log-concavity and Approximate Counting for Totally Unimodular Polytopes](https://arxiv.org/abs/2609.39917)

### 2026-09-30: Google's ERA-built model ranks #1 of 39 in the CDC FluSight 2025–26 flu-hospitalization forecasting challenge

**; Google Research; Science & math; confidence high; https://postcutoff.com/e/2026-09-30-google-era-flusight-flu-forecasting/

Google reported on Sept 30, 2026 that its best forecasting model ranked first of 39 eligible models in the CDC's FluSight challenge for the 2025–26 US flu season (October–May). FluSight forecasts weekly hospital admissions now and up to three weeks ahead. The forecasts were built with Empirical Research Assistance (ERA), Google's AI tool that writes and optimizes scientific code.

- Challenge: CDC FluSight, 2025–26 season; end-of-season analysis published Sept 30, 2026
- Google: 'of 39 eligible models, Google's best matched the season's observed hospital admissions' (current-week and up to three-weeks-ahead forecasts)
- Built with ERA (Empirical Research Assistance), the AI system behind the Gemini for Science computational-discovery tools

#### What happened
Google entered FluSight with forecasting models developed through ERA. At season's end its best model ranked first among the eligible
entries, which come from academic and public-health teams.

#### Why it matters
It is a real-world, prospectively scored test of AI-generated scientific software, not a retrospective benchmark.

Sources: [Google blog: Google's AI ranks #1 for predicting flu hospitalizations](https://blog.google/innovation-and-ai/models-and-research/google-research/google-science-ai-flu-forecasts/)

### 2026-10-01: Lehmer's 1965 permutation conjecture (a research problem in Knuth's TAOCP) proved; Claude Opus 5.5 found the short hypercube proof

****; Eindhoven University of Technology, Anthropic, Harmonic; Science & math; confidence high; https://postcutoff.com/e/2026-10-01-lehmer-permutation-conjecture-opus-5-5/

On Oct 1, 2026 Tom Verhoeff posted a proof of D. H. Lehmer's 1965 conjecture that the permutations of every multiset admit an imperfect Hamiltonian traversal by adjacent swaps. Knuth's The Art of Computer Programming lists the question as an unsolved research problem. After about six months of work with earlier Claude models had produced a proof of nearly 200 pages, Claude Opus 5.5 found a short proof "within a couple of hours", built on a partition of the words into hypercubes. Harmonic's Aristotle produced the Lean formalization.

- arXiv 2610.01240 (math.CO), Tom Verhoeff, 'A proof of Lehmer's permutation conjecture for neighbor-swap graphs', Oct 1, 2026
- Result: proves Verhoeff's 2017 reformulation, Hamiltonicity of the graph N(S) on non-stutter words, with two exceptional families that have only Hamiltonian paths. This implies Lehmer's conjecture
- AI role (paper): 'In that study phase, Claude Opus 5.5 proposed the representation that became the partition into hypercubes … It then developed this lead into the two theorems above, in successive multi-agent rounds: writing, independent recomputation of every construction, adversarial red-teaming, and hand refereeing'
- Acknowledgments: 'I worked with various Claude models — Opus 4.7, Opus 4.8, Fable 5 (and Fable 5.1), and Opus 5 — but progress was slow; after some six months the proof (then almost 200 pages of dense mathematics) and its Lean formalization (more than one hundred thousand lines) came together. Then Opus 5.5 appeared, and Anton asked it to come up with a simple proof from scratch, which it did within a couple of hours.'
- 'The hypercube proof at the heart of this article was found and developed by Claude Opus 5.5 in a multi-agent proof effort.' Harmonic's Aristotle produced the Lean formalization
- The problem was restarted after Anton Bakker's March 2026 challenge, made when Opus 4.7 came out: 'Software is done, and I bet mathematics as well'
- TU/e news feature (8 Oct): Verhoeff, now retired from TU/e, says 'The long proof emerged in close collaboration with me, the shorter proof was produced autonomously.' The long proof (~150 pages per TU/e; the paper's acknowledgments say almost 200) needed 100,000+ lines of Lean, the short (<30-page) proof about 35,000. DeepSeek V4.1 Flash turned proof steps into programs for computational checks

#### What happened
D. H. Lehmer conjectured in 1965 that the permutations of any multiset can be traversed by adjacent swaps, visiting every word,
with some words visited twice. Knuth's TAOCP lists it as an open research problem. In 2017 Verhoeff restated it as a
Hamiltonicity question. Working with successive Claude models from March 2026, he first reached a very long proof. Then Claude
Opus 5.5 proposed reading each word as a sequence of "dominoes". In that view the even-position swaps generate hypercube orbits,
and gluing their Gray cycles gives the Hamiltonian cycles. A multi-agent workflow developed this into the paper's main theorems.

#### Why it matters
This is a famous, decades-old combinatorics problem named in Knuth's TAOCP, and the paper credits the key proof idea directly to
Claude Opus 5.5. It is also a clear before-and-after comparison: earlier models helped produce a long proof slowly, while the
newer model found a short one quickly. The result is still an unrefereed preprint, so its status is pending.

Sources: [arXiv 2610.01240: A proof of Lehmer's permutation conjecture for neighbor-swap graphs](https://arxiv.org/abs/2610.01240); [TU/e: This mathematician worked on one problem for years and watched AI grow from assistant to grandmaster (8 Oct)](https://www.tue.nl/en/news-and-events/news-overview/08-10-2026-this-mathematician-worked-on-one-problem-for-years-and-watched-ai-grow-from-assistant-to-grandmaster); [IO+: How a mathematician watched AI grow from servant to grandmaster](https://ioplus.nl/en/posts/how-a-mathematician-watched-ai-grow-from-servant-to-grandmaster)

### 2026-10-01: Harvard physicist Matthew Schwartz releases BootLoops, an open-source harness that let Claude produce 36 manuscripts in 18 fields in three months

***; Harvard University, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-10-01-bootloops-claude-shaped-science/

On Oct 1, 2026 Anthropic published a guest post, "Claude-Shaped Science", by Harvard theoretical physicist Matthew Schwartz (a visiting researcher at Anthropic). He released BootLoops 1.0, an open-source toolkit that sits between a researcher and an LLM so that Claude can do exact quantitative calculations. With it, Schwartz and 19 co-authors produced 36 manuscripts in 18 fields in three months. Schwartz stresses that Claude's results needed expert checking.

- Output: 36 manuscripts in 18 fields with 19 co-authors over about three months, picked from ~400 candidate problems (Anthropic guest post)
- Particle physics: 30 integrals computed with the bootstrap method, 15 reproducing known results and 15 never computed before
- Economics (NBER w35782, Schwartz, Isaiah Andrews, Jesse M. Shapiro): the workflow ran 4,452 replication packages from five journals, flagged discrepancies in 3,460 articles or appendices, cut computation time >10x in 496, and proposed extensions in 923
- Other projects: population genetics on 5.7 billion mutation pairs from the 1000 Genomes Project; a word-stress database (AccStack) covering 6,072 languages; every US forest-inventory plot classified by life-history strategy
- BootLoops is owned and maintained by Schwartz; the work was funded by Anthropic but it is 'not an Anthropic project' (bootloops.ai)
- Caveats from Schwartz: Claude 'likes to declare victory too early', and 'done, with one asterisk' often means 'not done at all'; first findings were often 'technically correct but scientifically unremarkable' until domain experts steered them

#### What happened
Schwartz argues there is an "impedance mismatch" between how scientists work and what LLMs do best, so instead of making Claude imitate a
scientist he looked for "Claude-shaped problems": exact, checkable calculations that recur across fields. BootLoops packages the tools
and loops for such work. He reports that Claude reproduced one paper's results in about twenty minutes, a calculation that took him weeks
to code originally.

#### Why it matters
It is one of the broadest demonstrations so far of a single researcher using an AI agent to work across many unrelated fields, with code
released openly. The economics replication alone (thousands of papers re-run, thousands of discrepancies flagged) could matter for research
transparency. Schwartz's warnings about premature "victory" are a useful counterweight to autonomous-science claims.

Sources: [Anthropic (guest post by Matthew Schwartz): Claude-Shaped Science](https://www.anthropic.com/research/claude-shaped-science); [BootLoops: a toolkit for exact quantitative science](https://bootloops.ai/); [BootLoops source code (GitHub)](https://github.com/BootLoops-ai/bootloops); [NBER w35782: An LLM Workflow That Reproduces, Improves, and Extends Published Economics Research (Schwartz, Andrews, Shapiro)](https://www.nber.org/papers/w35782); [The Decoder: Open-source BootLoops harness supports AI models in precise scientific calculations (Oct 3)](https://the-decoder.com/open-source-bootloops-harness-supports-ai-models-in-performing-precise-scientific-calculations/)

### 2026-10-02: Kannan–Lovász–Simonovits (KLS) conjecture proved in two AI-assisted preprints: Song–Zhang's O(1) bound, then Bizeul–Klartag–Lehec ('most proofs … found by ChatGPT')

*****; Microsoft Research, Weizmann Institute of Science, OpenAI, Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-10-02-kls-conjecture-proved-ai-assisted/

The KLS conjecture (1995), which says isoperimetry in convex bodies is governed up to a universal constant by half-spaces, was claimed resolved twice in three days. Zhao Song and Xinzhi Zhang (arXiv 2610.01447, Oct 2; v2 Oct 4) prove an O(1) bound on the KLS constant with GPT-6 Astra, GPT-5.6 Sol and Claude Fable 5/5.1 after exploring 100+ approaches. Pierre Bizeul, Boaz Klartag and Joseph Lehec (arXiv 2610.05474, Oct 4) "present a proof" built on Song–Zhang's criterion, saying most proofs and ideas in it were found by ChatGPT.

- Song & Zhang, 'An O(1) Bound for the KLS Constant', arXiv 2610.01447 (math.PR), Oct 2, 2026, v2 Oct 4: 'We prove that ψ_n ≤ C for a universal constant C > 0, resolving the KLS conjecture'; also a universal Poincaré constant for isotropic log-concave measures
- Song–Zhang disclosure: 'The AI tools used in this work were GPT-6 Astra, GPT-5.6 Sol, Claude Fable 5, and Fable 5.1.' From July 28, 2026 they explored 'more than 100 approaches in collaboration with AI tools' to improve a log^(1/4) bound (from a paper that appeared on July 27, 2026) to log^(1/6); on Sept 30 the authors found a route to an f(log* n) bound, then sharpened it to a constant. 'All of the proofs have been carefully verified by the authors and several AI tools.'
- Bizeul, Klartag & Lehec, 'Presenting a proof of the Kannan-Lovasz-Simonovits conjecture', arXiv 2610.05474, Oct 4: 'The crucial step in the proof is the very recent criterion from Song and Zhang involving high-order derivatives of tilt averages'
- Bizeul–Klartag–Lehec disclosure: 'Most proofs and mathematical ideas in this paper were found by ChatGPT; a notable exception is the idea to use suspension which was suggested by the authors. The role of the authors has been mostly to understand these proofs and improve their exposition.' Their introduction says the conjecture now 'poses little challenge for contemporary AI tools'
- Background: Klartag and Lehec had earlier proved a polylog bound and (with others) resolved Bourgain's slicing problem; KLS implies the slicing and thin-shell results
- Verification status: two preprints, not peer-reviewed, no formal proof; Klartag–Lehec's independent write-up is a strong expert signal (confidence medium until reviewed)

#### What happened
KLS asks for a dimension-free isoperimetric constant for convex bodies. Since Eldan's stochastic localization (2013), bounds improved
from polynomial to polylogarithmic (Chen; Klartag–Lehec; Klartag). Song and Zhang, working with OpenAI and Anthropic models since late
July, posted a proof of about 138 pages of an O(1) bound on Oct 2. Two days later Bizeul, Klartag and Lehec, among the area's leading experts,
posted a shorter proof built on Song–Zhang's criterion, written mostly by ChatGPT.

#### Why it matters
KLS was among the best-known open problems in high-dimensional geometry, with consequences for sampling algorithms on convex bodies. The
same week saw AI-originated proofs of the linear Hadwiger conjecture and a refutation of the 3SUM/APSP hypotheses. Confidence is medium
only because neither paper has been refereed or formalized yet.

Sources: [arXiv 2610.01447: An O(1) Bound for the KLS Constant (Song, Zhang)](https://arxiv.org/abs/2610.01447); [arXiv 2610.05474: Presenting a proof of the Kannan-Lovasz-Simonovits conjecture (Bizeul, Klartag, Lehec)](https://arxiv.org/abs/2610.05474); [ETH Randomstrasse101: The KLS Conjecture (problem 30)](https://randomstrasse101.math.ethz.ch/posts/KLSConjecture/)

### 2026-10-02: Klartag and Moshe prove the ε-Dvoretzky conjecture (polynomial dependence on ε); the key probabilistic idea came from a ChatGPT discussion

****; Tel Aviv University, Weizmann Institute of Science, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-02-epsilon-dvoretzky-conjecture-klartag-moshe/

On Oct 2, 2026 Boaz Klartag and Shahar Moshe posted a short proof that the dependence on the accuracy ε in Dvoretzky's theorem is polynomial in 1/ε, not exponential. Their bound is n ≥ (C/ε)^((ℓ+1)/2)·|log ε| for an ℓ-dimensional section ε-close to an ellipsoid, through any interior point of any convex body. This settles a conjecture going back to V. Milman. The authors write that the idea of their new probability distribution on the Grassmannian "stemmed from a discussion between ChatGPT and the second named author".

- arXiv 2610.03204 (math.FA), 'A polynomial bound in Dvoretzky's theorem', Oct 2, 2026
- Theorem 1.1: if n ≥ (C/ε)^((ℓ+1)/2)·|log ε|, every convex body K ⊂ R^n with 0 in its interior has an ℓ-dimensional section with F ⊆ K∩E ⊆ (1+ε)F for a centred ellipsoid F; Theorem 1.2 gets a Euclidean ball with the same bounds up to the constant
- Previous best (centrally symmetric case only): exp(Cℓ|log ε|/ε) (Paouris–Valettas, improving Schechtman and Gordon); all proofs since Milman's 1971 concentration-of-measure approach had exponential dependence on 1/ε already for ℓ = 3
- Example: a 3-dimensional section within ε = C·√(log n / n) of an ellipsoid, instead of C·log log n / log n
- Proof: probabilistic but simple, using a new random model on the Grassmannian, Brunn–Minkowski in the space of operators and a union bound; also a simultaneous version for finitely many bodies
- AI disclosure (acknowledgements): 'The idea to use the above probability distribution on the Grassmannian stemmed from a discussion between ChatGPT and the second named author. ChatGPT was additionally used for finding references, for inspecting and improving the proofs and for language editing.'

#### What happened
Dvoretzky's theorem (around 1960, conjectured by Grothendieck) says that high-dimensional convex bodies have almost-Euclidean sections.
For more than fifty years, every proof needed the ambient dimension to grow exponentially in 1/ε for a fixed section dimension. Klartag
and Moshe give a short probabilistic proof with polynomial dependence that works for every convex body, symmetric or not. They name the
source of the key idea, a new random model for choosing the subspace, in their acknowledgements: a conversation between ChatGPT and
Moshe.

#### Why it matters
This is a classical problem in asymptotic geometric analysis, and the preprint comes from Boaz Klartag (Tel Aviv University and Weizmann Institute), a leading researcher in the field. The AI
contribution is stated modestly, but it concerns the central idea rather than editing. As of Oct 5 we found no Hacker News or press
coverage; the model version is not named.

Sources: [arXiv 2610.03204: A polynomial bound in Dvoretzky's theorem](https://arxiv.org/abs/2610.03204)

### 2026-10-02: Meta publishes six maths papers made by mathematicians working with Muse Spark in ordinary meta.ai chat, saying five answer open questions

***; Meta; Science & math; confidence medium; https://postcutoff.com/e/2026-10-02-meta-muse-spark-six-math-papers/

On Oct 2, 2026 Meta released six research papers written by mathematicians who used Muse Spark 1.1 and 1.2 in Thinking Mode through the regular meta.ai chat, with no custom scaffold. Results include a sharp threshold for Gaussian ellipsoid fitting, finite-time blow-up for the mass-critical biharmonic NLS (open since 2015), and counterexamples to Kida's semiabelian-groups conjecture and to a García-Martínez–Pérez-Rodríguez conjecture. Each paper marks which passages AI drafted, and a second group of mathematicians reviewed them. For several of the problems, other teams had posted independent solutions first.

- Announced Oct 2, 2026 on research.meta.ai ('Solving Open Research Problems Together') and by @AIatMeta (~304k views by Oct 3)
- Setup: Muse Spark 1.1 and 1.2 in Thinking Mode via the standard meta.ai chat interface, no custom research scaffold; mathematicians guided the work and a second group reviewed it
- Each paper labels passages drafted primarily by humans vs AI and credits independent solutions
- 1) Probability: strict threshold for Gaussian ellipsoid fitting (around n ≈ d²/4), lead Aykut Arslan; three other teams (Misiakiewicz & Wen; De la Cerda et al.; Koehler & Sohn) posted solutions in August 2026
- 2) PDEs: finite-time blow-up of radial negative-energy solutions of the mass-critical biharmonic NLS in dimension ≥ 2, open since 2015 (lead Leonard Dinh)
- 3) Group theory: 'Semiabelian groups need not be monomial', a 384-element counterexample to M. Kida's 2024 conjecture (Brennan, Golich); the AI agent 'Nilradical' had reported a different counterexample on Sept 16
- 4) Optimization: tightness of the cycle-based relaxation for completed length-three alpha-cycles, a question of Del Pia and Khajavirad (2026)
- 5) Arithmetic physics: p-adic string two-point function equals a height function on a curve, extending a Tate-curve result in a direction envisioned by Manin
- 6) Non-associative algebra: counterexample to a García-Martínez–Pérez-Rodríguez conjecture on solvable evolution algebras (lead Andres Barei); Hu and Wen independently reported counterexamples
- Meta frames it as a step after its models' 'gold-medal-level performance' in five maths, physics and chemistry competitions
- The papers are hosted on ai.meta.com/research/publications (no arXiv versions found as of Oct 4, 2026)

#### What happened
Meta joined OpenAI, Google and Anthropic in publishing AI-assisted research mathematics. Its emphasis is that the work used the
public chat product rather than an internal agent. Paper arXiv links were not given in the blog post; check them when they appear.

#### Why it matters
It adds Meta to the labs with AI-for-math research claims, and it shows how crowded the field has become: the same open problems are
now often solved independently by several AI-assisted teams within weeks.

Sources: [Tech Times: Meta AI closes five open math problems in consumer chat, not a research lab](https://www.techtimes.com/articles/328537/20261005/meta-ai-closes-five-open-math-problems-consumer-chat-not-research-lab.htm); [Meta AI Research: Solving Open Research Problems Together](https://research.meta.ai/blog/solving-open-research-problems-together); [AI at Meta on X: six papers from mathematician–Muse Spark collaborations](https://x.com/AIatMeta/status/2106099776035152231); [RuntimeWire: Meta publishes six math papers made with Muse Spark and human mathematicians](https://runtimewire.com/article/meta-muse-spark-six-mathematics-research-papers); [AlphaSignal: Meta's Muse Spark helped mathematicians solve five open research problems](https://alphasignal.ai/news/meta-s-muse-spark-helped-mathematicians-solve-five-open-research-problems); [Paper 1: The Strict Threshold for Gaussian Ellipsoid Fitting](https://ai.meta.com/research/publications/the-strict-threshold-for-gaussian-ellipsoid-fitting/); [Paper 2: Finite-Time Blow-Up of Radial Negative-Energy Solutions for the Mass-Critical Biharmonic NLS](https://ai.meta.com/research/publications/finite-time-blow-up-of-radial-negative-energy-solutions-for-the-mass-critical-biharmonic-nonlinear-schrodinger-equation/); [Paper 3: Semiabelian Groups Need Not Be Monomial](https://ai.meta.com/research/publications/semiabelian-groups-need-not-be-monomial/); [Paper 4: Tightness of the Cycle-Based Relaxation for Completed Length-Three Alpha-Cycles](https://ai.meta.com/research/publications/tightness-of-the-cycle-based-relaxation-for-completed-length-three-alpha-cycles/); [Paper 5: String Two-Point Function = Height Function on a Curve](https://ai.meta.com/research/publications/string-two-point-function-height-function-on-a-curve/); [Paper 6: On Solvable Evolution Algebras and a Conjecture by García-Martínez and Pérez-Rodríguez](https://ai.meta.com/research/publications/on-solvable-evolution-algebras-and-a-conjecture-by-garcia-martinez-and-perez-rodriguez/)

### 2026-10-02: ChatGPT Astra finds geometric triangle-free graphs with near-optimal chromatic number, the first improvement on Burling's 1965 box bound

***; Umeå University, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-02-tomon-triangle-free-box-graphs-astra/

On Oct 2, 2026 István Tomon (Umeå University) posted geometric constructions of triangle-free and high-girth graphs with rapidly growing chromatic number. The headline result is a triangle-free intersection graph of n boxes in R³ with chromatic number (log n)^(1−o(1)). He calls it the first improvement over Burling's double-logarithmic bound from 1965, and it nearly matches the O(log n) upper bound. The paper states: "The constructions presented in this paper were found by ChatGPT Astra after guidance from the author."

- arXiv 2610.03517 (math.CO), 'Geometric triangle-free graphs of large chromatic number', Oct 2, 2026
- Boxes in R³: triangle-free intersection graph with independence number n(log n)^(−1+o(1)), so χ = (log n)^(1−o(1)); answers a question of Walczak
- Unit-distance graphs: if R^d has a unit-distance graph with chromatic number r, it also has an induced one with girth ≥ g and the same chromatic number
- Lines in R³ and circle tangency graphs with girth ≥ g and χ = Ω_g((log n)^(1−o(1))), improving Davies and Davies–Keller–Kleist–Smorodinsky–Walczak and giving an alternative solution of Ringel's circle problem; ordered graphs motivated by the Gyárfás–Sumner conjecture
- AI declaration: 'The constructions presented in this paper were found by ChatGPT Astra after guidance from the author.' And: 'We instructed ChatGPT Astra to look specifically for parametric families, which quickly led to the main ideas presented in this paper.'

#### What happened
Burling's 1965 construction gave triangle-free box intersection graphs whose chromatic number grows only like log log n. Tomon's preprint
reaches (log n)^(1−o(1)), close to the known O(log n) ceiling, and adds several related girth results. He credits all the constructions to
ChatGPT Astra working under his guidance. In his words, "While this construction might seem like it came out of nowhere … we try to illuminate
how it uses ideas from known constructions … (So we can also claim some credit.)"

#### Why it matters
This is an exponential improvement on a 60-year-old bound in geometric graph colouring, and it is another case where the model supplied
the constructions rather than routine checks. It is a preprint with no reactions found as of Oct 5.

Sources: [arXiv 2610.03517: Geometric triangle-free graphs of large chromatic number](https://arxiv.org/abs/2610.03517)

### 2026-10-04: Linear Hadwiger conjecture proved: K_t-minor-free graphs are O(t)-colourable; proof found by GPT-6 Astra (Norin & Steiner), Lean-verified by Codex

*****; OpenAI, McGill University, ETH Zurich; Science & math; confidence high; https://postcutoff.com/e/2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra/

On Oct 4, 2026 Sergey Norin (McGill) and Raphael Steiner (ETH Zurich) posted arXiv 2610.05291, proving the linear Hadwiger conjecture: there is a constant C such that every graph with no K_t minor is Ct-colourable. The authors say the proof "was found by GPT-6 Astra, following the directions by the authors", and that almost none of their own proof ideas survived. A Lean 4 formalization of the main theorem, produced by OpenAI Codex (6 Sol and Astra), builds with no sorries.

- Paper: 'A Proof of the Linear Hadwiger Conjecture', Sergey Norin and Raphael Steiner, arXiv 2610.05291 (math.CO), Oct 4, 2026
- Result: ∃ C ∈ ℕ such that K_t-minor-free graphs are Ct-colourable (Hadwiger's 1943 conjecture asks for t−1 colours; the linear version had been a major open weakening; best prior general bound was O(t log log t), Delcourt–Postle)
- Disclosure: 'The proof presented in this paper was developed by OpenAI's ChatGPT model Astra 6, following research directions and feedback supplied by the authors. The authors suggested several possible approaches, but almost none of their specific proof ideas were retained in the final argument.'
- Process: asked first for the very dense case (v(G) ≤ C′t), which the model proved 'after only a couple of hours and some encouragement'; then an explicit dependence gave the conjecture for v(G) ≤ t(log t)^(1/2−o(1)); asked for 'an original idea to bridge the gap', it produced the bootstrap argument that closes it using Delcourt–Postle's reduction
- Ingredients: Reed–Seymour fractional linear Hadwiger, Delcourt–Postle reduction and chromatic separability, Pippenger–Frankl–Rödl matching theorem, LP duality, Liu–Luo-style contractions; the authors: the proof is 'contained in the convex hull of existing results. Yet it does not lie on a low-dimensional face of it'
- Lean 4: github.com/snorin239/LinearHadwiger formalizes Theorem 1 (plus Reed–Seymour, Theorem 4, Corollary 24), produced by 'OpenAI Codex versions 6 Sol and Astra, following guidance from the paper's authors'; no sorries, standard axioms only; release v0.1.0
- Verification status: preprint + Lean-verified main theorem; not yet peer-reviewed
- AI disclosure (abstract, arXiv 2610.05291): 'The proof was found by GPT-6 Astra, following the directions by the authors.'

#### What happened
Hadwiger's conjecture (1943) links graph minors and colouring; even its "linear" weakening, that O(t) colours suffice, resisted work
by Reed, Seymour, Norin, Postle, Delcourt and others, whose best bounds were slightly superlinear. Norin, one of the field's main
contributors, and Steiner directed GPT-6 Astra toward what they saw as the missing dense case. The model solved it quickly and, when
pushed for a new idea, found a bootstrap argument that finishes the proof. Codex then formalized the main theorem in Lean.

#### Why it matters
With the 3SUM/APSP refutation and the KLS proofs posted the same week, it marks a step change in AI-originated mathematics: famous problems
now fall with proofs largely written by models and checked by machines. Original Hadwiger (t−1 colours) remains open. Reactions had not
been found at 07:30 CEST on Oct 6.

Sources: [arXiv 2610.05291: A Proof of the Linear Hadwiger Conjecture](https://arxiv.org/abs/2610.05291); [GitHub: snorin239/LinearHadwiger (Lean 4 formalization)](https://github.com/snorin239/LinearHadwiger)

### 2026-10-05: Claude discovers an algorithm that refutes the 3SUM and APSP hypotheses: truly subquadratic 3SUM and truly subcubic APSP (Alman & Vassilevska Williams, Lean-verified)

*****; Anthropic, Columbia University, MIT; Science & math; confidence high; https://postcutoff.com/e/2026-10-05-claude-refutes-3sum-apsp-hypotheses/

On Oct 5, 2026 Josh Alman (Columbia) and Virginia Vassilevska Williams (MIT) posted arXiv 2610.06783, giving the first polynomial speedups over the textbook algorithms for 3SUM (O(n^1.9992)) and All-Pairs Shortest Paths (O(n^2.9995)), which refutes the 3SUM, APSP, Exact Triangle and Zero-Weight k-Clique hypotheses that underpin much of fine-grained complexity. The paper says an Anthropic internal research model (Claude) discovered the algorithm on its own, with no human input, while asked to check cryptographic constructions; Anthropic then certified the main theorems in Lean 4 (not every result in the paper).

- Paper: 'Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs', Josh Alman (Columbia) and Virginia Vassilevska Williams (MIT), arXiv 2610.06783, cs.DS, Oct 5, 2026
- Results: deterministic 3SUM on n polynomial-size integers in O(n^1.9992); APSP on directed graphs with polynomially bounded integer weights in O(n^2.9995); Exact Triangle in O(n^2.9983); Zero-Weight k-Clique in O(n^(k−0.0017⌊k/3⌋))
- Refuted (with known reductions): the 3SUM and APSP hypotheses, their real-valued versions, the Exact Triangle hypothesis, the Zero-Weight k-Clique hypotheses, and the three rectangular hinted Online Matrix–Vector conjectures of van den Brand, Nanongkai and Saranurak
- Core idea: an algorithm for thin matrix products that computes a sparse set of entries of XY polynomially faster than writing XY down, by modifying a variant of Coppersmith's rectangular matrix multiplication built from a ten-multiplication identity of Schönhage; as a graph algorithm it solves All-Edges Sparse Triangle on sparse lopsided tripartite graphs in truly subquadratic time
- How it was found (paper): 'An Anthropic employee used an internal research model to investigate open problems in the theory of cryptography' (constructions based on average-case hardness of Zero-k-Clique). 'Claude was tasked with verifying and improving the constructions, but instead developed this algorithm, first for the average case, then for the worst case. The session used 16M output tokens with no human input.'
- Anthropic shared the algorithm with the authors in September 2026 under a confidentiality agreement and offered compensation; the authors simplified, strengthened and extended it (the data-structure version and the Hinted OMv connection are theirs)
- Lean: after the paper was written, Anthropic used an internal model to certify Theorems 19 and 22 and the Zero-Weight case of Corollary 39 in Lean 4 + Mathlib, with all lemmas they rely on; repo anthropics/formal-math (3sum-apsp), checked with the Comparator tool, standard axioms only. Not formalized: real-input Las Vegas algorithms, Theorem 33, matrix-multiplication exponent values and the Section 5.1 running times
- Context: the O(n^2) 3SUM algorithm had only been improved by polylog factors (Grønlund–Pettie 2014 and later), and APSP by subpolynomial factors (Williams 2018); these hypotheses underpin many conditional lower bounds
- Verification status: preprint plus machine-checked Lean certificate of the main theorems; no peer review yet; reactions not yet found at 07:30 CEST Oct 6; by Oct 9 the Hacker News thread had 109 points and 49 comments, mostly summarising Claude's role, with one technical question on whether the sparse lopsided graph structure carries the subquadratic gain; no public expert assessment found
- Related independent work (Oct 7): Fischer (MPI-INF), Polak and Schmidt (Bocconi), '3SUM Is Really Hard: A Real-to-Integer Reduction' (arXiv 2610.10153), prove that a truly subquadratic Integer 3SUM algorithm yields a truly subquadratic Real 3SUM algorithm (comparison-addition model), answering an open question of Chan, Vassilevska Williams and Xu (STOC 2022). The paper does not cite 2610.06783 and has no AI disclosure; it was written under the assumption that 3SUM is hard

#### What happened
Fine-grained complexity rests on a few hypotheses: that 3SUM needs about n² time, APSP about n³ time, and so on. Many
"conditional lower bounds" assume them. Alman and Vassilevska Williams, both leading researchers in the area, now show polynomial
speedups for both, so all those bounds lose their basis. The improvement in the exponent is tiny (0.0008 for 3SUM), but any polynomial
improvement refutes the hypotheses.

Per the paper's "Acknowledgments and Methodology", the algorithm came from Claude: an Anthropic employee was using an internal research
model on cryptographic constructions built on the average-case hardness of Zero-k-Clique. Instead of verifying them, the model broke the
assumption, first on average and then in the worst case, in one session of 16M output tokens. Anthropic passed the algorithm to the two
authors in September, and certified the final paper's main theorems in Lean.

#### Why it matters
This is the most significant theoretical-computer-science result credited to an AI so far, and the first where a frontier model, unprompted,
overturned a widely believed conjecture rather than proving one that experts expected to be true. The Lean certificate addresses
correctness; whether the method generalises (e.g. to SETH or Orthogonal Vectors) is open. The paper was found on Oct 6 by a manual arXiv
scan; Anthropic had not yet announced it on its own channels at that time.

Sources: [arXiv 2610.10153: 3SUM Is Really Hard: A Real-to-Integer Reduction (Fischer, Polak, Schmidt)](https://arxiv.org/abs/2610.10153); [Hacker News discussion (109 points, Oct 6)](https://news.ycombinator.com/item?id=49977437); [arXiv 2610.06783: Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs](https://arxiv.org/abs/2610.06783); [GitHub: anthropics/formal-math, 3sum-apsp Lean formalization](https://github.com/anthropics/formal-math/tree/main/3sum-apsp)

### 2026-10-05: Google DeepMind's AlphaProtein Novo designs new-to-nature enzymes from scratch (piperidine synthesis, DEHP degradation), with Frances Arnold's lab

****; Google DeepMind, Caltech, University of Pittsburgh; Science & math; confidence high; https://postcutoff.com/e/2026-10-05-deepmind-alphaprotein-novo-enzyme-design/

On Oct 5, 2026 Google DeepMind posted two bioRxiv preprints on AlphaProtein Novo, a diffusion system that co-generates enzyme structure and sequence around a target reaction. The authors say it is the first time de novo enzyme design outperforms mining natural sequences on hard reactions: it produced nitrene transferases that make chiral piperidines (a common drug motif) and DEHP-degrading enzymes that work under denaturing conditions. Code is on GitHub (Apache 2.0); weights are under separate terms.

- Preprint 1: 'Designing enzymes for new-to-nature chemistry and non-natural substrates with AlphaProtein Novo' (Wu, Abramson, Frerix et al.; Google DeepMind, Caltech, Pitt), bioRxiv 10.64898/2026.10.01.756017
- Claim: 'the first time de novo enzyme design can outperform natural sequence mining' on challenging reactions; competitive activity on benchmark reactions; key ingredient: mechanism-inspired metrics based on AlphaFold 3 predictions
- Preprint 2 (with Frances Arnold's Caltech lab and Peng Liu's Pitt group): 'Design-Evolution Synergy Enables New-to-Nature Enzyme Chemistry', de novo hemoproteins for regio- and enantioselective C–H amination to chiral piperidines, combining motif scaffolding, redesign and directed evolution (bioRxiv 10.64898/2026.10.01.754751)
- DEHP (a phthalate plasticizer pollutant) degraders that remain active under denaturing conditions that inactivate natural enzymes
- Repository: google-deepmind/alphaprotein-novo, a diffusion pipeline with LigandMPNN redesign and AlphaFold 3 scoring; benchmarks include Kemp eliminase, serine esterase, DEHPase, carbene and nitrene transferases; code Apache 2.0, weights under separate terms of use
- Announced by Pushmeet Kohli on X (~31k views, Oct 5), crediting Jue Wang, Zachary Wu, Josh Abramson and the Arnold and Liu groups
- Secondary claim (Startup Fortune): one enzyme makes piperidine '99 times faster than current industrial methods'; not found in the abstracts, unverified

#### What happened
DeepMind extended its protein work (AlphaFold for structure, AlphaProteo for binders) to enzymes. AlphaProtein Novo generates a
protein's backbone and sequence together around a catalytic motif, redesigns sequences with LigandMPNN and filters designs with
AlphaFold 3-based, mechanism-aware metrics. With Frances Arnold's lab, designs were further improved by directed evolution into
efficient, stable enzymes for chiral piperidine synthesis.

#### Why it matters
Most industrial enzymes start from natural proteins and are improved by mutation. A generator that beats natural starting points for
non-natural chemistry would open enzyme routes to drug intermediates and pollutant clean-up that nature never evolved. Results are
preprints; claims such as "99x faster than industrial methods" come from secondary coverage and are not verified here.

Sources: [bioRxiv: Designing enzymes for new-to-nature chemistry and non-natural substrates with AlphaProtein Novo](https://www.biorxiv.org/content/10.64898/2026.10.01.756017v1); [bioRxiv: Design-Evolution Synergy Enables New-to-Nature Enzyme Chemistry](https://www.biorxiv.org/content/10.64898/2026.10.01.754751v1); [GitHub: google-deepmind/alphaprotein-novo](https://github.com/google-deepmind/alphaprotein-novo); [Pushmeet Kohli on X: AlphaProtein Novo preprint](https://x.com/pushmeet/status/2107128950975529455); [Startup Fortune: DeepMind unveils AI system that designs enzymes no living cell ever made](https://startupfortune.com/deepmind-unveils-ai-system-that-designs-enzymes-no-living-cell-ever-made/)

### 2026-10-05: A 15-year-old's computer-assisted preprint, coded with Claude, claims the rhombicosidodecahedron is not Rupert (Steininger–Yurkevich conjecture)

***; Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-10-05-rhombicosidodecahedron-not-rupert-claude/

On Oct 5, 2026 Gihyo Jung, described by Korean media as a 15-year-old ninth grader, posted "The rhombicosidodecahedron is not Rupert" on Zenodo with code on GitHub: a computer-assisted proof that this Archimedean solid cannot pass through a hole in a congruent copy of itself, settling a conjecture of Steininger and Yurkevich. He says Claude wrote the proof code. Not yet peer-reviewed or independently checked.

- Preprint: Gihyo Jung, 'The rhombicosidodecahedron is not Rupert', Zenodo 10.5281/zenodo.23159625, Oct 5, 2026 (about 7.8k views by Oct 14); code at github.com/swiri0921-bit/rhombicosidodecahedron-not-rupert
- Abstract: settles a conjecture of Steininger and Yurkevich, who built the first convex polyhedron without the Rupert property (the 'Noperthedron', arXiv 2508.18475, 2025). Because of the solid's symmetry, the standard subdivision method does not work alone, so the paper adds three new local certificates
- Method (README): implements Steininger–Yurkevich's global and local theorems; outward-rounded interval arithmetic with rigorous sin/cos; exact identities in Q(√5); LP (HiGHS) used only to find witnesses, all re-verified; cell cover of 211,680 cells (1.23e7 leaves) with 0 failures; a second implementation in exact rational arithmetic. About 38 hours of computation on a laptop
- Background: Steininger and Yurkevich conjectured (2021) that the rhombicosidodecahedron is not Rupert; their 2025 Noperthedron methods did not suffice for it. Benjamin Grimmer (Johns Hopkins) had told Quanta that two weeks of computer search on it 'just seems to resist any attempt'
- AI use (press, not in the Zenodo record or README): Jung told JoongAng Ilbo he used Claude 'to generate code for solving a difficult mathematical problem and then reviewing the results'; Korea Daily says he also used AI to check the proof and help write the paper. His Reddit post topped a Claude coding forum on Oct 10
- Jung says he chose the topic without an adviser, admits he cannot independently verify every detail, and says another study reached the same conclusion by a different method around the same time (not named; not found)
- Korean experts (JoongAng): Oum Sang-il (IBS Discrete Mathematics Group): posting 'establishes a public record of when it was released' but 'does not guarantee the validity of its contents'; Seo In-suk (SNU): the author must understand all details and answer questions; Choi Kyu-dong (UNIST): lack of training and a paper's value 'can be distinct'

#### What happened
A convex polyhedron is "Rupert" if a straight tunnel can be cut through it large enough for a congruent copy to pass through
(Prince Rupert's cube puzzle, c. 1693). For decades every convex polyhedron tested turned out to be Rupert, until Jakob
Steininger and Sergey Yurkevich proved in 2025 that a purpose-built polyhedron, the Noperthedron, is not. They had earlier
conjectured that the rhombicosidodecahedron (20 triangles, 30 squares, 12 pentagons) is not Rupert either, but their methods did
not settle it.

On Oct 5, 2026 Gihyo Jung posted a computer-assisted proof of that conjecture on Zenodo, with a GitHub repository of certificates
and two independent checkers. Korean newspapers (Korea Daily, Oct 13; JoongAng Ilbo, Oct 14) reported that he is a ninth grader
and that Claude wrote the code. Neither the Zenodo record nor the README contains an AI disclosure; the AI role is known only from
his interviews.

#### Why it matters
If it holds, it is the second named non-Rupert polyhedron and the first Archimedean solid shown not to be Rupert, from an author
with no formal training working with a coding agent. The Korean debate centres on authorship: whether a paper counts if its author
cannot check every step himself.

Sources: [Zenodo: G. Jung, The rhombicosidodecahedron is not Rupert (Oct 5, 2026)](https://zenodo.org/records/23159625); [GitHub: proof code and certificates](https://github.com/swiri0921-bit/rhombicosidodecahedron-not-rupert); [Korea JoongAng Daily: 15-year-old claims AI helped crack 300-year-old math puzzle (Oct 14)](https://www.koreajoongangdaily.com/korea/15yearold-claims-ai-helped-crack-300yearold-math-puzzle/12919724); [Korea Daily: Rupert's Hypothesis — South Korean middle schooler tackles 300-year-old math problem using AI (Oct 13)](https://en.koreadaily.com/ruperts-hypothesis-ai-math-middle-schooler/); [Steininger & Yurkevich: A convex polyhedron without Rupert's property (arXiv 2508.18475)](https://arxiv.org/abs/2508.18475); [Quanta: First shape found that can't pass through itself (Oct 2025)](https://www.quantamagazine.org/first-shape-found-that-cant-pass-through-itself-20251024/)

### 2026-10-05: Decreasing diagrams are complete for confluence: van Oostrom's 1993 question (RTA open problem 56) settled, machine-checked in Isabelle/HOL and Lean with Claude's help (Endrullis, Ivanov, van Raamsdonk)

**; Vrije Universiteit Amsterdam, Taras Shevchenko National University of Kyiv, Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-10-05-decreasing-diagrams-complete-for-confluence-rta-problem-56/

On 5 Oct 2026 Jörg Endrullis, Ievgen Ivanov and Femke van Raamsdonk posted arXiv 2610.06368: every confluent abstract rewriting system admits a locally decreasing labeling using only three labels, which is optimal. This answers Problem 56 of the RTA List of Open Problems, posed by Vincent van Oostrom in 1993 and previously known only for countable systems and up to cardinality ℵ_1. The abstract says: "We settle this thirty-three-year-old problem in full." The development is machine-checked in both Isabelle/HOL and Lean. The AI role is limited to formalization: "The mathematical content — the definitions, constructions, and proofs — is the authors' own; the formalizations were developed with the assistance of Anthropic's Claude, under the authors' direction and review."

- Paper: 'Decreasing Diagrams are Complete for Confluence', arXiv 2610.06368 [cs.LO], 5 Oct 2026; Jörg Endrullis (Vrije Universiteit Amsterdam), Ievgen Ivanov (Taras Shevchenko National University of Kyiv), Femke van Raamsdonk (Vrije Universiteit Amsterdam)
- Confluence: a nondeterministic computation 'always yields the same result, regardless of the order in which steps are taken'. Van Oostrom's decreasing diagrams technique reduces it to a local condition: 'a system is confluent whenever its transitions admit a locally decreasing labeling. Essentially all classical confluence criteria arise as corollaries.'
- Question (1993, RTA Problem 56): 'Does every confluent transition system admit a locally decreasing labeling?' Known before: yes for countable systems, 'and recently up to the first uncountable cardinal ℵ_1'
- Result: 'every confluent transition system admits a locally decreasing labeling using only three labels. This bound is optimal, as two labels do not suffice even at the first uncountable cardinality.'
- Paper: 'This outcome was not foregone: van Oostrom conjectured that decreasing diagrams are incomplete … and related questions about uncountable directed sets are independent of ZFC.'
- Paper: 'To the best of our knowledge, these are the first unconditional completeness results for methods of proving confluence.'
- Verification: 'the entire development is independently machine-checked in two proof assistants, Isabelle/HOL and Lean; both formalizations are archived on Zenodo'. The Isabelle proof shows the result holds in ZFC; the paper notes that 'a Lean proof alone would not give this'
- AI disclosure: 'the formalizations were developed with the assistance of Anthropic's Claude, under the authors' direction and review. The authors take full responsibility for the correctness and presentation of this work.' No Claude model is named

#### What happened
A rewriting system is confluent if, whenever a computation can branch, the branches can always be brought back together. It is
the property that makes the result of a program independent of evaluation order. Van Oostrom's decreasing diagrams method (1990s)
proves confluence by labeling the steps and checking only small local diagrams, and it subsumes nearly all older criteria. Whether
it can prove confluence of every confluent system was Problem 56 on the rewriting community's list of open problems. A positive
answer was known for countable systems; beyond that the question runs into set theory, and van Oostrom himself conjectured a
negative answer.

The preprint proves completeness at every cardinality by two transfinite constructions (a "fan" for regular cofinality and a
"double fan" for the singular case) and shows that three labels always suffice. Because the argument is delicate, the authors
formalized all of it twice, in Isabelle/HOL and in Lean, and say Claude assisted with both formalizations.

#### Why it matters
Here the AI did not find the proof. The entry is in the dataset as a sample of the other common use of models in 2026 mathematics:
making a full machine-checked formalization, in two proof assistants, cheap enough to ship with the first preprint of a
33-year-old problem's solution.

#### Not verified here
The formalizations were not run for this entry and the Zenodo archive was not opened. No expert reaction was found on 13 Oct 2026.

Sources: [arXiv 2610.06368: Decreasing Diagrams are Complete for Confluence](https://arxiv.org/abs/2610.06368); [Zenodo: Isabelle/HOL and Lean formalizations](https://doi.org/10.5281/zenodo.23160921)

### 2026-10-06: Artin–Davenport conjecture proved: every integral cubic form in 10+ variables has a nontrivial zero (Bonolis, Browning, Glas, Wang; key lemma via ChatGPT 6.0, Codex Lean auto-formalisation)

*****; TU Graz, ISTA, Imperial College London, Leibniz Universität Hannover, Academia Sinica, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-artin-davenport-cubic-forms-ten-variables/

On Oct 6, 2026 Dante Bonolis, Tim Browning, Jakob Glas and Victor Y. Wang posted arXiv 2610.08226, a 64-page proof that every integral cubic form in ten or more variables has a nontrivial integer zero. The bound of 10 is sharp, and the best previous results were Davenport's 16 variables (1963) and Heath-Brown's 14 (2007). The authors say the structure and most new ideas are theirs (June 2024 – September 2026), but a key dimension bound (Lemma 3.8) came "after prompting ChatGPT 6.0". A Codex Lean auto-formalisation of the main theorem, modulo five literature inputs, is public. It is an unrefereed preprint.

- Paper: 'The Artin–Davenport conjecture on cubic forms', Dante Bonolis, Tim Browning, Jakob Glas, Victor Y. Wang, arXiv 2610.08226 (math.NT), 6 Oct 2026, 64 pages
- Theorem 1.1: 'The Artin–Davenport conjecture is true': any integral cubic form in n ≥ 10 variables has a nontrivial integer zero
- History (paper): p-adic solubility for n ≥ 10 known since Demyanov (1950) and Lewis (1952); Davenport's circle method gave n ≥ 32 (1959), then n ≥ 16 (1963); Heath-Brown n ≥ 14 (2007). Heath-Brown (1983) and Hooley had settled smooth and isolated-singularity cases. Ten is optimal: a nonary norm-form example has only the trivial p-adic zero
- Method: geometric methods when the singular locus is large, otherwise a Kloosterman circle method with stratification of exponential sums (singular support: Beilinson, Hu–Yang, Raskin–Smith), a stratified geometric sieve and a cube-root Hensel lifting analysis
- AI statement (quote): 'After prompting ChatGPT 6.0, we were led to the improved bound dim I ⩽ ⌊(4n − 3)/3⌋ in Lemma 3.8 … ChatGPT 6.0 suggested an approach via the Galois descent procedure in Lemma 3.6'. Heath-Brown's private improvement to ⌊(3n − 3)/2⌋ would, the authors believe, only have handled eleven or more variables
- Other AI use (quote): 'ChatGPT 5.5, 5.6, 6.0 and 6.1 have been helpful with this process, as well as in proofreading our work, and conducting the Lean auto-formalisation'; the authors thank OpenAI's Mark Sellke for access to the Pro models in May 2026
- Lean: github.com/wangyangvictor/cubic-lean-mid, 'a Codex Lean auto-formalisation of Theorem 1.1 and its proof, modulo five inputs from the literature in elementary form' (not a complete formal proof)
- By-product: Theorem 1.3, a cubic hypersurface over Q whose singular locus has codimension at most ⌊(n−1)/3⌋ has a rational point
- Not in OpenAI's 6 Oct github.com/openai/math release (no cubic-form family in its catalogue)

#### What happened
On 6 October 2026, Dante Bonolis (TU Graz), Tim Browning (ISTA and Imperial College London), Jakob Glas (Leibniz Universität Hannover) and Victor Y. Wang (Université Paris-Saclay and Academia Sinica) posted a proof of the Artin–Davenport conjecture: every cubic form with integer coefficients in ten or more variables has a nontrivial integer zero. Since the early 1950s it has been known that ten variables are enough for solutions over every p-adic field and that ten is the least such number. Over the integers, Davenport's circle method needed 32 variables in 1959 and 16 in 1963, and Heath-Brown brought it down to 14 in 2007. The new 64-page paper closes the gap.

The proof splits into cases. If the cubic hypersurface has a large singular locus, geometric arguments produce a rational point. Otherwise a Kloosterman form of the circle method is used, and the key new input is to control where square-root cancellation fails in the relevant exponential sums by a stratification based on singular support. The authors add a stratified geometric sieve, estimates for "cubefull" moduli and a cube-root Hensel lifting analysis.

The AI statement is specific. The authors write that the structure of the proof and most of the new ideas are theirs, developed from June 2024 to September 2026. A key step needed an upper bound on the dimension of an incidence variety attached to the Hessian. They had proved ⌊(3n−2)/2⌋, and Heath-Brown privately improved it to ⌊(3n−3)/2⌋, which they think would have handled eleven variables. "After prompting ChatGPT 6.0, we were led to the improved bound dim I ⩽ ⌊(4n − 3)/3⌋ in Lemma 3.8", using a Galois-descent approach (Lemma 3.6) that ChatGPT 6.0 suggested. They also used ChatGPT 5.5, 5.6, 6.0 and 6.1 to combine estimates across parameter ranges and for proofreading, and Codex to produce a Lean auto-formalisation of the main theorem that still assumes five results from the literature. They thank OpenAI's Mark Sellke for Pro-model access given in May 2026.

As of 7 October the paper is an unrefereed preprint. No reactions from number theorists had been found yet; it appeared on the same day as OpenAI's 722-manuscript release, which drew most of the attention.

#### Why it matters
This is one of the oldest benchmark problems of the circle method, and a well-known group of analytic number theorists has now claimed it in full, at the sharp number of variables. It shows a pattern that differs from OpenAI's autonomous releases: a long human project in which a frontier model supplied one step the authors say they could not otherwise reach. For a model with an older cutoff, the record for cubic forms is no longer 14 variables.

Sources: [arXiv 2610.08226: The Artin–Davenport conjecture on cubic forms](https://arxiv.org/abs/2610.08226); [GitHub: wangyangvictor/cubic-lean-mid (Codex Lean auto-formalisation)](https://github.com/wangyangvictor/cubic-lean-mid)

### 2026-10-06: OpenAI model claims the free group factor problem: L(F₂) ≅ L(F₃), so all free group factors are isomorphic (Lean-formalized)

*****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-free-group-factors-isomorphism-openai/

Family 287 of OpenAI's 6 Oct 2026 math release claims an affirmative answer to the free group factor isomorphism problem, a central open question of von Neumann algebra theory. The 23-page manuscript says L(F₂) and L(F₃) are isomorphic, so by the Dykema–Rădulescu dichotomy all interpolated free group factors, including L(F∞), are isomorphic and their fundamental group is ℝ₊. The result is formalized in Lean by OpenAI, and a reasoning summary is published.

- Paper: 'An isomorphism of the free group factors' (dated 23 Sep 2026, 23 pages): 'We solve the free group factor isomorphism problem affirmatively by proving that L(F₂) and L(F₃) are isomorphic as tracial von Neumann algebras'
- Lean scope: 'interpolated free group factors with any parameters r,s>1, including the infinite parameter, are normally trace-preservingly isomorphic'; the fundamental-group consequence is not a separate formal statement
- First expert follow-up (14 Oct): Dima Shlyakhtenko (UCLA, free probability), arXiv 2610.19590, 'Building on OpenAI's proof of isomorphism of free group factors', gives a self-contained proof that the C*-algebra generated by a free semicircular n-tuple does not depend on n ∈ {2, 3, …, ∞}, which implies L(F_n) ≅ L(F_2). His reformulation centres on an 'Automorphism Approximation Property'. Statement: 'This work is based on [9] and proofs that are AI-generated. ChatGPT 6.0 Pro was used in the analysis of OpenAI's proof, and in preparation of this manuscript.'
- Reasoning summary (abridged chain of thought): the model first tried rigidity and free-entropy obstructions to an isomorphism (Popa, Boutonnet–Drimbe–Ioana–Popa, Peterson, Jung, Hayes) before finding the construction

#### What happened
OpenAI's catalogue says family 287 resolves the free group factor problem: the von Neumann algebras generated by the free
groups on two and three generators are isomorphic. A classical dichotomy then makes all of them isomorphic. The Lean
formalization covers the isomorphism of all interpolated free group factors.

On 14 Oct Dima Shlyakhtenko posted the first specialist follow-up ([arXiv 2610.19590](https://arxiv.org/abs/2610.19590)). He shows that OpenAI's argument can be modified to prove a formally stronger C*-algebra statement: C*(S₁, …, S_n) ≅ C*(S₁, S₂) for free semicircular systems and every n ∈ {2, 3, …, ∞}. Reduced free products of copies of the Jiang–Su algebra are likewise isomorphic for different n. Reduced free group C*-algebras, which by Pimsner–Voiculescu are not isomorphic for different ranks, still have strong free absorption properties. His proof is self-contained and "does not rely on the results of [9]". He writes that the work "is based on [9] and proofs that are AI-generated" and that ChatGPT 6.0 Pro was used to analyse OpenAI's proof.

#### Why it matters
The problem has been one of the best-known open questions in operator algebras for decades. An affirmative answer, especially one checked by machine, would reshape the field. Until
specialists confirm that the Lean definitions of group von Neumann algebras and interpolated factors match the classical ones,
and read the construction, it is a claim.

Sources: [Paper: An isomorphism of the free group factors](https://github.com/openai/math/blob/main/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026/An-isomorphism-of-the-free-group-factors-September-23-2026.pdf); [Reasoning summary: Isomorphism of free group factors](https://github.com/openai/math/blob/main/reasoning_traces/free-group-factor-isomorphism.pdf); [Shlyakhtenko: On the isomorphism of semicircular algebras (arXiv 2610.19590)](https://arxiv.org/abs/2610.19590); [Lean scope note for family 287](https://github.com/openai/math/blob/main/lean/docs/287.md)

### 2026-10-06: OpenAI release claims the rational Hodge conjecture for all CM abelian varieties, which would give the Tate conjecture for abelian varieties over finite fields (no Lean proof)

*****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-hodge-conjecture-cm-abelian-varieties-openai/

Family 032 of OpenAI's 6 Oct 2026 math release claims the rational Hodge conjecture for every complex abelian variety with complex multiplication, in every dimension and codimension. Through Milne's theorems this would give the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic. Companion manuscripts claim rational Hodge for arbitrary products of K3 surfaces and algebraicity of the Kuga–Satake correspondence; those two companions and the abelian-eightfold paper they relied on were withdrawn on 7 Oct after a sign error was found, while the main CM paper stands. OpenAI lists this work as an exception to its fixed automated procedure. No Lean formalization is given, so it rests on unrefereed manuscripts.

- Withdrawn 7 Oct (openai/math history.md): 'Algebraicity of Weil classes on split abelian eightfolds' (a sign error 'invalidates a stabilization-trace cancellation argument and the construction used by two dependent papers'), 'Algebraicity of Kuga–Satake Correspondences for K3 Surfaces' and 'The rational Hodge conjecture for products of K3 surfaces'. The main paper 'The rational Hodge conjecture for CM abelian varieties' remains, now listed with a 6 Oct version date; new family members 'Weil classes and Hodge classes on abelian powers' and 'Abelian covers, Gale correspondences, and the Hodge conjecture for powers' (both dated 6 Oct)
- Elliot Glazer flagged the eightfolds paper first on X (8 Oct 02:46 UTC, ~119k views), after having GPT-6 Astra check it 'based on a hot tip'
- Reactions: algebraic geometer Kiwamu Watanabe called the algebraic-geometry claims 'quite amazing' but said he cannot tell whether they are correct; Jared Duker Lichtman stressed the release 'is not the Millenium problem rumor'
- Main manuscript: 'The rational Hodge conjecture for CM abelian varieties' (dated 30 Sep 2026, 53 pages); abstract claims consequences 'the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic'
- Companions in the family: rational Hodge for every finite product of projective complex K3 surfaces (4 Oct); algebraicity of the Kuga–Satake correspondence for every K3 surface (3 Oct); Weil classes on split abelian eightfolds (18 Sep); Hodge on powers of abelian sixfolds of Weil type and abelian varieties of dimension ≤ 5 with imaginary-quadratic action; a conditional Kuga–Satake reduction (10 Sep)
- Family 001 (Milne's rationality conjecture) uses this theorem
- README: 'Exceptions to this fixed procedure include … proof of the Hodge Conjecture for CM abelian varieties' (how it was produced differently is not described)
- Not a Millennium Prize resolution: the full Hodge conjecture for all smooth projective varieties remains open; CM abelian varieties are a special, much-studied case
- No Lean scope note exists for family 032
- Expert diagnosis of the withdrawn papers (Yankı Lekili and Nick Sheridan, 'Rigidity, flexibility, and a sign error', arXiv 2610.30976, 22 Oct): the three withdrawn documents 'claimed to prove cases of the Hodge conjecture via homological mirror symmetry' and 'all broke down in the same way', when building an unobstructed Lagrangian immersion. They used [EEMS13, Theorem 1.1] to make all self-intersection points odd, but that construction 'always produces an immersion with a distinguished self-intersection point p of degree 2', and the immersions 'are typically obstructed'. Cause: OpenAI's papers used the standard sign convention while EEMS13 implicitly uses the opposite one, 'apparently first pointed out and reported to OpenAI by Nathan Humphrey'. The paper does not analyse the rest of the argument or the main CM paper
- Lekili–Sheridan on what remains: the HMS approach to Hodge cases 'is interesting and worth pursuing'; 'if this is overcome, the documents … may well provide a useful guide'; the general h-principle construction of unobstructed Lagrangians had made 'several researchers' sceptical before the withdrawal

#### What happened
OpenAI's catalogue lists "Hodge and Kuga–Satake results for all projective K3 surfaces" as family 032. Its central
manuscript proves the rational Hodge conjecture for CM abelian varieties, and companions extend it to products of K3 surfaces
and Weil-type abelian varieties. OpenAI's README names this work as one of two exceptions to the fixed procedure used for the
rest of the release. It does not say whether that meant more compute, human guidance or a different model setup.

#### Why it matters
The Hodge conjecture is a Millennium Prize problem. This claim covers an important special case, not the whole conjecture,
but its stated consequences, Tate for abelian varieties over finite fields and the Hodge standard conjecture for abelian
varieties, are long-standing goals in arithmetic geometry. There is no formal proof, so the claim depends entirely on expert
reading of a 53-page manuscript and its companions.

#### Expert post-mortem of the withdrawn companions (22 Oct)
Symplectic topologists Yankı Lekili and Nick Sheridan explained why the three withdrawn papers failed ([arXiv 2610.30976](https://arxiv.org/abs/2610.30976)). All three tried to reach Hodge cases through homological mirror symmetry and needed an unobstructed Lagrangian immersion. They built one with the Gromov–Lees h-principle and then cited a theorem of Ekholm–Eliashberg–Murphy–Smith to make every self-intersection point odd. That theorem, read with its own (opposite, unstated) sign convention, always leaves one self-intersection point of degree 2, and that point typically obstructs the Lagrangian. Nathan Humphrey had apparently found the convention mismatch first and reported it to OpenAI. The authors call the HMS route "interesting and worth pursuing" and say the error has clarified the EEMS13 sign convention. They do not discuss the main CM-abelian-variety paper, which OpenAI has not withdrawn.

Sources: [openai/math history.md: three Hodge-family withdrawals (7 Oct)](https://github.com/openai/math/blob/main/history.md); [Elliot Glazer on X: the eightfolds paper 'seems to be flawed'](https://x.com/ElliotGlazer/status/2108026240582246600); [Kiwamu Watanabe on X (in Japanese)](https://x.com/Kiwamu_Watanabe/status/2107621501193408756); [Paper: The rational Hodge conjecture for CM abelian varieties](https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026/paper.pdf); [Paper: The rational Hodge conjecture for products of K3 surfaces](https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/hodge-conjecture-products-k3.pdf); [openai/math README (exceptions to the fixed procedure)](https://github.com/openai/math); [arXiv 2610.30976: Lekili, Sheridan, Rigidity, flexibility, and a sign error](https://arxiv.org/abs/2610.30976)

### 2026-10-06: OpenAI releases 722 AI-written math manuscripts (372 result families) claiming hundreds of open problems, incl. quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors

*****; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/

On 6 Oct 2026 (22:19 UTC) OpenAI published github.com/openai/math: 722 manuscripts in 372 result families (dated 10 Sep–6 Oct, most 23–27 Sep, released together), written by an unreleased internal model that was posed about 4,000 problems, at an average of ~3 hours of ChatGPT Pro thinking compute per result. The catalogue claims to settle a long list of famous problems: the quasi-Riemann hypothesis (no zeta or Dirichlet L-function zeros with Re s > 7/8, hence no Landau–Siegel zeros), Khot's Unique Games Conjecture, the rational Hodge conjecture for CM abelian varieties, the isomorphism of free group factors, Kaplansky's direct-finiteness and zero-divisor conjectures (counterexamples), L = BPL, ω ≤ 9/4, Hilbert's tenth problem over ℚ and many more. A Lean library accompanies the release: 235 of the 372 families link a Lean scope note, the catalogue lists 162 manuscripts with a formalized main result, and OpenAI marks its review status 'unchecked'. OpenAI warns that unformalized results 'could have issues'. The release is real and checkable; the mathematics is mostly not yet independently verified.

- Repo: github.com/openai/math, single 'Initial commit' at 2026-10-06 14:58 PDT; ~3.1 GB, 722 manuscripts in preprints/ (PDF + source + BibTeX), overview.pdf ('OpenAI Research Catalog', 17 subject sections), CONTENTS.md manuscript map, Lean library in lean/, 10 reasoning summaries
- Manuscript dates run from 10 Sep to 6 Oct 2026; most are dated 23–27 Sep (≈560) and 4–5 Oct (≈136). OpenAI had claimed '100+ open problems' on 21 Sep (see 2026-09-21-openai-100-open-problems-claim)
- Procedure (README): 'The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model. On average, each result used three hours of ChatGPT Pro thinking compute with that model.' About 4,000 problems were posed; families were kept if they reached 'an appropriate level of significance'
- Stated exceptions to the fixed procedure: the zero-free region for the Riemann zeta function and the Hodge conjecture for CM abelian varieties; the Re(s) > 11/12 zero-free-region write-up was 'human edited for readability'
- Caveats (README): 'This collection includes results at different stages of verification. Not all have accompanying Lean formalizations… Some of the unformalized results could have issues. We will endeavor to fix any such issues quickly.' Corrections will be released as new versions, old versions kept
- Lean: lean/formalization.yaml lists 162 manuscripts with a formalized main result and 185 Comparator challenge configurations; scope 'Partial progress'; review status 'unchecked'. Challenges allow only the standard axioms (propext, Quot.sound, Classical.choice). It builds on Mathlib, PrimeNumberTheoremAnd (Kontorovich) and other community libraries. Per-family scope notes say exactly which statements were formalized (often the main theorem, not every consequence)
- Reasoning summaries for 10 families: two-point Chowla (007), irrationality exponent of π (017), Mahler conjectures (087), NP-hardness at the basic SDP threshold (102), quasipolynomial Szemerédi bounds (159), Kaplansky direct finiteness in char 2 (197), Mézard–Parisi formula (221), spontaneous magnetization in the quantum Heisenberg ferromagnet (271), free group factor isomorphism (287), 3D relativistic Vlasov–Maxwell (362)
- OpenAI blog 'Sharing AI progress in mathematics': the release drew on the 'advice and public recommendations' of the independent Advisory Group on Mathematics and AI (AGMAI, IAS); OpenAI will fund 'workshops, conferences, and special programs around the understanding of major results produced by AI' and is 'working to responsibly release the model that produced these results'
- AGMAI statement (6 Oct): 'an important event for mathematics'; its advisory role 'should not be interpreted as a judgment of the impact of these results or an endorsement of the process'; 'This release is the beginning, not the completion, of the process of human understanding'
- OpenAI's X post (6 Oct, 22:19 UTC): ~3.97M views, 18k likes on 7 Oct
- Famous claims, Lean-formalized per OpenAI's catalogue (not yet independently reviewed): quasi-Riemann hypothesis Re s > 7/8 for ζ and all Dirichlet L-functions (003, 199-page paper) and uniform exclusion of Landau–Siegel zeros; Unique Games Conjecture (102); all free group factors isomorphic, L(F2) ≅ L(F3) (287); irrationality exponent of π equals 2 (017); Catalan's constant irrational (005); Kaplansky zero-divisor conjecture false (196) and direct-finiteness false in char 2 and odd char, via a finitely presented nonsofic group, also refuting Gottschalk's surjunctivity conjecture (197); Thompson's group F nonamenable (248); Cannon's conjecture (246); ω ≤ 9/4 for matrix multiplication (107); symmetric and general Mahler conjectures (087); Falconer distance conjecture in all dimensions (073); Erdős's reciprocal-sum (Erdős–Turán) conjecture on arithmetic progressions (159); two-point Chowla (007); the plane is not 5-colourable (158); Borsuk fails in dimension 9 (156); Sidorenko's conjecture false (161); Barnette's conjecture (180); Seymour's second-neighbourhood conjecture (173); circulant Hadamard conjecture (179); Artin K(π,1) conjecture (254); Kadison similarity (288); hyperinvariant-subspace counterexample (293); hot-spots conjecture for simply connected planar domains (369); spontaneous magnetization of the 3D quantum Heisenberg ferromagnet (271); 3D relativistic Vlasov–Maxwell global regularity (362); Mézard–Parisi formula (221)
- Famous claims with NO Lean scope note (manuscript only): rational Hodge conjecture for CM abelian varieties, with the Tate conjecture for abelian varieties over finite fields and the Hodge standard conjecture as consequences (032); full BSD formula for Selmer corank 0/1 and Goldfeld's conjecture (002, 006); Hilbert's tenth problem over ℚ undecidable (004); L = RL = BPL (103); modularity of elliptic curves over imaginary quadratic fields (030); Fujita's freeness conjecture (038); Bloch's conjecture for surfaces (040); Kakeya maximal conjecture in 3D (074); abelian Zilber–Pink (016); the p-adic section conjecture (019); infinitude in Artin's primitive root conjecture (029); Hilbert–Smith conjecture (304); uniform limit-cycle bound in Hilbert's 16th problem (143; only the quintic Liénard case is in Lean); Hadwiger's conjecture false in fractional form (157; Lean covers only the companion linear list-Hadwiger bound)
- Several families overlap earlier AI-era results already in this dataset (k-server, Crouzeix, Mumford–Shah, Kahn–Kalai, Borsuk, linear Hadwiger)
- Overlap: family 181 (Erdős–Gallai cycle-decomposition conjecture, manuscript dated 24 Sep) proves the same theorem as Jaehoon Kim's arXiv 2610.07840 (posted 6 Oct, disclosing 'extensive use of ChatGPT (OpenAI) and Claude (Anthropic)')
- A model's reaction: Claude Opus 5.5, run on the cloned repo by @ereliuer_eteer and posted as its output 85 minutes after the release (~90k views; header 'Opus 5.5 on github.com/openai/math', and the text addresses the poster as 'you'), checked the Lean challenge statements ('Nothing hides in a home-made definition'), flagged that HarmonicGrowth.lean states its theorem as an axiom, and described 'a mix of awe and vertigo, plus a bit of grief'. We confirmed the flagged details in the repo; its '~120 formalized papers' undercounts formalization.yaml's 162
- Repo structure (checked 7 Oct): lean/ComparatorChallenges holds 405 challenge statements, each with a JSON config allowing only propext, Quot.sound and Classical.choice. Most state the theorem with `sorry` (e.g. quasi-RH: `riemannZeta s ≠ 0` for `7/8 < s.re`, against Mathlib's riemannZeta); HarmonicGrowth.lean instead uses `axiom mainStatement : MainClaim` with `theorem main := mainStatement`, presumably a stand-in since the comparator checks the solution module. The quasi-RH solution (OAI.NumberTheory.DirichletL.Nonvanishing) imports a module named Detector.FinalAssemblyUnconditional
- Viral readings checked against the repo (7 Oct): 'kakeya conjecture in R^3 has been resolved' (Rohan Pandey, ~96k views) mixes up two problems. The 3D Kakeya set conjecture was already proved by Hong Wang and Joshua Zahl in 2025 (arXiv 2502.17655). Family 074 claims the 3D Kakeya maximal conjecture and the 4D Hausdorff-dimension conjecture, both manuscript only. 'five-colours is no go' (Peter Gostev, ~59k) is accurate: family 158 (Lean: no_proper_five_coloring, arbitrary colourings) leaves 6 or 7 as the only possible chromatic numbers of the plane, partial progress on Hadwiger–Nelson (lower bound 5 since de Grey 2018)
- WSJ (7 Oct): OpenAI says it set its model on the Millennium Prize Problems, pouring 'millions of dollars in computing power' into the effort, only after hearing that Anthropic had already solved at least one. It declined AGMAI's request to stop testing on proprietary models: 'It is important to continue to evaluate our internal frontier models on mathematics and other sciences'
- First withdrawals (history.md, 7 Oct): a sign error in 'Algebraicity of Weil classes on split abelian eightfolds' 'invalidates a stabilization-trace cancellation argument and the construction used by two dependent papers'; OpenAI withdrew it together with 'Algebraicity of Kuga–Satake Correspondences for K3 Surfaces' and 'The rational Hodge conjecture for products of K3 surfaces' (all in the Hodge family 032, none Lean-formalized). The main CM-abelian-variety Hodge paper was not withdrawn. 14 other manuscripts were repaired, 13 updated citations; the catalogue now counts 719 manuscripts, with 300 of 719 top-line results (~42%) formalized after 6 new formalizations
- The flaw was flagged publicly first by Elliot Glazer (2:46 UTC, 8 Oct): 'Based on a hot tip: I had Astra check the OpenAI paper … and it seems to be flawed. This is one of the nonformalized results.' (~119k views). OpenAI's Dan Roberts announced the update at 5:20 UTC (~341k views); HN threads 337 and 224 points
- Independent human-led (often AI-assisted) proofs of problems in the catalogue appeared within days, with priority statements: Kittle & Kogler's exact overlaps conjecture (arXiv 2610.10511, found by GPT-6 Astra on 27 Sep; OpenAI family 148 dated 24 Sep), Hu, Liu & Wan's LeBrun–Salamon conjecture (2610.10410, 'obtained well before' OpenAI's family 062), Jaehoon Kim's Erdős–Gallai cycle decomposition (2610.07840, family 181), and Minzer, Fei & Wang's 4-to-1 games result posted on 14 Sep after hearing rumours (Quanta)
- More overlaps posted 7–8 Oct (sweep 2026-10-09): Engel & Mauri's human-written proof of the hyperkähler SYZ conjecture (arXiv 2610.12277; family 041; 'The structure of the proofs appears similar'); Qin & Sun's Ω(d² ln² d) Hilbert-number lower bound, obtained 'by the end of August' with GPT-5.6 Sol (2610.12218; complementary to family 143's finiteness claim); Ky's antipodal Hellinger conjecture (2610.10618; family 119). Family 119's Courtade–Kumar conjecture had already been proved three times on 21–22 Sep (see 2026-09-21-courtade-kumar-conjecture-proved)
- Expert refusal: Asaf Karagila, the leading specialist on the Partition Principle (family 244), wrote on 8 Oct that the preprint 'sucked. It is unclear, muddled' and should be desk-rejected, and that he will not review it; he called the release 'the equivalent of a Denial of Service' (HN 172 points; see 2026-10-08-karagila-openai-partition-principle)
- Organised opposition: the Association for Human Mathematics (statement dated 6 Oct, reposted as a guest post on Terence Tao's blog on 7 Oct) called the release 'not a demonstration of scholarship, but a demonstration of power' and urged mathematicians to 'discontinue their work with OpenAI'
- Autoformalisation critique: Bastounis, Circelli & Hansen (arXiv 2610.08144, 6 Oct; HN 334 points) argue that Lean verification of AI autoformalisations 'may offer no confidence in the original NL argument' and show that the Lean proof of OpenAI's Navier–Stokes blow-up does not correspond to its natural-language proof
- 'Lean checks at 42%' headlines (tech-insider.org, Developers Digest, 8–9 Oct) are a misreading: the figure is OpenAI's own README line 'The repository has ~42% top-line results formalized', i.e. history.md's 300 of 719 top-line results after the 7 Oct update. It is a share formalized, not a share that passed or failed an independent check
- Axios (8 Oct), 'OpenAI's math breakthrough points beyond math': maths, like programming, gives AI 'a way to tell when it is right', since proofs can be checked by experts and increasingly by computers; the piece quotes a Rutgers mathematician (Kontorovich's 'instant Fields Medal' line) alongside doubts about how original the results are
- 36Kr's 'Terence Tao Reposts Boycott Statement: Mathematics Community Fully Cuts Ties With OpenAI' (9 Oct) overstates events: it refers to the AHM statement Tao ran as a guest post on 7 Oct (already above), not a new statement by Tao or a field-wide boycott; the AHM text sets no enforcement mechanism
- Community reactions collected (Proofs and Prompts, '100+ reactions to 100+ solutions', posted Oct 8, updated Oct 10; roughly 80–100 signed reactions): Tao: 'many of the AI-generated proofs appear to introduce clever new ideas' but 'I am horrified by the many person-years of ongoing patient and deliberately slow research efforts … being casually disrupted or destroyed by such a release'; Scholze: 'mathematics is a marathon, not a sprint'; Agol: problems 'I might not have expected to be resolved in my lifetime'; Ben Green: the Re s > 7/8 proof 'is absolutely shocking to me'; Henry Wilton: 'If OpenAI wanted to destroy the mathematical community, this would be a great way to go about it.'
- Economist Joshua Gans ('OpenAI drops a bomb on maths', Substack, Oct 7): 'this is probably the biggest day of scientific advancement in history'; argues AI will split mathematicians' work into problem formulation and verification around machine proving
- The Verge long read (Robert Hart, 9 Oct; 'more than three dozen mathematicians'): Lichtman counts 'tens' of Fields-calibre results (progress toward RH, a Hodge special case, 4D Kakeya) but says all come 'out of existing methods and techniques'; Scott Armstrong says the targets show OpenAI is going after Yang–Mills and the mass gap; Buckmaster has heard of 'three people who had entire research programs obliterated'; Colva Roney-Dougal: 'friends and colleagues whose grant proposals have just been wiped out'
- Tao's Caltech lecture 'Math 2.0' (9 Oct, slides 10 Oct): 'proof abundance' makes 'further blind optimization of problem-solving alone … actively harmful to the long-term health of mathematics' (see 2026-10-09-tao-math-2-0-caltech-lecture)
- NBC News (Evan Bush, 11 Oct; mostly paywalled, opening read): headline 'Some OpenAI math discoveries are "jewels," if verified. But mathematicians are wary of "slop."'; one mathematician said the community should be 'dancing in the streets', another compared the release to a tsunami hitting her field; NBC counts potential solutions to roughly 370 problems
- Retraction Watch (Alicia Gallegos, Oct 8): an OpenAI spokesperson said the errors behind the three withdrawals were found in an internal audit: 'Where errors are identified, we will work to correct them promptly and withdraw papers if no fixes can be found.' Roughly half of the results were released without formal confirmation; OpenAI said its AGMAI advisers recommended releasing before 'full formalization'. Cornell's Alex Townsend: 'I think that OpenAI should have announced the manuscripts that were lean verified first'; he expects more errors. MIT's Andrew Sutherland called the withdrawals 'the responsible thing to do' and said many mathematicians are 'unhappy with how OpenAI has behaved up to this point'
- Sophie Huiberts, guest post on Tao's blog (Oct 12), 'This is normal bad work': 'This is poor quality work and these pdfs are all easy rejects'; 'October 6th was not a watershed moment for mathematics'; 'These pdfs are exhaust fumes from their internal benchmarking effort'. She criticises attribution, errors, missing code and 'salami-slic[ing] 372 results over 719 pdfs', and says a software supplier scooping its own ChatGPT users is 'a bad look' (post file 2026-10-12-huiberts-this-is-normal-bad-work)
- Second-wave press, Oct 12: franceinfo (via RTÉ) quotes Cédric Villani calling it 'the most radical change in mathematics in at least four centuries, perhaps even in 1,000 years', says Hugo Duminil-Copin called the shock 'monumental', said he cried and had been working on 16 of the problems, and quotes Northwestern's Antonio Auffinger (a year ago there were 'always hallucinations') and Benjamin Antieau; Cult of Mac (via Yahoo, citing The Washington Post) reports that NYU's Tristan Buckmaster now keeps original ideas out of Codex, while OpenAI says its investigation found his prompts could not have influenced its Navier–Stokes work. The Economist ('AI's mathematics blitz pushes science into strange new territory') and Quartz ('OpenAI's math data dump wiped out early-career researchers' programs, professor says') were not readable by our fetcher
- Critique of one manuscript (Ben Recht, UC Berkeley, arg min blog, 12 Oct; about 150 points on Hacker News): the release's claimed asymptotically faster Fourier transform is, in his reading, a 'galactic' algorithm. 'The estimated values for C were 10^100 when N=10^12', so the classical FFT stays faster at any usable size. He calls it 'less a breakthrough and more a malicious exploit' of big-O notation. He says he did not read the paper and examined it by questioning ChatGPT about it
- Second wave, Oct 12-13: Le Monde ('They have annihilated my field of research', Hugo Duminil-Copin; the article blocks automated access and was not read, quote from its headline and from IBT UK's summary), The Economist, Quartz (early-career programmes), Tom's Hardware, TechRadar ('mathpocalypse'), RTE and The Week. IBT UK's summary adds that Duminil-Copin said he believed OpenAI wanted to show it could 'massacre' mathematicians, and that Timothy Gowers declined to sign the Fields medallists' letter, saying a flood of results is not in itself a disaster while the loss of open problems could weaken the talent pipeline

#### What happened
Late on 6 October 2026, OpenAI published the list it had promised on 21 September, when it said an internal model had
"resolved" more than 100 open problems. The list turned out to be far larger: a public GitHub repository with 722
manuscripts in 372 "result families", each family grouping a main result with companions, consequences or alternative
proofs, classified into 17 subject areas from number theory to PDE.

The README says the model was posed about 4,000 problems during evaluation, "after performance on our existing mathematical
evaluations saturated". Each kept result used on average about three hours of ChatGPT Pro thinking compute. OpenAI names two
exceptions to this fixed procedure, the zeta zero-free region and the Hodge conjecture for CM abelian varieties, without
saying exactly how they differed. It also publishes abridged "summarized chain of thought" documents for ten families. These
show the model trying and discarding approaches; the free-group-factor summary, for example, lists failed attempts with
rigidity, Gaussian crossed products and free entropy before the final construction.

Verification is uneven, and OpenAI says so. A Lean 4 library ships with the release. `formalization.yaml` lists 162
manuscripts with a formalized main result and 185 Comparator challenges, and 235 of the 372 families in `CONTENTS.md` link
a Lean scope note. Each challenge states a theorem, and a checker confirms that OpenAI's proof proves exactly that statement
using only Lean's standard axioms. Whether each formal statement faithfully captures the famous problem is still a question
for human reviewers, and the catalogue's own review field reads "unchecked". The other families, among them the Hodge, BSD,
L = BPL and Hilbert's tenth claims, are unrefereed manuscripts.

The release followed advice from the independent Advisory Group on Mathematics and AI (AGMAI) at IAS, whose 29 September
guidelines asked labs to disclose compute, prompts and reasoning, formalize proofs and fund human understanding. AGMAI's
statement the same evening called it "an important event for mathematics" but said its role "should not be interpreted as a
judgment of the impact of these results or an endorsement of the process".

### The most famous claims
Status below: **Lean** means OpenAI's catalogue gives a formal proof of the main statement (machine-checked against a
published statement, not yet independently reviewed); **paper** means manuscript only.
- Quasi-Riemann hypothesis: ζ and every Dirichlet L-function have no zeros with Re s > 7/8, plus uniform exclusion of Landau–Siegel zeros (Lean). Separate entry: 2026-10-06-quasi-riemann-hypothesis-openai.
- Khot's Unique Games Conjecture, plus direct optimal Max-Cut and Vertex Cover hardness (Lean). Separate entry: 2026-10-06-unique-games-conjecture-openai.
- Rational Hodge conjecture for all CM abelian varieties, which would give the Tate conjecture for abelian varieties over finite fields (paper). Separate entry: 2026-10-06-hodge-conjecture-cm-abelian-varieties-openai.
- Free group factors: L(F₂) ≅ L(F₃), so all interpolated free group factors are isomorphic (Lean). Separate entry: 2026-10-06-free-group-factors-isomorphism-openai.
- Kaplansky's conjectures: a torsion-free group algebra over 𝔽₂ with zero divisors (196); direct finiteness fails in characteristic 2 and odd characteristic, through a finitely presented nonsofic group, which also refutes Gottschalk's surjunctivity conjecture and the group-ring Determinant Conjecture (197). All Lean.
- Irrationality exponent of π is exactly 2, which implies the Flint–Hills series converges (Lean for the exponent); Catalan's constant is irrational (Lean).
- Thompson's group F is nonamenable; Cannon's conjecture; Artin K(π,1) conjecture; a torsion-free hyperbolic group that is not residually finite (all Lean).
- Matrix multiplication ω ≤ 9/4 (Lean); integer multiplication and the DFT below n log n (109, 130); L = RL = BPL (paper); NP-hardness of 3-colouring with any constant number of colours (Lean).
- Combinatorics: Erdős's reciprocal-sum conjecture, i.e. any set of integers whose reciprocals diverge contains arbitrarily long arithmetic progressions (Erdős's best-known prize problem; Lean); the plane is not 5-colourable, Borsuk fails in dimension 9, Sidorenko's conjecture false, Seymour's second-neighbourhood conjecture, Barnette's conjecture, the circulant Hadamard conjecture (all Lean); Hadwiger's conjecture false in fractional form (paper; Lean covers only the companion linear list-Hadwiger bound, compare 2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra).
- Analysis and geometry: Falconer distance conjecture in all dimensions (Lean); Kakeya maximal conjecture in 3D (paper); symmetric and general Mahler conjectures (Lean); Hilbert's sixteenth problem, uniform bound on limit cycles in each degree (paper; Lean covers only the quintic Liénard case); hot-spots conjecture for simply connected planar domains (Lean); global regularity for 3D relativistic Vlasov–Maxwell (Lean).
- Number theory and arithmetic geometry (paper): full BSD formula for Selmer corank ≤ 1 and Goldfeld's conjecture; Hilbert's tenth problem over ℚ (undecidable); modularity over imaginary quadratic fields; abelian Zilber–Pink; the p-adic section conjecture; infinitude in Artin's primitive root conjecture.
- Operator algebras: Kadison similarity, strong Kadison–Kastler, the generator problem for II₁ factors, a counterexample to the hyperinvariant-subspace problem on Hilbert space (Lean); counterexamples to Baum–Connes (coefficient-free reduced) and Kadison–Kaplansky (paper).

#### Reactions (first ~7 hours; X view counts from fxtwitter, early 7 Oct)
- **Alex Kontorovich** (number theorist): "Quasi-RH?!?!???! Are you kidding me? If a human did this, it would be an instant Fields Medal, no questions asked… They got a zero free strip!!!! Insane" ([X](https://x.com/AlexKontorovich/status/2107609087902941646), ~420k views). A follow-up: "Yeah, no Siegel zeros either. So I guess two Fields medals…"
- **Levent Alpöge** (Anthropic): "Big, big, big, big props for quasiriemann and no Siegel zeroes". He called it "obviously the most significant moment in mathematical history", and also wrote: "There are some sad stories related to their users getting scooped / conflicts of interest (kakeya maximal comes to mind)" ([X](https://x.com/__alpoge__/status/2107616859595981117), ~112k). The Kakeya scooping allegation is unverified.
- **Daniel Litt** (Toronto): "Looks like mathematicians have a lot of exciting work to do". He called some framings of "progress" on famous conjectures "a stretch", and flagged that the Bloch's-conjecture paper "seems to cite some wrong existing work without mentioning it's wrong… but AFAICT doesn't meaningfully rely on it" ([X](https://x.com/littmath/status/2107603741222289427), [X](https://x.com/littmath/status/2107616410897678765)).
- **Jared Duker Lichtman**: "This output is absolutely historic… However, it is not the Millenium problem rumor". He said the cubic family of L-functions "appear to be essential for the proof" of the zero-free strip ([X](https://x.com/jdlichtman/status/2107609204722892924)).
- **Steven Strogatz**, on ω ≤ 9/4: "This leap in progress is like Bob Beamon's long jump" ([X](https://x.com/stevenstrogatz/status/2107606017706266686), ~118k).
- **Joe Bebel** (TCS): the release "includes a claimed proof and Lean formalization of the Unique Games Conjecture, which I have been personally working on for the last year and a half" ([X](https://x.com/joeintheory/status/2107612512288125096), ~99k). **Chris Peikert**: "Unique Games Conjecture, proved. L=RL, proved." ([X](https://x.com/ChrisPeikert/status/2107617657415782731)).
- **Kiwamu Watanabe** (algebraic geometer, in Japanese) called the algebraic-geometry list (Hodge for CM abelian varieties, Fujita, Nagata, log abundance, MMP termination) "quite amazing", but said he cannot tell whether it is correct ([X](https://x.com/Kiwamu_Watanabe/status/2107621501193408756), ~70k). **Isaac Kim** called the quantum-information claims "a shocking list of problems" ([X](https://x.com/Isaac__kim/status/2107607429437657563)).
- **Ken Ono**: "AI-assisted mathematical discovery at the highest level is no longer hypothetical" ([X](https://x.com/KenOno691/status/2107612779264000261)). **Gil Kalai**: "Certainly this is an amazing milestone for mathematics, and the results will needs to be verified" ([blog](https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/)).
- **Terence Tao**, about 30 minutes after WIRED reported the coming release (before it went live), on Mathstodon: solutions "are now being harvested at large scale in an unsustainable fashion", and a "Math 2.0" should "decenter the role of raw problem solving" ([Mathstodon](https://mathstodon.xyz/@tao/117395267721642920)). He had not commented on the content by early 7 Oct.
- **Andrew Sutherland** (MIT), to Scientific American: treat one-shot claims as unverified "until and unless they release the model… We should ask for receipts."
- OpenAI side: Sam Altman, "We are entering a new era of discovery now" ([X](https://x.com/sama/status/2107623610483720463), ~520k); Greg Brockman ([X](https://x.com/gdb/status/2107640886482104524)); Codex lead Thibault Sottiaux ([X](https://x.com/thsottiaux/status/2107597780352950624), ~356k). Former OpenAI contributor @AcerFur on family 109: "we can do integer multiplication faster than n log n" ([X](https://x.com/AcerFur/status/2107606747972309163), ~264k).
- Commentary: Zvi Mowshowitz, "simultaneously the biggest day in mathematical history, and also felt kind of slow"; François Chollet asked whether the jagged frontier is "mainly math + code" driven by RL with verifiable rewards; Gary Marcus said it "uses symbolic AI (Lean, etc)". High-reach posts by Dan McAteer (~388k), Will DePue (~282k) and AISafetyMemes ("90 of the 500 most important open problems", based on an LLM-ranked list at proofatlas.ai, not a mathematicians' ranking).
- **Independent Lean re-check:** within hours, a public repo (davegoldblatt/openai-zeta-proof-check, created 7 Oct 01:36 UTC) re-ran the Comparator check of the 7/8 zeta theorem. It reports acceptance by two kernels (Lean's and nanoda) with only the three standard axioms; the proof spans ~2,900 files and ~500k lines. This is one person's run; the paper itself was not reviewed.
- **Whole-library re-check (Oct 10–25):** Neal McBurnett rebuilt the entire OAI Lean library (122,458 AI-generated files, 26.1M lines, about eleven times Mathlib) at openai/math commit fd4aeeb2ee and ran Comparator on all 416 challenge configurations ([github.com/nealmcb/openai-math-lean-tooling](https://github.com/nealmcb/openai-math-lean-tooling), built with Claude Code). "Everything checks, with Lean's kernel": all 416 accepted, negative controls rejected. The independent nanoda kernel accepted 401 of 416 (with an enlarged stack); its failures were nanoda bugs, not proof errors, reported as nanoda_lib issue 55 (linked from the quasi-Riemann entry; opened Oct 15, closed Oct 21; 14 of 15 pass on newer nanoda, one SquareDifference declaration still overflowed). In his Oct 15 run both kernels accepted QuasiRiemannHypothesis (Lean replay ~6 min, nanoda ~81 s), a second independent two-kernel pass for the 7/8 theorem. Cost: ~3.8 days and ~410 CPU-hours on a laptop, or 13.8 hours and ~$85 on one 32-vCPU cloud machine. He stresses that a pass shows the Lean proof matches the Lean statement, not that the statement is faithful to the paper. We found no Lean Zulip or other forum thread re-verifying the 7/8 proof beyond these two repos.
- **High-reach explainers and memes** (views at 11:31 UTC on 7 Oct): kimmonismus posted an accurate summary of the README, "HOLY, the rumors were true" ([X](https://x.com/kimmonismus/status/2107597320028065793), ~195k). Hesamation wrote "mathematicians: frontier labs need to publish real proofs so we can check them, not blog posts / OpenAI: fine. here's 722 papers" ([X](https://x.com/Hesamation/status/2107600759953179030), ~122k). The meme "math is solved / it's so over" by @cgtwts reached ~182k ([X](https://x.com/cgtwts/status/2107597257684169010)).
- **Kakeya and Hadwiger–Nelson, as read on X:** Rohan Pandey's "kakeya conjecture in R^3 has been resolved" ([X](https://x.com/khoomeik/status/2107603513438335334), ~96k) conflates the 3D Kakeya *set* conjecture, proved by Hong Wang and Joshua Zahl in 2025 ([arXiv 2502.17655](https://arxiv.org/abs/2502.17655)), with what family 074 claims: the 3D *maximal* conjecture plus the 4D dimension conjecture, both manuscript only. Peter Gostev said he had spent "300 billion tokens on Hadwiger–Nelson problem (GPT-5.6 and Astra) and got absolutely nothing". He read family 158 correctly: "five-colours is no go… Not a full solution yet, but it's a biggie" ([X](https://x.com/petergostev/status/2107603534308864146), ~59k). The manuscript "The Euclidean plane is not five-colorable" (dated 23 Sep) has Lean comparator statements for no proper five-colouring and for a proper seven-colouring, so the chromatic number of the plane would be 6 or 7. The lower bound had been 5 since [de Grey (2018)](https://arxiv.org/abs/1804.02385).
- No expert had reported a mathematical error in a main proof by early 7 Oct.
- **First follow-up work:** on 7 Oct Doug Colkitt (@0xdoug) published conditional improvements to family 109's integer-multiplication exponent, from κ = 2^-182 to 2^-59 in three releases, done "with assistance from OpenAI Codex" and assuming OpenAI's unverified theorem ([GitHub](https://github.com/CrocSwap/integer-mult-bounds), [X](https://x.com/0xdoug/status/2107917690568327225)). Separate entry: 2026-10-07-colkitt-codex-integer-multiplication-exponent.

### Press
- WIRED (6 Oct, before the release): "OpenAI is pissing off a bunch of mathematicians again". It reports that mathematicians felt their advice was ignored (Bryna Kra: "Apparently, that input was ignored"; Nestor Guillen: "there's a perception of mobster behavior").
- NYT (Kenneth Chang, Siobhan Roberts): "OpenAI Releases Findings on 377 Math Problems, Further Roiling Field". Paywalled; we have only the preview, and the 377 figure is the NYT's count. Gizmodo also says 377; the repo itself says 372 families.
- Scientific American (Joseph Howlett), The Verge (Robert Hart; quotes AGMAI asking labs not to use releases as "marketing vehicles"), Gizmodo (Mike Pearl; notes AGMAI's 29 Sep request that labs "stop testing advanced mathematical problems on proprietary models").
- WSJ, "AI Solved One Math Problem and Everyone Freaked Out. It Just Cracked Hundreds More." (7 Oct, 01:09 UTC). We read it in full through HT Tech's syndicated copy; the WSJ original is paywalled. It describes "advances related to three of the five remaining Millennium Prize Problems, including the Riemann hypothesis". That is the WSJ's framing; Lichtman said the release is "not the Millennium problem rumor". According to the WSJ, OpenAI says it set its frontier model on the Millennium Prize Problems, "and poured millions of dollars in computing power into the effort", only after hearing that Anthropic had already solved at least one. OpenAI turned down AGMAI's request to stop testing on proprietary models: "It is important to continue to evaluate our internal frontier models on mathematics and other sciences, so we can accelerate developing the tools to advance those fields." The WSJ also reports that mathematicians rushed to post their own AI-assisted work before they could be scooped. It quotes Kontorovich by name and Alpöge's "obviously the most significant moment in mathematical history" as the words of "a researcher inside Anthropic".

### Day two (7–8 Oct): first withdrawals, rival proofs and a boycott call
- **Withdrawals.** On 7 Oct OpenAI updated the repository and its new `history.md`. A sign error in "Algebraicity of Weil classes on split abelian eightfolds" invalidated "a stabilization-trace cancellation argument and the construction used by two dependent papers", so OpenAI withdrew that paper, "Algebraicity of Kuga–Satake Correspondences for K3 Surfaces" and "The rational Hodge conjecture for products of K3 surfaces". All three belong to the Hodge family 032 and none had a Lean formalization. The headline CM-abelian-variety paper was not withdrawn. Fourteen other manuscripts were repaired (Lipschitz heights/Ashkin–Teller, Kähler MMP and abundance, taming/hypersymplectic, incompressible box transport, a stale citation in the BSD paper), and 13 more had their citations updated. Six new formalizations brought the count to 300 of 719 top-line results (~42%). Elliot Glazer had posted about two and a half hours before OpenAI's announcement that, "based on a hot tip", he had GPT-6 Astra check the eightfolds paper and "it seems to be flawed" ([X](https://x.com/ElliotGlazer/status/2108026240582246600), ~119k views). OpenAI's Dan Roberts announced the update: "We will continue to update the repo with new formalizations and with any errata we notice" ([X](https://x.com/danintheory/status/2108065033070789090), ~341k views). This was the first confirmed error in the release, and it fell in the unformalized part, as OpenAI's README had warned.
- **Rival and independent proofs.** Within a day, mathematicians posted papers on problems that also appear in the catalogue, each with a priority note. Kittle and Kogler proved the exact overlaps conjecture on the line (arXiv 2610.10511). Their disclosure says GPT-6 Astra found the first proof on 27 Sep, while OpenAI dates its family 148 manuscript 24 Sep. Hu, Liu and Wan's LeBrun–Salamon proof (2610.10410) says its argument was "obtained well before" OpenAI's family 062. Earlier, Jaehoon Kim had done the same for Erdős–Gallai (family 181). Quanta's Ben Brubaker reported that Dor Minzer's MIT group rushed out a 4-to-1 games result on 14 Sep, three days after hearing rumours of OpenAI's proof. Minzer: "There is a lot of value in failing and knowing why you failed. Using AI takes all of this out." Braverman and He (2610.10108) note that OpenAI's sub-n log n Fourier-circuit result also implies their refutation of the multiple-unicast conjecture.
- **Backlash and reflection.** The Association for Human Mathematics (see 2026-10-05-quanta-is-ai-the-end-of-math) issued a statement dated 6 Oct: "Releasing over 700 files at once is not a demonstration of scholarship, but a demonstration of power". It urged mathematicians to "discontinue their work with OpenAI and to return to a vision of science that centers human understanding". Terence Tao reposted it as a guest post on 7 Oct, followed by guest posts from Raghu Meka ("The barriers of perception": "the downside of not believing there is a barrier is far less than that of believing there is one") and Álvaro Lozano-Robledo ("What should we tell our students?": "Keep calm and carry on studying math"). Bastounis, Circelli and Hansen (arXiv 2610.08144) argued that Lean checks of AI autoformalisations need not certify the natural-language proof, using OpenAI's Navier–Stokes formalization as an example (see 2026-09-08-openai-navier-stokes-blowup). On 8 Oct Tao also shared the update to a parody he had posted before the release, "OpenAI Releases the Final Ten Minutes of 500 Previously Unreleased Films" (@Endings500), which now read "We Are Withdrawing Three Endings from Endings-500, Reaffirm Our Full Confidence in the Remaining 497" ([Mathstodon](https://mathstodon.xyz/@tao/117406062062897767)). TechCrunch's 8 Oct headline summed up the second-day mood: the solutions "aren't meeting the field's standards yet".
- **Outside mathematics.** Ethereum researcher Justin Drake cited the release ("mathematical superintelligence is upon us") in a 5.5M-view call for crypto "bunker mode"; see 2026-10-07-justin-drake-bunker-mode-ai-cryptography.

### Day three (8–9 Oct): expert triage, Lean doubts and the press wave
- **Erdős problems.** Thomas Bloom, who runs erdosproblems.com, listed the claims relevant to his site on 8 Oct ([blog](https://www.erdosproblems.com/forum/thread/blog:10)). He counts significant claims on 25 Erdős problems (#3, 90, 120, 138, 148, 172, 174, 181, 184, 293, 304, 371, 416, 431, 508, 551, 604, 802, 821, 952, 968, 970, 986, 1083, 1150). These include 12 problems from the FrontierMath Erdős list (#508 only partly), 21 of the 25 have Lean formalizations, and the PDFs total 1,347 pages. He has "not yet looked at in any serious way any of the proofs". His verdict: the papers are "very poorly written, in the usual AI fashion", and he expects most to be "(a) correct yet (b) capable of being substantially simplified", "very rich 'raw material'". He flags a gap for #3: the PDF claims quasipolynomial bounds for Szemerédi's theorem, but the Lean challenge file states only Erdős's qualitative conjecture, and the Lean code seems to prove a weaker bound. (OfficeChai's earlier count of 23 came from a first version of the list.)
- **Scientific American (8 Oct)** gave two pieces. In "The most exciting claims…" (Peter Hall, Joseph Howlett), Hector Pasten (PUC Chile) said of the Siegel-zero result: "This is not just progress on Siegel zeros—it settles the question. This is amazing." Roland Bauerschmidt (NYU) said of the sigma-model/Yang–Mills-type papers: "The second paper is horrible. If these results had been sent to me by a nobody, I would have deleted the e-mail." In "Mathematicians marvel, and grumble…", Michael Douglas (Harvard) said "Yesterday will be the day when a new era of math and mathematical physics began", and Andrew Sutherland (MIT) said "We should expect some of the proofs to contain mistakes, possibly serious ones". SciAm also noted that OpenAI has not said which of the ~4,000 problems posed the model failed on.
- **Unique Games, first-hand.** Scott Aaronson's "The Mathocalypse" (7 Oct) called the release "surely one of the biggest days in mathematical history". He noted that the claimed Unique Games proof lands directly on the research of his wife, Dana Moshkovitz. The Decoder (8 Oct) quotes her text message about the proof: "It feels like something written by someone who's on psychedelics."
- **Fortune (Jeremy Kahn, 7 Oct):** Daniel Litt called the results "great for mathematics". OpenAI gave reasoning summaries and compute estimates for only 10 problems. Its promise to fund "workshops, conferences, and special programs" still had no dates, hosts or budget by 9 Oct.
- **Boycott framing.** The Decoder (Manuel Uth, Matthias Bastian, 8 Oct) headlined that "some mathematicians call for OpenAI boycott", meaning the AHM statement. It describes Tao as chairing the AHM. Other reports call the statement the AHM communications group's text, run as a guest post that is not Tao's own position (XenoSpectrum, Futurism), and the AHM site lists no officers. Tao's chairmanship is unverified. Futurism (9 Oct) headlined "Mathematicians React With Fury".
- **Lean's own reliability.** On 9 Oct Thomas Hales posted a guest post on Tao's blog on what mathematicians should know about Lean. It recalls the July–August 2026 "Summer of Soundness Bugs", which included an AI-assisted Lean "disproof" of Collatz that passed two kernels, and says Lean proofs need a human audit "to ensure statement fidelity". See 2026-07-28-lean-kernel-soundness-bug-collatz. Zvi Mowshowitz's 9 Oct roundup ("New Math from OpenAI") collects similar worries, plus Josh Frisch's "main emotion thus far… has been loss: loss of meaning, loss of purpose".
- **The Conversation (Melissa Lee, Monash, 9 Oct; syndicated by Phys.org):** a colleague found one paper so unintelligible that as a journal editor "it would have gone straight into the bin". She quotes reactions ranging from "obviously the most significant moment in mathematical history" to "a giant pile of turd dumped at our doorstep", and suggests the release also serves OpenAI's AGI narrative ahead of its IPO.
- **Headlines we could not read (paywalled or blocked):** NYT, "'Breathtaking,' 'Devastating': Mathematics Reels After New OpenAI Release" (8 Oct); The Atlantic, "OpenAI Just Carpet-Bombed Mathematics"; Science, "AI for research mathematics has arrived"; New Scientist, "OpenAI has dumped 722 maths papers – now it must clean up the mess" and "OpenAI mistranslated mathematics into code for its Navier-Stokes proof" (about the Bastounis–Circelli–Hansen critique); The Guardian, "OpenAI's release of mathematical findings draws concerns from experts"; Sky News, "'mathpocalypse'". They are listed in links by title only.
- **Misreadings to avoid.** Some blogs (e.g. Pasquale Pillitteri, "solved parts of two more Millennium Problems") present the quasi-RH and the Hodge-for-CM-abelian-varieties results as Millennium Prize progress. Neither claims the Riemann hypothesis or the Hodge conjecture itself, and Lichtman called the release "not the Millennium problem rumor". Separately, Chinese and SEO sites re-ran older Anthropic results as new during the week (36Kr on θ(p_c)=0, see 2026-09-03-dying-percolation-theta-pc-zero; tech-insider on the 67.2% zeta-zeros result, see 2026-08-10-claude-riemann-zeta-zeros-two-thirds).

### Day four (9–10 Oct): the long reads
- **The Verge (Robert Hart, 9 Oct)** spoke to "more than three dozen mathematicians" ("Staggering", "Surreal", "Pure insanity"). Most said understanding the release could take years. Kevin Buzzard found about six results in algebraic number theory that "stood out", few if any formalized: "I either have to read possibly-not-correct slop, or wait for others to do the same, or wait for someone to formalise them". Brendan Hassett (Brown): "If this had been written by a person, I wouldn't spend any more time trying to understand it." Nalini Joshi (Sydney) noted "short bibliographies". Jared Duker Lichtman put "tens" of results at Fields-Medal level (progress toward RH, a Hodge special case, the 4D Kakeya conjecture) but said they all come "out of existing methods and techniques"; Álvaro Lozano-Robledo agreed. Scott Armstrong said "there were definitely some targets": Hong Wang's area and several Millennium problems, with a cluster in mathematical physics pointing at Yang–Mills and the mass gap. On the damage: Armstrong knew of a group whose programme was "wiped out", Buckmaster of "three people who had entire research programs obliterated", Colva Roney-Dougal (St Andrews) of colleagues "whose grant proposals have just been wiped out", and Francesco Fournier-Facio (Heriot-Watt) said parts of group theory were "bulldozed". Constantin Kogler (IAS) called it "the most important single moment in the history of mathematics", and Yang-Hui He (LIMS) compared the year to Euclid's *Elements*. The Verge notes OpenAI did not name the model or disclose prompts or the full problem list, against AGMAI's advice, and did not comment on the record.
- **Business Insider (Katherine Li, 10 Oct):** Buckmaster: "So we've had whole research programs wiped out on Tuesday by their dump". He also asked whether unpublished grant proposals, often run through LLMs during review, ended up in training data (a question, not evidence). Bryna Kra feared mathematicians will become more secretive. Kontorovich: "I don't think we have any interest in giving prizes to someone who pressed the button." OpenAI did not respond. Buckmaster repeated the "chilling effect" argument on MS NOW (Jacob Soboroff, 10 Oct).
- **The Independent (Andrew Griffin, 9 Oct)** read OpenAI's blog as "almost apologetic". It quoted the Proofs and Prompts reactions of Henry Wilton ("If OpenAI wanted to destroy the mathematical community, this would be a great way to go about it") and Hugo Duminil-Copin ("The shock is immense. I am paralysed"), and Alex Meiburg's blog on the "intrinsic value" of human comprehension. The Daily Mail, Chosun Ilbo (Duminil-Copin, Martin Bridson: "We should not limit ourselves to following in their footsteps", Buzzard, Tobias Osborne) and The Decoder ("How much beauty have we lost?", Sam Hughes; Matt Zaremsky's "blasted … with TNT") re-ran the same collection.
- **Opinion.** In Inside Higher Ed (9 Oct), Ahmed Abbes (CNRS) and Haynes Miller (MIT) called the wave "very much like an occupation by a foreign power" and said advisory committees give labs "a useful cover". They asked for government-guaranteed equal access to frontier models. Tom Whipple's Times column ("Has AI just killed maths?") is paywalled and was not read.
- **Older voices.** Fields Medallist Maxim Kontsevich told China Media Group (aired 10 Oct): "It's really faster than I expected", comparing AI proofs to "winning complicated chess in complicated position". The interview does not mention OpenAI. Tom Verhoeff (TU/e), who used Claude Opus 5.5, DeepSeek and Aristotle on Lehmer's permutation conjecture (see 2026-10-01-lehmer-permutation-conjecture-opus-5-5), told IO+ the OpenAI results look "truly esoteric".
- **Kakeya.** 36Kr/新智元 (10 Oct) pointed out that the catalogue's 4D Kakeya manuscript runs 175 pages and the 3D maximal-function version 97 pages, against Wang and Zahl's 127-page 3D set proof (Feb 2025). It reports no comment from Wang or Zahl.
- **Rehashes, not new findings.** "Mathematicians flag translation error in OpenAI's Navier–Stokes proof" (news.lavx.hu, 10 Oct; Chosun, 9 Oct, from TechCrunch) is the 6 Oct Bastounis–Circelli–Hansen preprint already covered above and in 2026-09-08-openai-navier-stokes-blowup. Crypto Briefing's "OpenAI claims breakthrough in solving Navier–Stokes" and Cape Times's "Touched on their hypotenuse" (Michael Sherman, 7 Oct) restate the September dispute.
- **Tao's "Math 2.0" lecture** (Caltech, 9 Oct) is a separate entry: 2026-10-09-tao-math-2-0-caltech-lecture.

### Follow-up papers by outside mathematicians (8–9 Oct arXiv)
- Krzysztof Burdzy, co-author of the 1999 Burdzy–Werner counterexample to the general hot spots conjecture, posted a ChatGPT-generated note ([arXiv 2610.12622](https://arxiv.org/abs/2610.12622)) extending OpenAI's smooth-domain hot-spots theorem to non-smooth simply connected domains. His commentary: "The OpenAI model that generated [3] missed the theorem presented in this note", and "It is highly surprising that [3] did not cite the spectacular paper of de Dios Pont", which shows that the conjecture fails for convex domains in high dimensions. Rodrigo Bañuelos produced a near-identical result with ChatGPT at the same time.
- Baiying Liu re-tuned the parameters of the quasi-Riemann proof to get Re s > 0.874957 in place of 7/8, with a Lean formalization ([arXiv 2610.12234](https://arxiv.org/abs/2610.12234)).

### Expert expositions and follow-ups (10–13 Oct arXiv)
Several specialists posted human rewrites of release proofs, the first outside checks of individual families:
- **Thompson's group F (248):** Sang-hyun Kim and Yash Lodha, "a human written exposition of OpenAI's recent proof that Thompson's group F is nonamenable", with a generalization; "No part of this writing … was produced by AI" ([arXiv 2610.18360](https://arxiv.org/abs/2610.18360); own entry 2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition).
- **Landau–Siegel zeros (003):** William D. Banks, explicit constant (1 − β) log q ≥ 0.003, written with Claude and checked line by line ([arXiv 2610.15483](https://arxiv.org/abs/2610.15483); own entry 2026-10-11-landau-siegel-zeros-banks-explicit-account).
- **Deligne–Drinfeld conjecture (008):** Thomas Willwacher, "a simplified and more human-friendly version of the proof … No originality is claimed", written "for the convenience of myself and fellow human readers who found [Ope26] hard to follow"; AI tools helped him understand and restructure the argument ([arXiv 2610.15592](https://arxiv.org/abs/2610.15592)).
- **Builds on release results:** Goldbach exceptional set X^{7/12+ε} from the 7/8 half-plane (Pintz & Althöfer, [2610.15559](https://arxiv.org/abs/2610.15559)); 7/8 → 0.8748867 (Sang, [2610.15291](https://arxiv.org/abs/2610.15291)); unconditional Ellenberg–Venkatesh ℓ-torsion bounds (Frei, Sofos & Widmer, [2610.18491](https://arxiv.org/abs/2610.18491)); natural length equals the exact Hausdorff measure for SLE (Zhan, [2610.15066](https://arxiv.org/abs/2610.15066), family 230); truth-table degrees rigid under Baire-measurable automorphisms, adapting the Turing-degree rigidity proof (Kjos-Hanssen, [2610.16059](https://arxiv.org/abs/2610.16059), family 241, with Claude and a Lean proof by Aristotle); a dynamical two-point Chowla generalization (Wang & Yi, [2610.16394](https://arxiv.org/abs/2610.16394), family 007); Erdős #824 conditionally on the "Weighted dilation graphs" preprint (Li & Zhuang, [2610.15674](https://arxiv.org/abs/2610.15674), Lean by ChatGPT/Codex).
- **14 Oct listing:** follow-ups to the release (Penrose/Bray et al., Hardy–Littlewood–Chowla, Shlyakhtenko and others) are in [the mid-October catalogue](/e/2026-10-14-ai-assisted-results-mid-october-2026/) (moved there on 2026-10-15 to keep this page within the size budget).
- **Reading the release as a source of tools:** Ying Xie, "What the Machine Invented" ([arXiv 2610.17090](https://arxiv.org/abs/2610.17090)), extracts new lemmas and constructions from 38 families (extraction with Claude Fable 5.1). The qtBench paper (Cainelli, Luccioli, Iraci, Paolini, [2610.16555](https://arxiv.org/abs/2610.16555)) reports OpenAI researchers' "promising candidate solutions" to several of its 29 algebraic-combinatorics problems and asks whether "AI discovery at scale" is ethical and how results should be released.

#### Why it matters
If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced
by an AI system in about six weeks. Machine-checked Lean proofs for famous statements such as Unique Games and the
quasi-Riemann hypothesis make parts of it verifiable far faster than refereeing. The scale also creates a new problem, the one
AGMAI flagged: the community must now absorb, check and understand hundreds of results it did not choose. Until that happens,
treat each individual claim as unverified unless a Lean proof exists and experts have confirmed that its formal statement
matches the problem.

Sources: [NBC News: Some OpenAI math discoveries are 'jewels,' if verified. But mathematicians are wary of 'slop.' (11 Oct)](https://www.nbcnews.com/tech/tech-news/openai-math-discoveries-are-jewels-verified-mathematicians-are-wary-sl-rcna602148); [OpenAI on X: 'We're releasing a broad range of new mathematical results…'](https://x.com/OpenAI/status/2107596713791767021); [GitHub: openai/math (722 manuscripts, Lean library)](https://github.com/openai/math); [OpenAI: Sharing AI progress in mathematics](https://openai.com/index/sharing-ai-progress-in-mathematics/); [OpenAI Research Catalog (overview.pdf)](https://github.com/openai/math/blob/main/overview.pdf); [Lean formalization catalogue (formalization.yaml)](https://github.com/openai/math/blob/main/lean/formalization.yaml); [AGMAI: On OpenAI's Release of Mathematical Results (6 Oct)](https://agmai.org/statement-oct6/); [Hacker News discussion (652+ points)](https://news.ycombinator.com/item?id=49984923); [Alex Kontorovich on X: 'instant Fields Medal'](https://x.com/AlexKontorovich/status/2107609087902941646); [Levent Alpöge on X](https://x.com/__alpoge__/status/2107616859595981117); [Daniel Litt on X](https://x.com/littmath/status/2107603741222289427); [Joe Bebel on X (Unique Games)](https://x.com/joeintheory/status/2107612512288125096); [Steven Strogatz on X (matrix multiplication)](https://x.com/stevenstrogatz/status/2107606017706266686); [Claude Opus 5.5 reads the repo (posted by @ereliuer_eteer)](https://x.com/ereliuer_eteer/status/2107618065869574208); [Sam Altman on X: 'a new era of discovery'](https://x.com/sama/status/2107623610483720463); [Terence Tao on Mathstodon (pre-release)](https://mathstodon.xyz/@tao/117395267721642920); [Gil Kalai: Sharing AI progress on mathematics – amazing](https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/); [Independent Lean re-check of the 7/8 zeta proof (davegoldblatt)](https://github.com/davegoldblatt/openai-zeta-proof-check); [WIRED: OpenAI is pissing off a bunch of mathematicians again](https://www.wired.com/story/openai-is-pissing-off-a-bunch-of-mathematicians-again/); [NYT: OpenAI Releases Findings on 377 Math Problems, Further Roiling Field](https://www.nytimes.com/2026/10/06/science/openai-math-problems.html); [Scientific American: OpenAI unleashes hundreds more math results](https://www.scientificamerican.com/article/openai-unleashes-hundreds-more-math-results-upon-a-field-already-in-shock/); [The Verge: OpenAI math release on GitHub](https://www.theverge.com/ai-artificial-intelligence/1005004/openai-math-release-github); [Gizmodo: OpenAI dumps 377 new math results on GitHub](https://gizmodo.com/openai-dumps-377-new-math-results-on-github-publishes-hand-wringing-blog-post-2000822613); [WSJ: AI Solved One Math Problem and Everyone Freaked Out. It Just Cracked Hundreds More.](https://www.wsj.com/tech/ai/openai-ai-math-problems-millennium-prize-23d14511); [kimmonismus on X: 'HOLY, the rumors were true' (~195k views)](https://x.com/kimmonismus/status/2107597320028065793); [Hesamation on X: 'OpenAI: fine. here's 722 papers' (~122k views)](https://x.com/Hesamation/status/2107600759953179030); [Rohan Pandey on X: 'kakeya conjecture in R^3 has been resolved' (needs correction, ~96k views)](https://x.com/khoomeik/status/2107603513438335334); [Peter Gostev on X: Hadwiger–Nelson, 'five-colours is no go' (~59k views)](https://x.com/petergostev/status/2107603534308864146); [Douglas Colkitt: conditional κ = 2^-59 improvement of family #109 (integer multiplication), with OpenAI Codex](https://github.com/CrocSwap/integer-mult-bounds); [Wang & Zahl (2025): the Kakeya set conjecture in three dimensions (arXiv 2502.17655)](https://arxiv.org/abs/2502.17655); [openai/math history.md: 7 Oct withdrawals, fixes and formalizations](https://github.com/openai/math/blob/main/history.md); [Dan Roberts (OpenAI) on X: 6 new Lean formalizations, 19 modifications, 3 withdrawals](https://x.com/danintheory/status/2108065033070789090); [Elliot Glazer on X: Astra check finds the abelian-eightfolds paper flawed](https://x.com/ElliotGlazer/status/2108026240582246600); [HN: OpenAI withdraws three mathematical results (224 pts)](https://news.ycombinator.com/item?id=50002650); [AHM Statement on OpenAI's October 6 Release of Mathematical Documents](https://www.ahmath.org/statements); [AHM statement as guest post on Terence Tao's blog (7 Oct)](https://terrytao.wordpress.com/2026/10/07/ahm-statement-on-openais-october-6-release-of-mathematical-documents/); [Raghu Meka (guest post on Tao's blog): The barriers of perception](https://terrytao.wordpress.com/2026/10/07/the-barriers-of-perception/); [Álvaro Lozano-Robledo (guest post on Tao's blog): What should we tell our students?](https://terrytao.wordpress.com/2026/10/08/what-should-we-tell-our-students/); [Bastounis, Circelli & Hansen: Navier–Stokes lost in translation (arXiv 2610.08144)](https://arxiv.org/abs/2610.08144); [Quanta: As AI Closed In on 'Unique Games' Proof, Researchers Raced to Beat the Machines](https://www.quantamagazine.org/as-ai-closed-in-on-unique-games-proof-researchers-raced-to-beat-the-machines-20261007/); [TechCrunch: OpenAI's math solutions aren't meeting the field's standards yet](https://techcrunch.com/2026/10/08/openais-math-solutions-arent-meeting-the-fields-standards-yet/); [Implicator: OpenAI posts 372 AI math results, then withdraws three papers over a sign error](https://www.implicator.ai/openai-posts-372-ai-math-results-then-withdraws-three-papers-over-a-sign-error/); [Thomas Bloom: Erdős problems in the OpenAI proofs (8 Oct)](https://www.erdosproblems.com/forum/thread/blog:10); [Scientific American: The most exciting claims from OpenAI's heap of new proofs (8 Oct)](https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/); [Scientific American: Mathematicians marvel, and grumble, at OpenAI's trove of new results (8 Oct)](https://www.scientificamerican.com/article/mathematicians-marvel-and-grumble-at-openais-trove-of-new-results/); [Scott Aaronson: The Mathocalypse (7 Oct)](https://scottaaronson.blog/?p=10169); [Fortune: OpenAI publishes solutions to more than 370 outstanding math challenges (7 Oct)](https://fortune.com/2026/10/07/openai-math-controversy-solutions-370-outstanding-challenges-published-criticisms-celebration/); [The Decoder: Some mathematicians call for OpenAI boycott after AI-generated proofs flood their field (8 Oct)](https://the-decoder.com/some-mathematicians-call-for-openai-boycott-after-ai-generated-proofs-flood-their-field/); [Futurism: Mathematicians React With Fury as OpenAI Releases Hundreds of New AI-Generated Proofs (9 Oct)](https://futurism.com/artificial-intelligence/mathematicians-openai-furious); [The Conversation (Melissa Lee): Is this the 'mathocalypse'? (9 Oct)](https://theconversation.com/is-this-the-mathocalypse-why-openais-latest-results-dump-has-left-mathematicians-in-shock-293837); [Zvi Mowshowitz: New Math from OpenAI (9 Oct)](https://thezvi.substack.com/p/new-math-from-openai); [Thomas Hales (guest post on Tao's blog): What mathematicians should know about the Lean Theorem Prover (9 Oct)](https://terrytao.wordpress.com/2026/10/09/what-mathematicians-should-know-about-the-lean-theorem-proverquestions-of-reliability-and-ai/); [NYT: 'Breathtaking,' 'Devastating': Mathematics Reels After New OpenAI Release (paywalled, not read)](https://www.nytimes.com/2026/10/08/science/mathematicians-respond-openai-release.html); [The Atlantic: OpenAI Just Carpet-Bombed Mathematics (not read)](https://www.theatlantic.com/technology/2026/10/openai-math-proofs-future/688940/); [Nature News (Davide Castelvecchi): OpenAI posts 700 maths preprints online: mathematicians are up in arms (7 Oct; standfirst: 'The firm used an unreleased model to tackle hundreds of high-profile problems.')](https://www.nature.com/articles/d41586-026-03196-8); [Nature News (Kaia Glickman): Will AI scoop your science? Some researchers see a gloomy future (8 Oct)](https://www.nature.com/articles/d41586-026-03128-6); [Science: AI for research mathematics has arrived (not read)](https://www.science.org/doi/10.1126/science.ael6628); [New Scientist: OpenAI has dumped 722 maths papers – now it must clean up the mess (not read)](https://www.newscientist.com/article/2592684-openai-has-dumped-722-maths-papers-now-it-must-clean-up-the-mess/); [The Guardian: OpenAI's release of mathematical findings draws concerns from experts (not read)](https://www.theguardian.com/technology/2026/oct/07/openai-mathematical-findings-concerns); [Sky News: OpenAI sparks 'mathpocalypse' (not read)](https://news.sky.com/story/openai-sparks-mathpocalypse-as-it-publishes-full-or-partial-solutions-to-372-outstanding-problems-13596633); [Axios: OpenAI's math breakthrough points beyond math (8 Oct)](https://axios.com/2026/10/08/openai-math-proofs-ai-model); [openai/math README ('~42% top-line results formalized')](https://github.com/openai/math/blob/main/README.md); [Developers Digest: OpenAI Math Repo, what the Lean proofs do and don't verify (8 Oct)](https://www.developersdigest.tech/blog/openai-math-repo-lean-proofs-what-they-verify-2026); [Proofs and Prompts: 100+ reactions to 100+ solutions (Oct 8)](https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/); [Joshua Gans: OpenAI drops a bomb on maths (Oct 7)](https://joshuagans.substack.com/p/openai-drops-a-bomb-on-maths); [The Verge (Robert Hart): 'Pure insanity': Mathematicians will need years to make sense of OpenAI's latest drop (9 Oct)](https://www.theverge.com/ai-artificial-intelligence/1008726/openai-mathematics-solutions-chaos); [The Independent (Andrew Griffin): Mathematicians are very, very angry at OpenAI. Here's why (9 Oct)](https://www.independent.co.uk/tech/mathematicians-are-very-very-angry-at-openai-here-s-why-b3064273.html); [Business Insider via Yahoo (Katherine Li): NYU professor says OpenAI's mass release wiped out early-career researchers' projects (10 Oct)](https://tech.yahoo.com/ai/articles/nyu-professor-says-openais-mass-001630910.html); [Inside Higher Ed (Ahmed Abbes, Haynes Miller): The AI Invasion Has Come for Mathematics (opinion, 9 Oct)](https://www.insidehighered.com/opinion/views/2026/10/09/ai-invasion-has-come-mathematics-opinion); [Bastille Post: Fields Medalist Kontsevich says speed of AI advances in math took him by surprise (CMG interview, 10 Oct)](https://www.bastillepost.com/global/article/6240625-fields-medalist-kontsevich-says-speed-of-ai-advances-in-math-took-him-by-surprise); [Chosun Ilbo: AI's math breakthroughs frustrate Fields Medalists (9 Oct)](https://www.chosun.com/english/industry-en/2026/10/09/5FGROQRWCRHJJHMCHFBWKL2STU/); [Daily Mail: The AI 'mathpocalypse' (9 Oct)](https://www.dailymail.com/sciencetech/article-16195709/AI-Mathpocalypse-mathematician-obsolete-fears.html); [The Times (Tom Whipple): Has AI just killed maths? (column, 9 Oct; paywalled, not read)](https://www.thetimes.com/comment/columnists/article/ai-killed-maths-boffins-crunching-numbers-krcmg9gf7); [Retraction Watch: OpenAI withdraws three preprints a day after releasing 722 manuscripts on unsolved math problems (Oct 8)](https://retractionwatch.com/2026/10/08/openai-withdraws-preprints-722-manuscripts-unsolved-math-problems/); [Sophie Huiberts (guest post, Terence Tao's blog): This is normal bad work (Oct 12)](https://terrytao.wordpress.com/2026/10/12/this-is-normal-bad-work/); [franceinfo via RTÉ: Why mathematicians are in 'crisis' due to AI advancements (Oct 12)](https://www.rte.ie/news/a-european-perspective/2026/1012/1595084-ai-maths/); [Cult of Mac via Yahoo Tech: A mathematician shared his ideas with AI. Now, he's keeping his research away from OpenAI (Oct 12)](https://tech.yahoo.com/ai/chatgpt/articles/mathematician-shared-ideas-ai-now-220026003.html); [The Economist: AI's mathematics blitz pushes science into strange new territory (Oct 12; paywalled, not read)](https://www.economist.com/science-and-technology/2026/10/12/ais-mathematics-blitz-pushes-science-into-strange-new-territory); [Quartz: OpenAI's math data dump wiped out early-career researchers' programs, professor says (Oct 12; not readable, 403)](https://qz.com/openai-math-dump-early-career-researchers-wiped-out-101226); [Ben Recht (arg min): Faster Fourier Transforms](https://www.argmin.net/p/faster-fourier-transforms); [Hacker News: A look at the math that OpenAI claims computes faster Fourier transforms](https://news.ycombinator.com/item?id=50065932); [Le Monde: OpenAI's latest blow to mathematicians: 'They have annihilated my field of research' (Oct 12; not read, automated access blocked)](https://www.lemonde.fr/en/science/article/2026/10/12/openai-s-latest-blow-to-mathematicians-they-have-annihilated-my-field-of-research_6758515_10.html); [IBT UK: Fields Medallist says OpenAI's 700 AI-generated maths papers have 'annihilated' his research field (Oct 13)](https://www.ibtimes.co.uk/openai-ai-generated-proofs-mathematicians-backlash-1825190)

### 2026-10-06: OpenAI model claims the quasi-Riemann hypothesis: no zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized)

*****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/

Family 003 of OpenAI's 6 Oct 2026 math release claims a zero-free half-plane Re s > 7/8 for the Riemann zeta function, every Dirichlet L-function and every finite-order Hecke L-function over ℚ(√−3). This is the 'quasi-Riemann hypothesis', open since Riemann (1859): before this, no fixed half-plane Re s > θ with θ < 1 was known to be free of zeros. A companion manuscript claims uniform exclusion of Landau–Siegel zeros. Both are formalized in Lean by OpenAI; experts have not yet confirmed them.

- Reactions: Alex Kontorovich, 'If a human did this, it would be an instant Fields Medal… They got a zero free strip!!!!' (~420k views); Levent Alpöge, 'Big, big, big, big props for quasiriemann and no Siegel zeroes'; Jared Duker Lichtman: the cubic family of L-functions 'appear to be essential for the proof'. An independent re-run of the Lean check (davegoldblatt/openai-zeta-proof-check, 7 Oct) reports acceptance by two kernels with only the standard axioms (~2,900 files, ~500k lines)
- Second independent two-kernel check: in Neal McBurnett's whole-library rebuild of openai/math (commit fd4aeeb2ee, all 416 Comparator configurations), `QuasiRiemannHypothesis` was accepted by Lean's kernel (replay ~6 min) and by nanoda (~81 s) on Oct 15 (nealmcb/openai-math-lean-tooling, results/kernel-pair-sweep.jsonl). This checks the Lean proof against the 169-byte Lean statement, not the paper
- Main manuscript: 'The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8' (dated 30 Sep 2026, 199 pages)
- Companion: 'The Quasi-Riemann Hypothesis (alternate 11/12 proof)' (5 Oct), whose write-up OpenAI says was 'human edited for readability'; the README lists the zeta zero-free-region work as an exception to its fixed automated procedure
- Companion: 'Uniform exclusion of Landau–Siegel zeros' (1 Oct): an absolute c > 0 such that every real zero β of every primitive nonprincipal real Dirichlet L-function of conductor q ≥ 3 satisfies (1 − β) log q ≥ c (the Lean scope note says no explicit value of c is given)
- Lean: Comparator challenge QuasiRiemannHypothesis states `riemannZeta s ≠ 0` for 7/8 < Re s (Mathlib's riemannZeta), with separate challenges for Dirichlet L-functions, Hecke L-functions and Siegel zeros; only the axioms propext, Quot.sound and Classical.choice are allowed; review status 'unchecked'
- Context: before the release, the best unconditional zero-free regions for ζ narrow towards Re s = 1 at large height (Vinogradov–Korobov type); the Riemann hypothesis itself (θ = 1/2) remains open
- Science magazine news story (Zack Savitsky, 15 Oct 2026, vol. 394, issue 6821, pp. 254–255), 'AI takes a bite out of math's hardest problem', with the standfirst 'A trove of AI papers includes progress on the Riemann hypothesis, a famous puzzle about prime numbers' (title and standfirst from Crossref metadata; the article body was not readable to us, HTTP 403)
- Follow-on (Oct 12): Cureton and Thorner, with AxiomProver generating the proofs and Lean code, extend the strategy to every Dedekind zeta function, zero-free for Re s > 31/32 (Hexagon 2610.00200; see 2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover)
- Hector Pasten (PUC Chile) to Scientific American (8 Oct), on the Siegel-zero consequence: 'This is not just progress on Siegel zeros—it settles the question. This is amazing.' (expert reaction, not a verification)
- Quanta (Konstantin Kakaes, 19 Oct 2026): the 199-page Sept 30 version (zeros between 1/8 and 7/8) was formally verified in Lean; the 49-page Oct 5 version (1/12 to 11/12) is not yet formally verified and is the one mathematicians study, because 'it's shorter' (Radziwiłł, per Quanta); it is the only paper of the release a person edited, and OpenAI won't say who ('The paper is the result of work from everyone at OpenAI'); 7–13-page write-ups by other mathematicians circulate privately
- Quanta quotes: Jared Duker Lichtman (Stanford), 'the most significant AI-for-math achievement so far, by quite a margin'; Peter Sarnak won't consider it 'a Theorem' until humans check all details and is repurposing a course to study it; Maksym Radziwiłł (NYU): 'In a sense, the vast majority of consequences of RH already follow from quasi-RH' but the papers 'are horribly written'; Alexander Dunn (Georgia Tech): 'The proof is a surprising synthesis of well-known ideas', and on the metaplectic-form family, 'Nobody expected this'; Will Sawin (Princeton) doubts the methods reach 1/2; Thomas Bloom: 'At this point nothing would surprise me'
- Quanta: a 2021 formula of Dunn and Radziwiłł is a key ingredient; Henryk Iwaniec narrowly missed this family of functions in 2002

#### What happened
Family 003 of OpenAI's release says it resolves the quasi-Riemann hypothesis. A 199-page manuscript proves that no
Dirichlet L-function, ζ included, vanishes in Re s > 7/8. An alternative, human-edited proof reaches 11/12, and a third
paper excludes Landau–Siegel zeros uniformly. The Lean library states the 7/8 result for Mathlib's `riemannZeta` and for
Dirichlet and Hecke L-functions, and states the Siegel-zero gap. The Lean scope note says the formal Siegel result "does not
rule out real zeros elsewhere in (0,1)"; it excludes them only in the window 1 − c/log q < β < 1.

### Follow-up: slightly better constants (8 Oct)
Baiying Liu ([arXiv 2610.12234](https://arxiv.org/abs/2610.12234)) follows "the same strategy" as OpenAI's paper and varies its two free parameters (b = 1/8, ℓ = 1/6 in the original). This moves the zero-free half-plane from Re s > 7/8 = 0.875 to Re s > 34999/40000 = 0.874975, and with algebraic parameters to Re s > (1507 − 2√921)/1653 ≈ 0.874957. The note says the result "has been formalized by Lean" and discloses: "OpenAI Codex (GPT 6.1 Sol Medium-Ultra) assisted with drafting, source checking, and the search for improved bounds. Anthropic Claude (Opus 5.5 Medium) assisted with auditing." It is a third-party paper that builds on the AI proof, and so depends on it being correct.

### Follow-ups of 10–13 Oct (arXiv)
- **Siegel zeros, explicit:** William D. Banks (Missouri) wrote out the Siegel-zero argument as a direct proof with explicit constants, (1 − β) log q ≥ 3.0 × 10⁻³ for all q ≥ 3, and says he "verified the mathematical arguments in their entirety" (Claude drafted the first report and expansion). Own entry: [2026-10-11-landau-siegel-zeros-banks-explicit-account](/e/2026-10-11-landau-siegel-zeros-banks-explicit-account/).
- **Goldbach exceptional set:** János Pintz and Ingo Althöfer ([arXiv 2610.15559](https://arxiv.org/abs/2610.15559)) prove E(X) ≪ X^{7/12+ε} for the number of even integers up to X that are not sums of two primes, assuming any common zero-free half-plane for Dirichlet L-functions, "The uniform 7/8 theorem reported in the OpenAI mathematics release supplies this input." Disclosure: ChatGPT proposed and developed the conditional exponent g(θ), checked algebra, expanded the circle-method argument and prepared the LaTeX; Pintz supplied the major-arc method.
- **Constant 0.8748867:** Hejian Sang ([arXiv 2610.15291](https://arxiv.org/abs/2610.15291)), assuming the analytic inputs of OpenAI's work, moves 7/8 to 8748867/10000000 for the same Hecke and Dirichlet families; generative AI was used "to read and understand the framework of [OAI26]" and to develop parts of the construction.
- **Applications:** Frei, Sofos and Widmer ([arXiv 2610.18491](https://arxiv.org/abs/2610.18491)) draw on "recent breakthroughs of OpenAI and Cureton–Thorner" to make the Ellenberg–Venkatesh ℓ-torsion bounds for class groups unconditional and prove an effective Chebotarev theorem conditional on quasi-RH for Dedekind zeta functions ("No artificial intelligence tools were used in this work").

### Follow-up of 14 Oct (arXiv)
- **Prime gaps, conditionally:** Matt Visser (Victoria University of Wellington), "Primes between 8th powers?" ([arXiv 2610.20058](https://arxiv.org/abs/2610.20058)), works out what the claimed 7/8 half-plane would imply: primes in (n⁸, (n+1)⁸), in (n⁸, n⁸ + n⁷) for every n ≥ 1, and in Hoheisel-style intervals (x, x + x^{7/8}). Everything is "contingent on the validity of the claimed 7/8 version of the QRH". He is openly cautious about the release: "Most of the 722 preprints are unreadable and would earn an automatic desk reject if submitted to a scientific journal", and he questions how reliable the autoformalization is.

### Follow-up of 21 Oct (arXiv)
- **Gauss sphere problem:** Chiara Bellotti and Bryce Kerr ([arXiv 2610.29636](https://arxiv.org/abs/2610.29636)) improve Heath-Brown's error term for lattice points in a 3D ball to O(R^{55/42+ε}) unconditionally, and "incorporating the recent 7/8-quasi-GRH result announced by OpenAI" to O(R^{2274811/1743872+ε}). One of the first applications of the OpenAI claim to a classical counting problem; the improved exponent is conditional on the claim. AI use: "The authors thank ChatGPT for assistance in editing and proof reading the manuscript."

#### Why it matters
If correct, this is the largest advance on the zeros of ζ since the 19th century, and it would remove the Siegel-zero
obstruction that runs through modern analytic number theory. Because the core statement is short and in Mathlib's language,
the Lean proof can be checked by anyone with enough compute. Experts will still need to read the argument, and to check that
the formal definitions match the classical ones.

Sources: [nanoda_lib issue 55: kernel failures on openai/math exports (fixed Oct 21)](https://github.com/ammkrn/nanoda_lib/issues/55); [Quanta: AI Proved the Quasi-Riemann Hypothesis. What Happens Now? (19 Oct 2026)](https://www.quantamagazine.org/ai-proved-the-quasi-riemann-hypothesis-what-happens-now-20261019/); [Banks: On the OpenAI proof that Landau-Siegel zeros do not exist (arXiv 2610.15483)](https://arxiv.org/abs/2610.15483); [Pintz & Althöfer: The Goldbach exceptional set, exponent 7/12 (arXiv 2610.15559)](https://arxiv.org/abs/2610.15559); [Liu, Slightly improved zero-free half-planes for the quasi-Riemann hypothesis (arXiv 2610.12234)](https://arxiv.org/abs/2610.12234); [Alex Kontorovich on X](https://x.com/AlexKontorovich/status/2107609087902941646); [Independent Lean re-check (davegoldblatt/openai-zeta-proof-check)](https://github.com/davegoldblatt/openai-zeta-proof-check); [Whole-library Lean + nanoda re-check incl. QuasiRiemannHypothesis (nealmcb/openai-math-lean-tooling)](https://github.com/nealmcb/openai-math-lean-tooling); [openai/math: CONTENTS.md, family 003](https://github.com/openai/math/blob/main/CONTENTS.md); [Paper: The Quasi-Riemann Hypothesis, Re s > 7/8](https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf); [Paper: Uniform exclusion of Landau–Siegel zeros](https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf); [Lean scope note for family 003](https://github.com/openai/math/blob/main/lean/docs/003.md); [Comparator challenge: QuasiRiemannHypothesis.lean](https://github.com/openai/math/blob/main/lean/ComparatorChallenges/QuasiRiemannHypothesis.lean); [Scientific American: The most exciting claims from OpenAI's heap of new proofs (8 Oct)](https://www.scientificamerican.com/article/the-most-exciting-claims-from-openais-heap-of-new-proofs/); [Science (news, Zack Savitsky, 15 Oct): AI takes a bite out of math's hardest problem](https://www.science.org/doi/10.1126/science.aen2355); [arXiv 2610.29636: Bellotti, Kerr, On the Gauss sphere problem (uses the 7/8 quasi-GRH)](https://arxiv.org/abs/2610.29636)

### 2026-10-06: OpenAI model claims a proof of Khot's Unique Games Conjecture, with Lean formalization

*****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-unique-games-conjecture-openai/

Family 102 of OpenAI's 6 Oct 2026 math release, 'The Unique Games Theorem', claims a deterministic polynomial-time reduction from 3SAT to Unique Games over a fixed alphabet, with completeness ≥ 1 − ε and soundness ≤ δ. That would prove Subhash Khot's 2002 conjecture. Companion papers give direct proofs that Max-Cut is NP-hard beyond the Goemans–Williamson ratio and Vertex Cover NP-hard below factor 2. OpenAI formalized them in Lean; the community has not yet verified them.

- Quanta (Ben Brubaker, 7 Oct): after rumours of an OpenAI proof reached Dor Minzer (MIT) on 11 Sep, he and graduate students Yumou Fei and Shuo Wang posted a 4-to-1 version of Khot's 2-to-1 games conjecture, 'with nearly identical implications', on ECCC on 14 Sep, with the disclaimer that the manuscript 'is complete mathematically, but it is not in the shape we wished to share in'; their breakthrough had come in April 2026 after 'five failures'. Minzer: 'There is a lot of value in failing and knowing why you failed. Using AI takes all of this out.' Quanta describes OpenAI's release as including 'a Lean-verified proof of the 2-to-1 conjecture' in a manuscript without human editing or independent review
- Reactions: Joe Bebel, 'a claimed proof and Lean formalization of the Unique Games Conjecture, which I have been personally working on for the last year and a half' (~99k views); Chris Peikert: 'Unique Games Conjecture, proved. L=RL, proved.'
- Paper: 'The Unique Games Theorem' (dated 23 Sep 2026, 58 pages): for every fixed ε, δ ∈ (0, 1/2), a deterministic polynomial-time reduction from 3SAT to Unique Games over a fixed finite alphabet
- Companions: optimal Max-Cut hardness (beyond α_GW ≈ 0.878), Vertex Cover hardness below factor 2, constant-factor hardness for Min-UnCut and directed feedback vertex set, 'using established PCP and Label Cover hardness results'
- Lean: the scope note formalizes the reduction from binary 3SAT to unweighted bipartite unique games with alphabet 𝔽₂^s, the Max-Cut gap, the Vertex Cover threshold, Min-UnCut and DFVS; 'No assumption that P≠NP is built into the statement'
- Context: the 2-to-2 Games theorem (Khot–Minzer–Safra, 2018) was the strongest prior evidence; UGC implies optimal inapproximability for many problems
- ECCC TR26-179 (Fei, Minzer & Wang, 'On the Hardness of 4-to-1 Games with Perfect Completeness'; revision 2 posted 8 Oct): proves Khot's 4-to-1 Games Conjecture (CCC 2002); stated consequences include NP-hardness of finding a proper k-colouring of a 3-colourable graph for every constant k, and of finding an independent set with a δ fraction of the vertices in a 2-colourable 3-uniform hypergraph
- The release also lists the 2-to-1 Games Conjecture with perfect completeness (family 105) and NP-hardness of colouring 3-colourable graphs with any constant number of colours (106)
- Scott Aaronson ('The Mathocalypse', 7 Oct) noted that the claim lands on the research of Dana Moshkovitz; The Decoder (8 Oct) quotes her text message about the proof: 'It feels like something written by someone who's on psychedelics.'

#### What happened
OpenAI's release includes "The Unique Games Theorem", which claims a reduction from 3SAT to Unique Games. If valid, it
proves the Unique Games Conjecture outright, with no reliance on unproven assumptions. A reasoning summary covers the related
result on NP-hardness at the basic semidefinite threshold. Lean statements for all the hardness results are published as
Comparator challenges.

#### Why it matters
UGC sits at the centre of hardness of approximation. A proof would make the Goemans–Williamson Max-Cut algorithm and the
factor-2 Vertex Cover algorithm provably optimal unless P = NP. A formal proof makes checking faster, but encoding
complexity-theoretic statements (machines, polynomial time, encodings) faithfully in Lean is subtle, and reviewers will need
to check that the formal statement matches the conjecture.

Sources: [Joe Bebel on X](https://x.com/joeintheory/status/2107612512288125096); [Chris Peikert on X](https://x.com/ChrisPeikert/status/2107617657415782731); [Paper: The Unique Games Theorem](https://github.com/openai/math/blob/main/preprints/The-Unique-Games-Theorem-September-23-2026/paper.pdf); [Lean scope note for family 102](https://github.com/openai/math/blob/main/lean/docs/102.md); [Comparator challenge: UniqueGamesTheorem.lean](https://github.com/openai/math/blob/main/lean/ComparatorChallenges/UniqueGamesTheorem.lean); [Reasoning summary: NP-hardness at the basic semidefinite threshold](https://github.com/openai/math/blob/main/reasoning_traces/basic-semidefinite-threshold-np-hardness.pdf); [Quanta: As AI Closed In on 'Unique Games' Proof, Researchers Raced to Beat the Machines](https://www.quantamagazine.org/as-ai-closed-in-on-unique-games-proof-researchers-raced-to-beat-the-machines-20261007/); [ECCC TR26-179: Fei, Minzer & Wang, On the Hardness of 4-to-1 Games with Perfect Completeness](https://eccc.weizmann.ac.il/report/2026/179/); [Raghu Meka (guest post on Tao's blog): The barriers of perception](https://terrytao.wordpress.com/2026/10/07/the-barriers-of-perception/); [Scott Aaronson: The Mathocalypse (7 Oct)](https://scottaaronson.blog/?p=10169); [The Decoder: Some mathematicians call for OpenAI boycott after AI-generated proofs flood their field (8 Oct)](https://the-decoder.com/some-mathematicians-call-for-openai-boycott-after-ai-generated-proofs-flood-their-field/)

### 2026-10-06: Chen, Tserunyan & Tucker-Drob solve the Jackson–Kechris–Louveau 'finite index over treeable' problem (2002); ChatGPT 6 Astra Ultra supplied a 'crucial observation'

****; University of Warsaw, McGill University, University of Florida, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-06-finite-index-over-treeable-jkl-problem/

On 6 Oct 2026 Ruiyuan Chen (Warsaw), Anush Tserunyan (McGill) and Robin Tucker-Drob (Florida) posted arXiv 2610.08986, dedicated to Alexander Kechris's 80th birthday. They prove that every countable Borel equivalence relation containing a finite-index Borel treeable subequivalence relation is Borel treeable, answering Jackson–Kechris–Louveau's Open Problem 6.4(B) (2002) in its strongest, purely Borel form. As a corollary, strong treeability of locally compact groups passes to finite-index overgroups (new even for countable discrete groups). Their "Declaration of AI use" says they consulted ChatGPT 6 Astra Ultra after conceiving the approach and that "GPT provided a crucial observation which ensures the coherence of the forest obtained at stage n + 1 with that obtained at stage n".

- Theorem 1.2: if F ⊆ E are countable Borel equivalence relations, F has finite index in E and F is Borel treeable, then E is Borel treeable. Previously open even for index 2 in the probability-measure-preserving setting (treeing only on a conull set)
- Method: a median-subalgebra filtration on products of trees, extending the median-graph treeing algorithm of Chen–Poulin–Tao–Tserunyan (2025) to well-behaved increasing unions; answers Question 1.1, how to 'canonically combine' finitely many trees on the same vertex set into one tree-like structure
- Corollary 1.3: a locally compact second countable group with a strongly treeable finite-index open subgroup is strongly treeable (also for measure and Borel strong treeability); for unimodular groups this implies fixed price
- AI disclosure: 'We conceived of the approach to proving Theorem 1.2 described above prior to consulting GPT.' GPT 'supplied the following observation that we missed: for a given Q_{n+1}-class D, the partial equivalence relation Q_n|D is invariantly separable by half-spaces in M_D', which in the final proof 'remains implicit but becomes essentially definitional' (Theorem 4.1(c))
- Writing: 'The article was entirely written by the three named authors'; ChatGPT gave editorial suggestions after the first draft
- No expert reactions found on X, HN or blogs as of 9 Oct; not in OpenAI's 6 Oct catalogue

#### What happened
Treeable equivalence relations play the role that free groups play among groups. Jackson, Kechris and Louveau asked in 2002
whether treeability survives finite-index extensions. They had shown that free actions of virtually free groups give
treeable relations, and the general question became a driver of geometric methods in the field.

Chen, Tserunyan and Tucker-Drob build the final tree in stages from a filtration of median algebras on a product of trees.
Their AI declaration is unusually precise: they had the staged approach and its index-2 setting, and ChatGPT 6 Astra Ultra
pointed out that each stage's relation is invariantly separable by half-spaces in the next stage's median graph. This was
the coherence property they needed, and after reformulation it became a definition.

#### Why it matters
It settles a named problem from a standard reference on countable Borel equivalence relations, with direct consequences in
measured group theory (fixed price, strong treeability). The disclosure is a clear example of the "AI supplies the missing
lemma" pattern in research-level mathematics.

Sources: [Chen, Tserunyan & Tucker-Drob: Finite index extensions of treeable equivalence relations are treeable (arXiv 2610.08986)](https://arxiv.org/abs/2610.08986)

### 2026-10-06: Woodin's question answered: GCH below a strongly compact cardinal implies GCH everywhere (Zhixing You, 4-page proof with GPT-6 Astra)

****; Hebrew University of Jerusalem, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-gch-above-strongly-compact-woodin-question/

On Oct 6, 2026 Zhixing You (Hebrew University of Jerusalem) posted arXiv 2610.07537, a four-page paper answering a long-standing question of Hugh Woodin: if κ is a strongly compact cardinal and the generalized continuum hypothesis holds below κ, then GCH holds everywhere. The analogous facts for supercompact cardinals, and Solovay's theorem for singular strong limits, were classical; Apter had shown the answer can be negative without the axiom of choice. The acknowledgement says the work "used AI assistance from GPT-6-Astra with the ultra reasoning setting, with GPT-6.1-Sol and GPT-6-Astra invoked dynamically". Unrefereed.

- Paper: 'The generalized continuum hypothesis above a strongly compact cardinal', Zhixing You, arXiv 2610.07537 (math.LO), 6 Oct 2026, 4 pages; preprint dated 5 Oct
- Theorem A: if κ is strongly compact and GCH holds below κ, then GCH holds (in ZFC). Remark 3.3: the proof does not use crit(j) = κ, so it also works for ω₁-strongly compact κ
- Background (paper): Solovay — above a strongly compact, 2^λ = λ⁺ for singular strong limit λ; Scott — measurable κ with GCH below gives 2^κ = κ⁺; supercompact κ with GCH below gives full GCH (Kanamori, Prop. 22.2). Woodin asked whether strong compactness suffices (Apter, 'On a problem of Woodin', Arch. Math. Logic 2000), and Apter gave a negative answer without choice
- AI statement (quote): 'This work used AI assistance from GPT-6-Astra with the ultra reasoning setting, with GPT-6.1-Sol and GPT-6-Astra invoked dynamically during the investigation. The author takes full responsibility for the paper.' The paper does not say which steps came from the models
- Author: Zhixing You, Einstein Institute of Mathematics, Hebrew University of Jerusalem (a set theorist with earlier work on strong compactness, e.g. arXiv 2302.13171)
- Not in OpenAI's 6 Oct github.com/openai/math catalogue (no strongly-compact or GCH family)
- Verification: unrefereed preprint; no expert reaction found as of 7 Oct 2026. A four-page answer to a 25-year-old large-cardinal question will need checking by set theorists

#### What happened
In ZFC, the continuum function λ ↦ 2^λ is almost unconstrained at regular cardinals (Easton's theorem), but large cardinals force regularity higher up. Solovay showed that above a strongly compact cardinal κ, 2^λ = λ⁺ at every singular strong limit λ. It was also known that if κ is supercompact and GCH holds below κ, then GCH holds everywhere. Hugh Woodin asked whether the weaker hypothesis of strong compactness is enough. In 2000 Arthur Apter showed that without the axiom of choice it is not.

On 6 October 2026 Zhixing You (Hebrew University of Jerusalem) posted a four-page paper answering Woodin's question positively in ZFC: if κ is strongly compact and GCH holds below κ, then GCH holds. The argument works with elementary embeddings witnessing λ-strong compactness and does not use that the critical point is κ, so it also applies to ω₁-strongly compact cardinals.

The acknowledgement reads: "This work used AI assistance from GPT-6-Astra with the ultra reasoning setting, with GPT-6.1-Sol and GPT-6-Astra invoked dynamically during the investigation. The author takes full responsibility for the paper." The paper does not say which ideas came from the models.

#### Why it matters
Woodin's question is a well-known open problem about large cardinals and the continuum function, and a short positive answer would be a notable result in set theory. It also adds mathematical logic to the fields where frontier-model help is now disclosed, after the Medvedev-logic undecidability result. A short proof of a long-open question is also exactly the kind of claim experts will want to check before accepting.

Sources: [arXiv 2610.07537: The generalized continuum hypothesis above a strongly compact cardinal](https://arxiv.org/abs/2610.07537); [Apter (2000): On a problem of Woodin, Archive for Mathematical Logic 39](https://doi.org/10.1007/s001530050146)

### 2026-10-06: OpenAI model claims the irrationality exponent of π is exactly 2 (Lean-formalized), which implies the Flint–Hills series converges

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-irrationality-exponent-pi-openai/

Family 017 of OpenAI's 6 Oct 2026 math release is "The irrationality exponent of π is 2" (manuscript dated 24 Sept 2026, 23 pages, author "OpenAI"). It claims that for every ν > 2 and all large enough q, |π − p/q| ≥ q^(−ν) for every integer p, so π is approximated by fractions no better than a typical number. The best previous bounds were 7.1032 (Zeilberger–Zudilin, 2019) and 7.1019 (Bai, Sept 2026). A corollary is that the Flint–Hills series Σ 1/(n³ sin² n) converges. OpenAI's Lean library formalizes the exponent statement against Mathlib's Real.pi. No public expert assessment was found by 12 Oct 2026.

- Catalogue text (openai/math CONTENTS.md, family 017): 'Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational p/q satisfies |π − p/q| ≥ q^(−2−ε). This also proves convergence of the Flint–Hills series Σ 1/(n³ sin² n), with angles in radians.'
- Paper: 'The irrationality exponent of π is 2', author 'OpenAI', dated 24 Sept 2026, 23 pages. Theorem 1.1 is ineffective: 'The threshold may depend on ν, and the argument does not give it effectively.' The paper stresses that exponent 2 does not mean bounded continued-fraction partial quotients
- History given in the paper: Mahler (1953) exponent 42; Mignotte 20; Hata 8.0160…; Salikhov 7.6063…; Zeilberger–Zudilin 7.103205334137…; Bai (Sept 2026) 7.101862832357. 'These quantitative bounds are context, not inputs to our proof.'
- Method (paper's outline): 'a multivariable interpolation theorem to construct a nonzero determinant', then two incompatible estimates for its absolute value; related to Laurent's interpolation determinants and to Roth's many-variable method, with an interpolation statement 'adapted to logarithmic curves'
- Flint–Hills: Alekseyev showed convergence requires μ(π) ≤ 5/2 and Meiburg that μ(π) < 5/2 suffices; Theorem 1.1 supplies this. More generally Σ 1/(n^a |sin n|^b) converges if and only if a > max{1, b} (Corollary 5.2)
- Earlier claims (paper): a convergence claim by Mantzakouras and López Zapata 'was withdrawn in September 2026'; Carella's claim of exponent two with bounded partial quotients 'contains a sign obstruction'. Neither is used
- Lean: the scope note says 'The formalization proves that the irrationality exponent of π is exactly two'; the Comparator challenge PiExponent.lean states it for Mathlib's Real.pi, including the supremum characterization. 'The paper's convergence consequence for the Flint–Hills series is outside this selected statement.'
- Status in OpenAI's own log: history.md (7 Oct 2026) lists three withdrawn manuscripts and 14 repaired ones; family 017 is in neither list
- Follow-up by an outsider (9 Oct): a pseudonymous 'Dan Brown' (GitHub jdb19937) published 'The logarithms of rational numbers have irrationality exponent 2', claiming μ(log α) = 2 for every positive rational α ≠ 1 (so μ(log 2) = 2), with a Lean proof described as 'a module-by-module modification of OpenAI's formalization of μ(π) = 2'. Hacker News, 22 points; the author's comment: 'This is a Lean-verified variation of the OpenAI proof for pi.' Not peer-reviewed; the author's identity is unconfirmed
- Not found as of 12 Oct 2026 (web and Hacker News search; X not searched): public assessments by specialists such as Wadim Zudilin, Frank Calegari or Terence Tao, or an independent re-run of this Lean proof
- Second follow-up, on arXiv: Jingwen Liu, Kai Jiang and Pingwen Zhang (first author at Wuhan University), 'Irrationality exponents of logarithms of positive rational numbers' (arXiv 2610.10192, dated 7 Oct 2026, v2 8 Oct), state μ(log r) = 2 for every positive rational r ≠ 1. 'The present proof adapts the separated-weight method of [1, Section 2]', where [1] is OpenAI's π manuscript; the link is the exponential map. The result is ineffective ('no effective value for it is obtained here'). The PDF has no AI-use statement. Preprint, not reviewed, and it rests on a method from an unrefereed manuscript

#### What happened
The irrationality exponent of a number measures how closely fractions can approximate it. Almost every real number has exponent 2,
and π was expected to, but proofs were far from it: the record stood at about 7.10 after decades of refinements of integral
constructions. Among the 722 manuscripts OpenAI published on 6 Oct 2026, family 017 claims the exact value 2 by a different route, an
interpolation-determinant argument in many variables. The result is ineffective, meaning it gives no explicit point after which
the inequality holds.

The claim settles a question that became popular outside research mathematics: whether the Flint–Hills series converges. That
depends on how often sin n is very small, which is the same as how well π is approximated by fractions.

OpenAI's Lean library contains a proof of the exponent statement, written against Mathlib's definition of π. Three days after the
release, a pseudonymous programmer published a modification of that Lean proof which, he says, gives exponent 2 for the logarithm
of every rational number, including log 2. If both hold, the technique is general.

#### Why it matters
Among the release's claims this is one of the easiest to state and one with a long, well-known history of small improvements,
including an AI-assisted one in September 2026. The Lean statement is short and uses only standard Mathlib definitions, so a
successful independent kernel check would leave little room for doubt about the theorem itself. As of 12 Oct no such check and
no specialist's assessment had been published, so the entry's status is pending.

Sources: [Paper: The irrationality exponent of π is 2 (openai/math, 24 Sept 2026)](https://github.com/openai/math/blob/main/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026/paper.pdf); [openai/math: CONTENTS.md, family 017](https://github.com/openai/math/blob/main/CONTENTS.md); [Lean scope note for family 017](https://github.com/openai/math/blob/main/lean/docs/017.md); [Comparator challenge: PiExponent.lean](https://github.com/openai/math/blob/main/lean/ComparatorChallenges/PiExponent.lean); [openai/math README (procedure and exceptions)](https://github.com/openai/math/blob/main/README.md); [openai/math history.md (withdrawals and fixes, 7 Oct 2026)](https://github.com/openai/math/blob/main/history.md); ['Dan Brown': The irrationality measure of the logarithm of a rational number is 2 (Lean)](https://jdb19937.github.io/log-irrationality-measure/); [Hacker News: The logarithms of rational numbers have irrationality exponent 2](https://news.ycombinator.com/item?id=50027884); [Liu, Jiang & Zhang: Irrationality exponents of logarithms of positive rational numbers (arXiv 2610.10192)](https://arxiv.org/abs/2610.10192)

### 2026-10-06: OpenAI's math release claims the optimal approximation for metric k-median: a polynomial-time (1 + 2/e + ε) algorithm matching the known hardness, with a Lean formalization

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e/

Among the 719 manuscripts OpenAI published on Oct 6, 2026 (github.com/openai/math), family 125, "The metric k-median approximation threshold and recovery" (two manuscripts dated Sept 24, 2026), claims a deterministic polynomial-time (1 + 2/e + ε)-approximation for finite metric k-median with specified candidate facilities, for every fixed ε > 0. Because approximating below 1 + 2/e is NP-hard, this would make 1 + 2/e ≈ 1.736 the exact threshold, where the best published human bound was about 2. OpenAI lists Lean comparator challenges stating the algorithm and the threshold. The EPFL-led better-than-2 paper of Oct 13 cites the claim as concurrent work.

- Source: OpenAI's math repository, CONTENTS.md family 125: 'Gives a deterministic polynomial-time (1+2/e+ε)-approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming P≠NP, the optimal infimum approximation factor is 1+2/e.'
- Manuscripts (both dated Sept 24, 2026): 'The approximation threshold for metric k-median' and 'Single-exponential recovery and bounded-price strictness for metric k-median'; the second also gives a randomized (2 − σ)-approximation for an absolute σ > 0
- Lean: lean/docs/125.md lists four comparator statements (MetricKMedian, KMedianThreshold, KMedianRecovery, KMedianRefinedRecovery); MetricKMedian.lean states the result with Lean's TM2ComputableInPolyTime (a polynomial-time Turing-machine algorithm), permitted axioms propext, Quot.sound and Classical.choice; the solution module OAI.Combinatorics.KMedian has about 1,300 files (repo tree, Oct 15)
- Context: k-median cannot be approximated below 1 + 2/e in polynomial time unless P = NP (Guha–Khuller / Jain–Mahdian–Saberi); before October 2026 the best guarantees were (2 + ε) (Cohen-Addad et al. 2025; Byrka et al. 2026), as summarised in arXiv 2610.18144
- Concurrent human work: arXiv 2610.18144 (EPFL, Google Research, CMU, NYU; Oct 13) reaches better than 2 and describes OpenAI's (1 + 2/e + ε) algorithm as extending Byrka et al.'s iterative rounding; a group around Jarosław Byrka and Shi Li independently reached 2 − σ
- Production: per the repo README, most results came from an unreleased internal OpenAI model, about three hours of ChatGPT Pro thinking compute per result, from about 4,000 posed problems
- Verification status: OpenAI's own Lean formalization; no referee report, expert blog post or independent build check found as of Oct 15

#### What happened
OpenAI's Oct 6, 2026 release of model-written mathematics ([github.com/openai/math](https://github.com/openai/math)) includes a
family of two k-median papers dated Sept 24. The headline claim is a deterministic polynomial-time algorithm that, for any fixed ε > 0,
finds k facilities whose total client distance is within (1 + 2/e + ε) of the optimum. Approximating k-median below 1 + 2/e is NP-hard,
so if correct this settles the problem's approximation ratio in the model with specified candidate facilities. A companion paper gives
a randomized (2 − σ)-approximation through a recovery argument.

OpenAI's Lean catalogue links four "comparator" statements for the family. The main one, `MetricKMedian.lean`, states the algorithm's
existence and polynomial running time formally in Lean (via Mathlib's Turing-machine complexity definitions). The proof module is large,
about 1,300 Lean files.

The result received little attention in the first week. It surfaced through the EPFL-led paper of Oct 13 (arXiv 2610.18144), whose own
human-proved algorithm breaks factor 2 and which mentions the OpenAI work as concurrent. No independent expert has yet published an
assessment of the OpenAI proofs or checked the formalization build.

#### Why it matters
Closing the gap between the 1 + 2/e lower bound and the factor-2 upper bound for k-median was a well-known open question in
approximation algorithms. A formalized, model-generated answer would be one of the strongest computer-science items in OpenAI's
release, but it has the same caveats as the rest: self-published, unrefereed, and checked so far only by OpenAI's own Lean work.

Sources: [OpenAI math repository: CONTENTS.md (family 125, metric k-median)](https://github.com/openai/math/blob/main/CONTENTS.md); [OpenAI math repository: Lean scope note for family 125](https://github.com/openai/math/blob/main/lean/docs/125.md); [Lean comparator statement: MetricKMedian.lean](https://github.com/openai/math/blob/main/lean/ComparatorChallenges/MetricKMedian.lean); [arXiv 2610.18144: A Better-than-2 Approximation for k-Median Part I (cites OpenAI's result as concurrent work)](https://arxiv.org/abs/2610.18144)

### 2026-10-06: OpenAI's maths release also claims the 3D Fourier restriction, Bochner–Riesz and local smoothing conjectures (families 077–080): the core of Stein's harmonic-analysis programme in three dimensions, manuscript only

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/

OpenAI's 6 Oct 2026 release of AI-written mathematics (github.com/openai/math) contains four harmonic-analysis families that claim central open problems in three dimensions. Family 077 claims Stein's Fourier restriction conjecture for the sphere in its bounded-data form (the extension operator maps L∞(S²) to Lᵖ(ℝ³) for every p > 3, 103 pages) and the diagonal Lᵖ → Lᵖ form for all compact positively curved surfaces (25 pages). Family 078 claims the 3D Bochner–Riesz conjecture (114 pages), 079 Sogge's local smoothing conjecture for the 3D wave equation (165 pages) and 080 the sharp Sobolev endpoint of Carleson's Schrödinger convergence problem (83 pages for the planar H^(1/3) case). The papers are dated 24 Sept and credited to "OpenAI". The catalogue gives none of them a Lean link, and we found no expert assessment. The first citation on arXiv is Korobenko and Sawyer (15 Oct), who call the diagonal theorem "claimed" and build a probabilistic analogue for the Schrödinger equation.

- Family 077 (catalogue): 'Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions … bounded from Lᵖ(Σ) to Lᵖ(ℝ³) for every p > 3.' Two papers dated 24 Sept 2026: 'Elliptic capacity propagation and Fourier restriction to the sphere' (103 pp.; 'the Fourier extension operator maps L∞(S²) boundedly into Lᵖ(ℝ³) for every p > 3. This is the full conjectured open range, and the threshold p = 3 is sharp') and 'Diagonal Fourier extension for positively curved surfaces in three dimensions' (25 pp.)
- Prior state of the art, as the sphere paper itself summarises it: Guth's polynomial partitioning gave the bounded-data estimate for p > 13/4, Hong Wang's broom method p > 42/13, and Wang–Wu the diagonal estimate for p > 22/7. The claim closes the remaining gap to p > 3
- Dependencies: the diagonal paper 'uses two theorem inputs': the packet propagation theorem of the sphere paper and 'The Kakeya maximal theorem [12, Theorem 1.1]', i.e. OpenAI's family 074 manuscript 'The Kakeya maximal conjecture in three dimensions' (23 Sept). The restriction claim therefore stands or falls with the Kakeya maximal claim, which Levent Alpöge linked to a scooping dispute
- Family 078: 'Resolves the three-dimensional Bochner–Riesz conjecture in its strict range' (paper 'Bochner–Riesz multipliers in three dimensions', 114 pp.; main estimate: Tδ bounded on L³(ℝ³) for every δ > 0, then interpolation and duality)
- Family 079: 'Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions' (paper 'Critical local smoothing for the three-dimensional wave equation', 165 pp.; critical L³ estimate with every positive Sobolev loss)
- Family 080: 'Proves almost-everywhere convergence e^{itΔ}f → f … for every f ∈ H^{n/(2(n+1))}(ℝⁿ) and every dimension n ≥ 2', the sharp Sobolev endpoint of Carleson's problem (planar paper 'Endpoint convergence for the planar Schrödinger equation', 83 pp.)
- Verification: the catalogue entries for 074, 077, 078, 079 and 080 carry no Lean link (073 Falconer and 075 L log L Fourier convergence do), lean/docs/077.md returns 404, and neither 077 PDF mentions Lean. Korobenko–Sawyer's remark that OpenAI proved the diagonal case 'with L∃∀N certification' is not supported by the repository as checked on 19 Oct
- First follow-up: Lyudmila Korobenko (Reed College) and Eric T. Sawyer (McMaster), 'Probabilistic Fourier extension and the Schrödinger equation', arXiv 2610.22634 (submitted 15 Oct, 8 pp.): 'Just recently, OpenAI has claimed to prove the diagonal inequality p = q in the case of d = 3 dimensions'; their averaged Sobolev regularity result for the Schrödinger equation is the probabilistic counterpart of 'the deterministic analogue in three dimensions … obtained recently by OpenAI'. The paper has no AI-use statement
- Coverage gap: our main release entry listed Kakeya maximal and Falconer but not these four families; a viral 7 Oct summary post had listed 'Fourier restriction conjecture — PARTIAL'. No press article, Hacker News thread or expert blog post on the restriction or Bochner–Riesz claims was found on 19 Oct

#### What happened
OpenAI's 6 Oct 2026 release (see the [main entry](/e/2026-10-06-openai-math-release-722-manuscripts/)) is so large that some of its
biggest claims went unnoticed here for two weeks. Families 077–080 sit in the analysis section of the catalogue, next to the Falconer
(073) and Kakeya (074) families, and claim most of the classical "Stein programme" in three dimensions:

- **Fourier restriction (077).** Elias Stein's restriction problem (late 1960s) asks for which exponents the Fourier extension
  operator of the sphere is bounded. In ℝ³ the conjecture is that L∞(S²) → Lᵖ(ℝ³) holds for every p > 3. The best human results, as
  OpenAI's own paper lists them, were p > 13/4 (Guth, polynomial partitioning), p > 42/13 (Hong Wang, "brooms") and the diagonal
  estimate for p > 22/7 (Wang and Wu). OpenAI's 103-page paper claims the full range, and a 25-page companion claims the diagonal
  Lᵖ(Σ) → Lᵖ(ℝ³) bound for every compact positively curved surface and every p > 3.
- **Bochner–Riesz (078).** A 114-page paper claims boundedness of the Bochner–Riesz means on L³(ℝ³) for every positive order,
  which by interpolation gives the conjectured range.
- **Local smoothing (079).** A 165-page paper claims Sogge's local smoothing conjecture for the wave equation in three space
  dimensions.
- **Carleson's Schrödinger problem (080).** The catalogue claims almost-everywhere convergence at the sharp Sobolev exponent
  n/(2(n+1)) in every dimension; the planar paper (83 pages) treats H^(1/3)(ℝ²).

The diagonal restriction paper says it depends on the sphere paper and on OpenAI's 3D Kakeya maximal theorem (family 074). These
four families have no Lean formalization in the repository, unlike the release's showcase results.

On 15 Oct Lyudmila Korobenko and Eric Sawyer posted the first arXiv paper we found that cites one of these claims. They prove a
probabilistic ("martingale transform") version of Fourier extension estimates for the Schrödinger equation, and describe OpenAI's
diagonal theorem as the deterministic analogue that has been "claimed". They do not check it.

#### Why it matters
If they hold, these papers settle, in dimension three, problems that have organised harmonic analysis since the 1970s, and that
Bourgain, Wolff, Tao, Guth, Wang and others advanced step by step. They are long, unformalized and unreviewed, and they rest on
the equally unreviewed Kakeya maximal manuscript. They are a natural next target for the kind of expert check that confirmed
OpenAI's Brennan proof. Readers should treat them as claims.

Sources: [OpenAI math catalogue (CONTENTS.md, families 074–080)](https://github.com/openai/math/blob/main/CONTENTS.md); [OpenAI: Elliptic capacity propagation and Fourier restriction to the sphere (24 Sept 2026, PDF)](https://github.com/openai/math/blob/main/preprints/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026/Elliptic-capacity-propagation-and-Fourier-restriction-to-the-sphere-September-24-2026.pdf); [OpenAI: Diagonal Fourier extension for positively curved surfaces in three dimensions (PDF)](https://github.com/openai/math/blob/main/preprints/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026/Diagonal-Fourier-extension-for-positively-curved-surfaces-in-three-dimensions-September-24-2026.pdf); [OpenAI: Bochner–Riesz multipliers in three dimensions (PDF)](https://github.com/openai/math/blob/main/preprints/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026/Bochner-Riesz-Multipliers-in-Three-Dimensions-September-24-2026.pdf); [OpenAI: Critical local smoothing for the three-dimensional wave equation (PDF)](https://github.com/openai/math/blob/main/preprints/Critical-local-smoothing-for-the-three-dimensional-wave-equation-September-24-2026/paper.pdf); [OpenAI: Endpoint convergence for the planar Schrödinger equation (PDF)](https://github.com/openai/math/blob/main/preprints/Endpoint-convergence-for-the-planar-Schrodinger-equation-September-24-2026/paper.pdf); [arXiv 2610.22634: Probabilistic Fourier extension and the Schrödinger equation (Korobenko, Sawyer)](https://arxiv.org/abs/2610.22634)

### 2026-10-06: doubleAI's 'Double Agent' claims explicit regular graphs of girth (8/5)·log n, beating the 4/3 constant of Margulis and Lubotzky–Phillips–Sarnak (1988)

***; doubleAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-doubleai-high-girth-regular-graphs-8-5/

On 6 Oct 2026 doubleAI, the Israeli company founded by Mobileye co-founder Amnon Shashua, published "d-Regular Graphs of Girth 8/5 log_{d−1} n", authored by its AI system "Double Agent". For every prime power q it constructs explicit (q³+1)-regular graphs with girth at least (8/5)·log_{q³} n − O(1), improving the constant 4/3 of Margulis and of Lubotzky, Phillips and Sarnak, the best known since 1988 (the Moore bound caps it at 2). The graphs are quotients of a Bruhat–Tits tree of a special unitary group in four variables. A Lean proof is posted; per press reports the company says the run was autonomous and cost under $2,000. No independent expert assessment yet.

- Paper: 'd-Regular Graphs of Girth 8/5 log_{d−1} n', Double Agent, doubleAI, October 2026, 40 pages (doubleai.com/papers/girth85.pdf); Lean proof as a zip (doubleai.com/papers/girth85-lean.zip); no arXiv version found
- Abstract: 'For every prime power q we construct an explicit infinite family of (q³+1)-regular graphs whose girth satisfies girth ≥ 8/5 log_{q³} n − O(1) … This improves the constant 4/3 of Margulis and of Lubotzky, Phillips and Sarnak, which has been the best known since 1988.' Degrees: 9, 28, 65, … (q = 2, 3, 4, …)
- Method: quotients of the Bruhat–Tits tree of SU in four variables of Witt index one, by congruence subgroups of an arithmetic group from a degree-4 division algebra with an involution of the second kind; the girth bound 'rests on a certificate' bounding det(w − I) for congruence elements
- For one explicit family of 28-regular graphs the paper also proves a matching upper bound: girth = (8/5)·log₂₇ n + O(1)
- Company claims (Calcalist, 8 Oct; Startup Fortune, 15 Oct): 'the first improvement in 38 years on a central problem in extremal graph theory'; produced autonomously, 'No human steering the math', for under $2,000 of compute. The paper lists no human authors and does not describe the process
- Startup Fortune notes that Lean checks logical consistency, not whether the theorem is stated as the field intends or whether the construction is new, and that the result is not peer-reviewed; Calcalist presents it as a company claim
- Reach: an HN submission of the PDF (7 Oct) drew 1 point; no expert reactions found as of 15 Oct

#### What happened
doubleAI, which so far had been known for its WarpSpeed GPU-kernel optimizer and the FormulaOne benchmark, posted a 40-page
paper whose only listed author is its AI system, "Double Agent". The paper builds explicit infinite families of regular
graphs whose shortest cycle (girth) grows like (8/5)·log n. The previous record for explicit constructions, 4/3, came from
the Ramanujan graphs of Margulis and of Lubotzky, Phillips and Sarnak in 1988. A Lean formalization is offered as a
download. The company told Calcalist that the system worked autonomously for under $2,000 of compute.

We found the result through a 15 Oct Startup Fortune article in the news sweep, nine days after the announcement. There was
no arXiv preprint, the HN post got no traction, and the company's page has no human authors to search for.

#### Why it matters
High-girth regular graphs are used in coding theory (LDPC codes) and in expander constructions, and the 4/3 constant was a
long-standing benchmark. If experts confirm the result, it is a solo AI result on a classic problem from a small company,
outside the big US labs. Until then it remains a company claim, formally checked by the company's own Lean files.

Sources: [doubleAI: Advancing a decades-old problem in extremal graph theory](https://www.doubleai.com/research/advancing-a-decades-old-problem-in-extremal-graph-theory); [Paper (PDF): d-Regular Graphs of Girth 8/5 log_{d−1} n](https://www.doubleai.com/papers/girth85.pdf); [Lean proof (zip)](https://www.doubleai.com/papers/girth85-lean.zip); [Calcalist Tech: OpenAI is solving decades-old math problems. An Israeli startup is taking a different route (8 Oct)](https://www.calcalistech.com/ctechnews/article/hswjujwr5); [Startup Fortune: An Israeli AI startup says it cracked a 38 year old graph theory bound for $2,000 (15 Oct)](https://startupfortune.com/an-israeli-ai-startup-says-it-cracked-a-38-year-old-graph-theory-bound-for-2000)

### 2026-10-06: Walter Trump's 1979 packing of 11 unit squares proved optimal, with a full Lean verification (AI involvement reported, unconfirmed)

***; Open-source community; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-eleven-square-packing-optimality-lean/

On Oct 6, 2026 the GitHub project 11SquaresFormalized reported a complete Lean 4 verification (7,920 modules, zero admissions) that Walter Trump's 1979 packing of eleven unit squares in a square is optimal, with side s(11) ≈ 3.87708359. It formalizes a computer-assisted certificate proof posted on Sept 29. Startup Fortune reported that OpenAI's Astra and Anthropic's Claude were credited as contributors, but the repositories' acknowledgements reviewed here do not mention AI models.

- Result: the minimum side of a square holding 11 unit squares is T = (6u+4)/(1+2u−u²) ≈ 3.8770835900228, where u is the root in (9/25, 37/100) of 5u⁸−10u⁷−2u⁶+14u⁵+12u⁴−6u³+2u²+2u−1 = 0; rotations and boundary contact allowed
- Underlying proof (repo 11SquaresOptimal, Sept 29, 2026, by GitHub user Queuingtheorydotcom): a center cover reduces any smaller packing to 2,184 cell patterns; exact certificates exclude 2,180; symmetry reduces the last four to one case, which is closed by a branch cover and an isolation argument. The README says it is 'not a completed Lean/formal proof' and not peer-reviewed
- Formalization (11SquaresFormalized, verification report dated 2026-10-06): 7,920 Lean modules accepted, 0 sorry/admissions, 2,234 audited theorem targets, status OPTIMALITY_PROVED_WITH_NATIVE_CERTIFICATES; Lean 4.34.1 + pinned Mathlib
- Trust model: numerical certificates use native_decide, so the theorem trusts Lean's kernel and native compiler, 'not a kernel-only verification claim'
- Credits in the repo: EvolvingPrograms and @ctjlewis (verification runner), @wand125 (n11-optimality-lean formalization), Benjamin Gurevitch, @Julian-JJ, @Guzhou0806, Queuingtheorydotcom (the proof); Walter Trump for the packing
- AI role: Startup Fortune (Oct 7) says the work credits 'OpenAI's Astra and Anthropic's Claude' as direct contributors. The README, ACKNOWLEDGEMENTS and PROVENANCE files and recent commit messages checked on 2026-10-07 do not mention AI models. Treat the AI claim as unverified
- History: Trump found the tilted arrangement in 1979; the problem was popularized by Martin Gardner and Erich Friedman's 'Squares in Squares' pages, and became an internet meme through xkcd #2740 ('Square Packing', 2023)

#### What happened
How tightly can eleven unit squares be packed into a square? Walter Trump's 1979 answer uses a tilted cluster and had never been proved
best. On Sept 29, 2026 a GitHub user posted a computer-assisted certificate proof. Within a week, several contributors turned it into a full
Lean 4 development, which passed a complete verification run on Oct 6. The proof relies on many numerical certificates checked with
`native_decide`, so it trusts Lean's compiled code as well as its kernel.

#### Why it matters
It settles a well-known open case in packing theory, widely known from xkcd. If the reported AI credit is confirmed, it would be an example
of models from two rival labs helping to formalize a decades-old result. That needs a primary source (an X post or the contributors' own
statement) before it can be relied on.

Sources: [GitHub: Queuingtheorydotcom/11SquaresFormalized (Lean formalization)](https://github.com/Queuingtheorydotcom/11SquaresFormalized); [Verification report 2026-10-06](https://github.com/Queuingtheorydotcom/11SquaresFormalized/blob/main/docs/VERIFICATION_20261006.md); [GitHub: Queuingtheorydotcom/11SquaresOptimal (computer-assisted proof, Sept 29)](https://github.com/Queuingtheorydotcom/11SquaresOptimal); [GitHub: wand125/n11-optimality-lean](https://github.com/wand125/n11-optimality-lean); [Startup Fortune: AI models formally proved Walter Trump's 1979 square packing is optimal](https://startupfortune.com/ai-models-formally-proved-walter-trumps-1979-square-packing-is-optimal/); [Erich Friedman: Squares in Squares](https://kingbird.myphotos.cc/packing/squares_in_squares.html)

### 2026-10-06: Explicit tensors of border rank 3n − o(n), past the 'linear flattening barrier'; proofs obtained with GPT-6 Astra (Benjamin Lovitz)

***; Concordia University, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-06-explicit-tensors-border-rank-3n-gpt-6-astra/

On 6 Oct 2026 Benjamin Lovitz (Concordia University) posted arXiv 2610.08504. It constructs explicit n × n × n tensors of border rank at least 3n − o(n), improving the previous record of (2 + ε)n by Landsberg and Michałek, and proves that linear flattenings, the standard lower-bound method, cannot show more than 2n − 1 (the earlier barrier was 6n − 4). The paper's "Use of artificial intelligence" section says the author "used GPT-6 Astra interactively throughout every stage of preparing this manuscript, including to obtain proofs of the main results, which the author subsequently verified and simplified". Unrefereed preprint.

- Paper: 'Explicit tensors beyond the linear flattening barrier', Benjamin Lovitz (Concordia University, Montreal), arXiv 2610.08504 [cs.DS], 19 pages, 6 Oct 2026
- Theorem 1.1: 'There is a polynomial-time explicit tensor T_n ∈ (C^n)^⊗3' with border rank at least 3n − 11n^(3/4). Previous record: (2 + ε)n for any 0 < ε < 1/42 and large n (Landsberg–Michałek)
- Barrier (Theorem 1.2, per the abstract): 'the linear flattening method cannot be used to establish lower bounds on border rank beyond 2n − 1', improving the 6n − 4 barrier of Garg et al. and Buczyński; for odd n Koszul flattenings reach 2n − 1, for even n they reach 2n − 2, 'leaving an open gap of size one'
- Context (paper): the maximum border rank of an n × n × n tensor is about n²/3, so explicit lower bounds remain linear; border rank of the matrix-multiplication tensor governs the exponent ω
- AI disclosure: 'The author used GPT-6 Astra interactively throughout every stage of preparing this manuscript, including to obtain proofs of the main results, which the author subsequently verified and simplified. The author takes full responsibility for the correctness, exposition, and attribution in this manuscript.' The author thanks Harm Derksen for feedback on a draft
- Not found as of 12 Oct 2026: expert reactions or press coverage

#### What happened
Border rank measures how hard a tensor is to compute approximately, and for the matrix-multiplication tensor it decides how fast
matrices can be multiplied. Proving that a specific, explicitly written tensor has high border rank is a standard hard problem of
complexity theory. The best bounds sat near 2n, obtained with "flattenings" (turning the tensor into a matrix and
measuring its rank).

Lovitz's paper does two things. It proves that the linear flattening method cannot go beyond 2n − 1, a much lower ceiling than
the 6n − 4 known before. And it builds explicit tensors with border rank about 3n by another route, so these tensors are provably
out of reach of flattenings. The author states that the proofs of the main results were obtained interactively with GPT-6 Astra
and then checked and simplified by him.

#### Why it matters
It moves a record in a core area of algebraic complexity and explains why the old method stalled.
It is another example of a specialist using a frontier model as the main source of proofs.

Sources: [arXiv 2610.08504: Explicit tensors beyond the linear flattening barrier](https://arxiv.org/abs/2610.08504)

### 2026-10-06: Why SAT solvers fail on multiplication: first exponential resolution lower bound for multiplier associativity; 'main ideas and the core proof … are due to GPT-5.6-Sol' (Liew)

***; OpenAI, Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-multiplier-associativity-hard-for-resolution-gpt-5-6-sol/

On 6 Oct 2026 Vincent Liew posted arXiv 2610.08988, "Associativity of Multiplication Is Hard for Resolution". It shows that general resolution proofs of the associativity of n-bit multiplication need size 2^Ω((n/log n)^(1/4)), for a broad class of multiplier encodings that includes the array and Wallace-tree multipliers used by SMT solvers. The abstract calls it the first lower bound of this kind and says it "resolves an open problem of Beame and Liew". The paper's "AI Methodology" section states: "The main ideas and the core proof of the associativity lower bound are due to GPT-5.6-Sol." Claude Fable 5 helped rewrite the proofs and Claude Fable 5.1 generated a Lean 4 formalization of the main theorems. Unrefereed preprint.

- Paper: 'Associativity of Multiplication Is Hard for Resolution', arXiv 2610.08988 [cs.CC], 6 Oct 2026, 33 pages; Vincent Liew
- Result: resolution proofs of associativity of n-bit multiplication 'require size 2^Ω((n/log n)^(1/4))'; the bound 'holds for a broad class of multiplier encodings based on partial product summation, including the standard array and Wallace-tree multipliers used to bit-blast multiplication in SMT solvers'
- Motivation (abstract): 'SAT solvers are empirically known to perform poorly when reasoning about multiplication. Yet for over a decade we have lacked a theoretical explanation for this phenomenon. CDCL SAT solvers implicitly search for resolution proofs, and no lower bound on proof size has ruled out the existence of short proofs that solvers simply fail to find.'
- Method: a reduction from a perfect-matching principle on bounded-degree bipartite expander graphs, which Itsykson, Slabodkin and Sokolov proved hard for resolution, to multiplier associativity
- Extension: combined with Håstad's recent lower bound for the perfect-matching principle of the odd grid, the same reduction 'yields an exponential lower bound for multiplier associativity in the stronger bounded-depth Frege proof systems'
- AI Methodology: 'The main ideas and the core proof of the associativity lower bound are due to GPT-5.6-Sol. This materially affected Sections 5, 7, 8, and 9, and Appendix A. During the verification process, the author reformulated and rewrote the definitions, lemma statements, and proofs with the assistance of Claude Fable 5, and added attributions along the way.'
- Also: 'The Lean formalization (Appendix B) was generated by Claude Fable 5.1.' and 'The author verified the correctness and originality of all content, including references, and takes full responsibility for the paper's content.'
- Caveat on the Lean proof (paper): the formalization includes its own definitions of the circuits; 'While we have inspected these definitions ourselves, we caution that they are a potential source of error.'
- Lean 4 code: github.com/dysfunctor/associativity

#### What happened
Modern SAT solvers (CDCL solvers) can be seen as searching for proofs in a simple system called resolution. They are fast on many
hardware-verification tasks but notoriously slow on anything involving multiplication, such as checking that two multiplier
circuits compute the same function or that (xy)z = x(yz) for n-bit numbers. It was not known whether this is a weakness of the
solvers' search or a real limit, meaning that short resolution proofs do not exist at all.

Liew's preprint proves the second for associativity: any resolution proof has size exponential in roughly the fourth root of n.
The proof embeds a perfect-matching principle on expander graphs, already known to be hard for resolution, into the associativity
statement. With a recent result of Håstad the argument extends to bounded-depth Frege systems, which are stronger than resolution.

The paper says that the main ideas and the core proof come from GPT-5.6 Sol, that the author checked and rewrote them with
Claude Fable 5, and that Claude Fable 5.1 produced a Lean 4 formalization of the main theorems.

#### Why it matters
It gives the first theoretical explanation for an effect that practitioners in verification have worked around for more than a
decade, on a question posed by specialists in proof complexity. It is also a clear instance of the 2026 pattern in theoretical
computer science: the model supplies the reduction, a second model formalizes it, and the human author's work is verification
and exposition.

#### Not verified here
The Lean development was not run for this entry. The paper gives no affiliation for the author and none is stated here. No expert
reaction or Hacker News discussion was found on 13 Oct 2026.

Sources: [arXiv 2610.08988: Associativity of Multiplication Is Hard for Resolution](https://arxiv.org/abs/2610.08988); [Lean 4 formalization (dysfunctor/associativity)](https://github.com/dysfunctor/associativity)

### 2026-10-06: Optimal bound for the polynomial Littlewood–Offord problem, 'discovered autonomously by GPT-6 Pro' (exposition by Grebennikov)

***; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-polynomial-littlewood-offord-gpt-6-pro/

On Oct 6, 2026 Alexandr Grebennikov posted arXiv 2610.08708, an exposition of a proof he says "was discovered autonomously by GPT-6 Pro". It shows that a degree-d multilinear polynomial with r disjoint top-degree monomials vanishes on random ±1 inputs with probability O_d(r^{-1/2}). This is optimal, removes the polylog factor in the Meka–Nguyen–Vu bound, and resolves a conjecture attributed to H. Nguyen and Vu. Its key lemma bounds the total influence of bounded-degree rational functions, resolving a conjecture of Kothari, Kovacs-Deak, Wang and Yang. The ChatGPT log is public.

- Paper: 'Optimal bound for the polynomial Littlewood–Offord problem', Alexandr Grebennikov, arXiv 2610.08708 (math.CO), 6 Oct 2026, 8 pages
- Result: P[F(ξ₁,…,ξₙ) = 0] = O_d(r^{-1/2}) for Rademacher ξᵢ, where F is degree-d multilinear with r degree-d monomials on disjoint variable sets; previously (log r)^{O_d(1)} r^{-1/2} (Meka, O. Nguyen, Vu); known before only for d = 1 (Erdős), d = 2 (Kwan–Sauermann) and bounded Chow rank (Grebennikov–Kwan)
- AI statement (quote): 'The proof was discovered autonomously by GPT-6 Pro (the chat logs can be found here), and the author does not claim any credit for the proof ideas. The argument was subsequently verified, digested, and rewritten from scratch by the author.'
- Chat log: chatgpt.com/share/6ab4f17d-7a3c-83eb-a4e6-a99920bd9384 (linked from the PDF)
- A footnote notes that a similar linear-algebra statement appears implicitly in the AI-assisted Chvátal-conjecture proof (Chang, H. Liu, M. Liu) and its follow-ups
- Not part of OpenAI's 6 Oct github.com/openai/math release

#### What happened
The Littlewood–Offord problem asks how likely a random ±1 combination is to hit a fixed value. Erdős settled the linear case, and the polynomial version has been studied since Costello, Tao and Vu's work on random symmetric matrices. Hoi Nguyen and Van Vu conjectured that a degree-d multilinear polynomial with many nonzero top-degree coefficients vanishes with probability O(m^{-1/2}). The best general bound, by Meka, Oanh Nguyen and Vu, was off by a polylogarithmic factor.

On 6 October 2026 Alexandr Grebennikov, who had earlier worked on the bounded-Chow-rank case with Matthew Kwan, posted an eight-page exposition of an argument that gives the optimal O_d(r^{-1/2}) bound. The key step is an estimate for the total influence of functions p/q with p and q of bounded degree, which also resolves a recent conjecture of Kothari, Kovacs-Deak, Wang and Yang. The author states that the proof "was discovered autonomously by GPT-6 Pro … and the author does not claim any credit for the proof ideas". He then verified it and rewrote it from scratch, and he links the full ChatGPT conversation.

#### Why it matters
It is another case where a consumer-tier model (ChatGPT's GPT-6 Pro) is credited with the whole idea behind an optimal bound in a well-studied area, with a public transcript. The same day, OpenAI's internal-model release dominated attention.

Sources: [arXiv 2610.08708: Optimal bound for the polynomial Littlewood–Offord problem](https://arxiv.org/abs/2610.08708); [ChatGPT shared conversation (GPT-6 Pro chat log)](https://chatgpt.com/share/6ab4f17d-7a3c-83eb-a4e6-a99920bd9384)

### 2026-10-06: Preprint claims a 4.268 upper bound for the random 3-SAT threshold, down from 4.4898, with a Lean 4 proof; 'large language models drove the research'

***; Independent; Science & math; confidence medium; https://postcutoff.com/e/2026-10-06-random-3-sat-threshold-upper-bound-4-268-llm/

Fedor Vorobyev's arXiv preprint 2610.10599 (Oct 6, 2026) claims that a random 3-SAT formula with 4.268n clauses is unsatisfiable with high probability. The best earlier rigorous upper bound was 4.4898, and the cavity method of statistical physics predicts a threshold near 4.267. The paper says "Large language models drove the research leading to this paper" and that the proof is formalized in Lean 4. It has no independent expert review yet.

- Claim (abstract): 'a random 3-SAT formula with n variables and ⌊4.268n⌋ clauses is unsatisfiable with high probability'
- Earlier state, as the abstract gives it: 'successive upper bounds reached 4.4898 in the work of Díaz, Kirousis, Mitsche, and Pérez-Giménez, while the cavity method predicts a value near 4.267'
- Method (abstract): builds on interpolation methods, 'including the energetic approach of Achlioptas and Menchaca-Méndez'; bounds the smallest number of clauses any assignment must violate by comparison with a simpler system of constraints on single variables; ends in a finite calculation checked in exact rational arithmetic
- AI disclosure (Section 9, verbatim): 'Large language models drove the research leading to this paper. Their contribution included developing the mathematical argument, generating and revising the Lean formalization and numerical code, and drafting and revising the manuscript. The author directed the work and worked with the models to improve the explanations and presentation.' The models are not named
- Verification statement (verbatim): 'Lean checks the proof of the formal theorem; the correspondence between the formal definitions and the intended model, and between the formal proof and this exposition, requires human inspection. We do not claim an independent human verification of every step.'
- 16 pages, single author; Lean sources and certificate code at github.com/yamelton/sat-upper-4268 (Apache-2.0), archived on Zenodo (doi 10.5281/zenodo.23120101)
- Reactions: none found. A Hacker News search on Oct 13 returned no thread; X was not searched in this quick run

#### What happened
In random 3-SAT, each clause picks three of n variables at random. As the ratio of clauses to variables grows, formulas go
from almost always satisfiable to almost always unsatisfiable. Physicists' cavity method puts the switch near 4.267, but
proofs lag: the rigorous upper bound stood at 4.4898.

Fedor Vorobyev's preprint claims 4.268, a hair above the predicted value. The argument compares the random formula with a
simpler system by interpolation and reduces the bound to a finite computation with rational numbers. The author states that
language models developed the mathematics, wrote the Lean 4 formalization and drafted the paper, and that he directed them.

#### Why it matters
If it holds, this closes almost all of the gap between what is proved and what physics predicts for one of the best-known
problems on random constraint satisfaction. It is also a test of a pattern that is becoming common: a single author, unnamed
models, and a Lean proof in place of peer review.

The caution is the paper's own. Lean certifies the formal theorem, and a human has to confirm that the formal statement is
the real random 3-SAT model. Nobody outside has done that in public yet. The author's background was not checked, and the
models used are not named. Treat the result as a claim until specialists respond.

Sources: [Vorobyev: An upper bound of 4.268 for the random 3-SAT satisfiability threshold (arXiv 2610.10599)](https://arxiv.org/abs/2610.10599); [Lean 4 formalization and certificate code (GitHub)](https://github.com/yamelton/sat-upper-4268)

### 2026-10-07: Outside researchers using AI crowdsource a tighter exponent in OpenAI's integer-multiplication result #109, from 2^-182 to about 2^-8.4 (conditional)

****; Independent researchers, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/

Within two days of OpenAI's 722-manuscript math release, outsiders using AI coding agents raised the exponent saving κ in OpenAI's family #109, "Integer multiplication below n log n" (T(n) = O(n (log n)^(1−κ)), originally κ = 2^-182), by a factor of about 2^168. Doug Colkitt (@0xdoug, with OpenAI Codex) reached 2^-59 on 7 Oct and 2^-34 early on 8 Oct; his repository then became a crowdsourced effort with more than 80 pull requests in a day and reached a reviewed κ ≈ 5.10×10^-5 (> 2^-15). In a parallel repository Swapnil Jain (@SJ_Swapnil_Jain, with Claude) posted seven rounds ending at κ ≈ 6.40×10^-5 (> 2^-14) on the evening of 8 Oct. The community repo kept climbing: its reviewed value passed 6.6×10^-4 on 9 Oct and reached κ ≈ 1.518×10^-3 (about 2^-9.36) on 12 Oct, roughly 30 times the 8 Oct figure, using constructions built on the E8 lattice and the [8,4,4] Hamming code; the organizers then said they had hit a wall and asked geometers for new structures. Everything is conditional on OpenAI's unverified manuscript; Lean checks cover only finite certificates and arithmetic, and the gains are asymptotic, not practical speedups. On 23 Oct the wall broke: a new reviewed record κ ≈ 0.00302 (about 2^-8.4, +80% over the 13 Oct value), built on a helper-register idea and independently verified.

- Upstream: OpenAI family #109, 'Integer multiplication below n log n' (manuscript dated 23 Sep 2026, released 6 Oct), claims T(n) = O(n (log n)^(1−κ)) with κ = 2^-182 on a fixed multitape Turing machine; Colkitt pins it at openai/math commit adc7f12. #109 was not among the three manuscripts OpenAI withdrew on 7 Oct
- Colkitt (repo CrocSwap/integer-mult-bounds, git history, UTC): 7 Oct 02:22 κ = 5.8×10^-33; 13:35–13:48 κ = 2^-78 (direct nonadjacent axis routing); 19:27 κ = 2^-59 (paired-block circuits sharing intermediate sums); 8 Oct 00:27 κ = 83/10^12 > 2^-34 (compact control bits instead of whole windows); 01:10 κ = 2^-31 (complex-network compression); 13:10 2^-30 checkpoint (ternary construction)
- Community phase (same repo, 8 Oct): Colkitt merged Rohan Arun's PR #39, κ = 971668963/(2.5×10^13) ≈ 3.887×10^-5 > 2^-15 (release 'community-kappa-15', audit 15:12 UTC; X post 15:30 UTC). By 19:03 UTC the reviewed witness was κ = 25508460085039/(5×10^17) ≈ 5.1017×10^-5 (PRs #57/#61/#62 by eumemic, Alejandro Zarzuelo Urdiales and Avi Eisenberg). The repo had 88 pull requests by 21:12 UTC; open, unreviewed PRs claimed up to κ ≈ 5.28×10^-5 (#74)
- Swapnil Jain (repo Swapnil-jain/integer-mult-kappa, commits in UTC, 8 Oct): 07:17 κ = 249/(5×10^10) ≈ 4.98×10^-9; 07:29 κ = 7499/10^12 ≈ 7.5×10^-9 (≈ 2^-27, announced in his X 'second update' at 08:15; we did not find his first X update); 10:03 κ ≈ 4.19×10^-6 > 2^-18; 11:10 κ ≈ 1.197×10^-5 > 2^-17; 12:38 κ ≈ 1.548×10^-5 > 2^-16; 14:01 κ ≈ 3.667×10^-5 > 2^-15; 20:37 κ = 1599247689723/(2.5×10^16) ≈ 6.397×10^-5 > 2^-14 (X post 20:56 UTC)
- Current best claimed (8 Oct, 21:00 UTC): Jain's κ ≈ 6.397×10^-5 > 2^-14, self-published and not reviewed by anyone else; best reviewed in Colkitt's community repo: κ ≈ 5.102×10^-5 > 2^-15
- The two series are not fully independent: Jain's README says it 'builds on OpenAI's Integer multiplication below n log n and on Douglas Colkitt's integer-mult-bounds' and uses results from open PRs there (#7, #24, #36, #41, #44, #62), cited as 'prior work, not collaborators'. Colkitt's README and release notes credit 'Aurel Prosz (Paureel) and Swapnil Jain' for 'attributed two-stage development and paid copied-stream endpoints'
- What Lean checks: Jain's lean/Round5–7.lean (core Lean 4, `decide`) verify that the child-width histograms sum to the total rank, the moment bounds and the κ assembly inequalities. The README: 'The analytic facts behind those bounds … are stated premises, not formalised'; round seven's stage two 'is assumed, not machine-checked'; 'Full upstream multiplication theorem: Assumed'; 'Independent review / formalisation: Not supplied'. The histograms come from Python generators. In Colkitt's repo, Lean packages (Ryan S's PR #26 for the 83/10^12 arithmetic; PR #61's scoped certificate) 'verify their stated finite/arithmetic contracts' and 'do not formalize the whole multiplication algorithm'
- AI disclosures, verbatim. Colkitt: 'The research, implementation, and drafting were performed with assistance from OpenAI Codex.' Jain: 'Research, implementation and drafting were done with assistance from Claude (Anthropic).' (no model version named). The community NOTICE file credits contributors' use of GPT-6 Astra, Codex, GPT-6.1-Sol Ultra, Claude and Claude Opus 5.5
- X reach (fxtwitter, 8 Oct ~21:00 UTC): Colkitt 2^-59 post ~695k views, 2^-34 post ~249k, 2^-15 community post ~81k; Jain: round three ~135k, round four ~117k, round six ~102k, round five ~66k, 'second update' ~63k, round seven ~760 (an hour after posting). Aurel Prosz's live tracker beyond-n-log-n.netlify.app ~23k
- Reaction: little outside X. Hacker News had only a 3-point 'They just merged my PR' story (8 Oct); no press or named expert assessment found by 8 Oct evening
- 9–12 Oct, community repo (README checkpoints and commit times, UTC): on 9 Oct the reviewed value moved through 7.79×10^-5, 9.26×10^-5, 3.146×10^-4, 4.609×10^-4 and 4.72×10^-4 to the 'entrance-bank' checkpoint κ ≈ 6.619×10^-4 (published 12:57); 10 Oct 14:08: 'gen4 circuits and common-delivery frames', κ ≈ 7.236×10^-4 (DreamingOfClouds' PR #276 with Rohan Arun's #279); 12 Oct 01:55: κ = 1517760977820331/10^18 ≈ 1.518×10^-3 ≈ 2^-9.36 (icekylinx's #531 on #523, with eumemic's #493), '2.10 times the previously selected exponent saving'
- The 12 Oct README: 'a conditional finite witness: the all-size compiler, common-label/mutable-basis realization, normalized residual and hosting interfaces, restored rows, precision/tape and analytic transfer remain assumptions. It is not a practical runtime ratio or a complete Lean proof'; the same batch adds DreamingOfClouds' Lean proofs of two lemmas (G and A)
- Scale on 12 Oct (GitHub API): PR numbers above 530, 195 stars, 85 forks, 26 accounts in the contributor list (one is the 'claude' account); CONTRIBUTORS.md says AI-assistance disclosures are retained per contribution
- Rohan Arun (12 Oct, X): his open PR #527 claims κ ≈ 1.467×10^-3, capped by the 'complex supplier'; 'we hit a hard wall again and progressed only 0.7% since last night'. He asks experts in lattices, coding theory and finite geometry for 'a close relative of E8 or the [8,4,4] code', about 100–120 well-separated directions in 8 to 16 dimensions; Barnes–Wall lattices and Reed–Muller codes are the current ideas, and the Leech lattice was 'tested and ruled out as a drop-in'. He calls the effort 'decentralized auto-research'
- Jain's parallel repo (description on 11 Oct): 'a conditional witness kappa = 6685733/10^10 (> 2^-11)', i.e. ≈ 6.69×10^-4, below the community repo's reviewed value
- Jain's series after round seven (his repository's commits, UTC, and README): 9 Oct 04:13 round eight, κ ≈ 1.2613×10^-4 > 2^-13; 05:22 round nine, κ ≈ 4.6637×10^-4 > 2^-12 (through the cover assembly of community PR #144); 09:38 round ten, κ ≈ 6.1534×10^-4 > 2^-11; 13:01 round eleven, κ = 6685733/10^10 ≈ 6.6857×10^-4 ≈ 2^-10.547, 'about 89200 times our first witness'. No further round by 12 Oct. On 11 Oct he linked a paper, 'Ceilings and Impossibility Results for Finite Witnesses of Sub-n log n Integer Multiplication' (v1 on Zenodo, corrected v2 on ResearchGate), and the README now lists routes that do not improve κ (docs/CLOSED_IDEAS.md). Still self-published and conditional; later rounds reuse modules from community PRs (#124, #143, #144, #162, #166, #168, #184) with credit
- 13 Oct, community repo (README and GitHub API, read 13 Oct ~20:00 UTC): the reviewed value passed through κ = 0.001584931…, 0.001602278… (Jeremy Kothe's #588 with eumemic's #543), 0.001635183… (#617) and 0.001635527… (Chafik Boukhalfa's #626) to κ = 836330098538459/(5×10^17) = 0.001672660197076918, about 2^-9.22, from Rohan Arun's PR #1094 ('rare-pattern T16 exchanges', merged 13 Oct 18:47 UTC). The README calls it 'about 0.02823% more exponent saving' than the previous #1042 checkpoint, so the curve has flattened: +10% over the 12 Oct value in about 41 hours. The same conditions remain ('All-size/compiler, common-label/mutable-source realization, whole-call hosting, precision/tape, exact recovery and analytic obligations remain conditional'). The repo now directs AI agents to an autoresearch.md task guide ('Donating inference or asking an agent to contribute?'); 247 stars, PR numbers above 1,090
- First arXiv paper on the #109 κ race (2610.14263, posted 9 Oct, cs.CC): Volkan, Zerrin and Dağhan Dağlı, 'Bipartite Carrier Compression and Machine-Certified Bounds in Schönhage Trilinear Multiplication Tensors', claim κ ≈ 4.131×10^-5 by removing 185 carrier roles at h = 23 and h = 25, 'strictly surpassing the preceding record on this architecture (κ ≈ 4.123864×10^-5)', with the wire-reduction identity checked in Lean 4. That is far below the community repo's value, which on 15 Oct (README read ~20:20 UTC) was still κ = 0.001672660197076918, while the repo kept merging (PR numbers past #2370). The paper has no AI-use statement
- 17 Oct (GitHub API and README, read ~17:30 UTC): the README's selected, reviewed record is still κ = 0.001672660… (Rohan Arun's #1094, 13 Oct). Higher values are conditional on PRs still under review: #3148 (merged 17 Oct, 'κ ≈ 0.001729777', a packing-cap construction), and Rohan Arun's open #3184 (κ = 0.001730189…) and #3190 (κ = 0.001733427…, which equals the 'bit-bound' ceiling another PR (#2423) reports for the selected construction). #3184 is the first proposal Rohan Arun credits to a free model, OpenCode's 'Space Bunny Max' (an anonymous model community analysis links to MiniMax M3.1, unconfirmed), with Claude Opus 5.5 helping verify it; he says Opus kept improving on Space Bunny's idea, that models get stuck in 'local minima' so switching models injects fresh ideas, and that 'the barrier of entry for frontier auto-research just dropped to 0'. Progress had slowed: about +3.5% over the reviewed record in four days, after a 30-fold rise in the first week
- 19 Oct (GitHub API, read ~09:00 UTC): the free-model claim did not hold. Rohan Arun wrote on 18 Oct that 'most of the free OpenCode model's ideas didn't pass verification' and that 'frontier models are much better at verifying their own work first'; #3184 (the Space Bunny Max proposal) was closed without merging. He now runs GPT-6 Astra (high) coordinating a fleet of agents with context-sharing and 'forgetting' strategies against 'local minima' (open-sourced in rohanarun/Auto-research-agent), and on 19 Oct opened #3934–#3959 with conditional κ up to 0.001736738… (#3959). The README's reviewed record is still κ = 0.001672660… (#1094, 13 Oct); a maintainer queue pass on 18 Oct merged ~196 PRs into the research archive without changing the selected result
- 19 Oct: OpenCode's stealth model 'Space Bunny' was revealed as MiniMax M3.1 Flash ($0.15/$0.60 per 1M tokens, weights due 23 Oct), see the MiniMax entry
- 23 Oct (GitHub API and README, read ~21:00 UTC): the wall broke. On 22 Oct zxv's idea (helper registers that end holding a finished transform of their own input count as output instead of being undone) took the conditional value to ~0.0023, and 1337isnot1337's #4470 (merged 22 Oct) to 0.00302015. On 23 Oct the maintainer admitted Cao Yifan (intqwq)'s #4610, independently verified in Chris R (Clear1nsight)'s #5241, as the new selected reviewed record: κ = 15108879/(5×10^9) = 0.0030217758 > 2^-9 (about 2^-8.37), +80% over the 13 Oct record 0.001672660 (#1094). All eleven stages of the independent admission replay passed; the same five conditional assumptions remain. Rohan Arun (Oct 23, ~3.8k views) says the team sees several paths to a conditional κ = 0.01 and a possible 0.006, which need a rebuilt construction and days of verification (unverified)
- 24 Oct (README, certificates/selected-result.json and GitHub API, read ~morning UTC): new selected reviewed record κ = 605457647/(2×10^11) = 0.003027288235 from Chris R (Clear1nsight)'s #5413 (opened 02:07, merged 02:54 UTC; promoted 03:30 UTC), completing Dillon Amburgey's #5402 coupled provider with Ali Romman's #5398 controller; provider audit #5407 and independent companion #5414. About 0.18% above #4610; same five conditional assumptions; the README says it is not a complete Lean proof or independent human verification of the theorem

#### What happened
OpenAI's 6 Oct release (see 2026-10-06-openai-math-release-722-manuscripts) included family #109, a manuscript claiming that
two n-bit integers can be multiplied in time O(n (log n)^(1−κ)) with κ = 2^-182. That would beat the O(n log n) bound of
Harvey and van der Hoeven (2019), widely believed to be optimal, though only by a tiny power of log n.

Doug Colkitt (X @0xdoug; X bio: founder of SLX, founding contributor to Fogo, former quant) called integer multiplication
"probably the most shocking result" of the release at 00:35 UTC on 7 Oct. Under two hours later he pushed
github.com/CrocSwap/integer-mult-bounds, "Research draft by Douglas Colkitt — conditional on the underlying manuscript". It
keeps OpenAI's files unchanged under `upstream/` and ships each improvement as a patch, with Python certificates for the
arithmetic:
- **02:22 UTC**: tuned parameters and a smaller finite network (h = 46) give κ = 5.8×10^-33, "a nearly 2^75 fold increase". He also proved a ceiling of κ < 5.838×10^-33 for that network family.
- **13:35–13:48 UTC**: direct nonadjacent axis swaps, which the manuscript already allowed, route around the cubic bottleneck behind that ceiling: κ = 2^-78.
- **19:27 UTC**: paired-block circuits that share intermediate sums and scratch space, an exact early-stopping guard width and a smaller Gaussian width: κ = 2^-59 (minimum margin about 0.4%). The README calls this "a 2^123-fold increase in exponent saving over the original".

The README states the limits plainly: the upstream theorem "remains an assumption", the scripts "do not constitute a formal
proof of the complete algorithm", and there is no Lean formalization. Disclosure, verbatim: "The research, implementation, and
drafting were performed with assistance from OpenAI Codex. This assistance is not independent review or endorsement by OpenAI.
No priority claim is made."

On X, @bubbleboi's quote post (~25k views) said: "This guy has 6.1 Astra running in a loop and is breaking the record for
integer multiplication algorithms every few hours lmaooooo." The model attribution is bubbleboi's claim; the repository credits
OpenAI Codex and names no model. We found no press or Hacker News coverage by the evening of 7 Oct.

### 8 October: Colkitt's second day and the community repo
At 00:27 UTC Colkitt published κ = 83/10^12 > 2^-34, "a roughly 48 million fold improvement over the previous result",
by "moving compact control bits instead of entire windows" ([X](https://x.com/0xdoug/status/2107994110644588859), ~249k
views), and at 01:10 UTC a 2^-31 bound from compressed complex networks. Others joined overnight: icekylinx, Zhihao Chen,
RaD, eumemic, Rohan Arun and others opened pull requests; Pranav Balakrishnan posted an independent κ > 2^-31 from his
own repository; Ryan S (GitHub princezuda) added Lean checks of the 2^-34 certificate arithmetic (PR #26); and X user
@platypii started a project to formalize the results in Lean ("A lot of ground work needed"). At 12:23 UTC Colkitt wrote that it had "taken off in terms of now having a real community
effort" and asked contributors to open PRs so their work could be counted. At 15:30 UTC he "validated and merged
Rohan's PR", κ > 2^-15: "This result was a true community effort that combined techniques from a number of contributors"
([X](https://x.com/0xdoug/status/2108218367962124423), ~81k). Rohan Arun's own post said the improvements "started
getting faster than we could run the verifications" and that he "Mostly used Codex here but Claude collaborating on the
other side might have helped". By 19:03 UTC the repository's reviewed witness was κ ≈ 5.1017×10^-5, combining
Avi Eisenberg's interval strips (#62), eumemic's joint frame compiler (#57) and Alejandro Zarzuelo Urdiales's exact
refinement with a scoped Lean certificate (#61). The README credits about 20 contributors, and its NOTICE file records
each one's AI assistance (Codex, GPT-6 Astra, GPT-6.1-Sol Ultra, Claude, Claude Opus 5.5). Aurel Prosz built a live
tracker (beyond-n-log-n.netlify.app). Open PRs that evening claimed up to κ ≈ 5.28×10^-5, not yet reviewed.

### Swapnil Jain's parallel series
Swapnil Jain (X @SJ_Swapnil_Jain; bio "Backend & AI Engineer", founder of First Impress Lab, MS Georgia Tech; ~970
followers) ran his own repository, github.com/Swapnil-jain/integer-mult-kappa, "Conditional research draft by Swapnil
Jain", which "builds on OpenAI's Integer multiplication below n log n and on Douglas Colkitt's integer-mult-bounds". He
posted numbered updates on X through 8 Oct: κ ≈ 2^-27 at 08:15 UTC ("building on Doug Colkitt's results"), > 2^-18 at
10:05 ("we have now entered 2^-1x territory", ~135k views), > 2^-17 at 11:10, > 2^-16 at 12:39 ("The assembly and moment
certificates are now also checked in Lean's kernel"), > 2^-15 at 14:03 (κ ≈ 3.667×10^-5, "Both interchange certificates
and the full assembly are checked in Lean's kernel", ~102k) and > 2^-14 at 20:56 (κ ≈ 6.397×10^-5, "a 2¹⁶⁸ fold
improvement over the original OAI result"). We did not find his first X update (the 4.98×10^-9 witness committed at 07:17
UTC). His rounds combine his own constructions (two-stage interchange with a common flag basis, copied centres, retained
point totals, deferred readouts) with results from open PRs on Colkitt's repository, which he takes as given. Colkitt's
repository in turn credits Jain, with Aurel Prosz, for two-stage development, so the two efforts feed each other.

Jain's disclosure, verbatim: "Research, implementation and drafting were done with assistance from Claude (Anthropic).
AI assistance is not independent review." His README is explicit about scope. The Lean files check, with `decide` in core
Lean 4, that the child-width histograms sum to the total rank, that the moment bounds hold and that the κ assembly
inequalities hold. "The analytic facts behind those bounds … are stated premises, not formalised"; in round seven, stage
two "is assumed, not machine-checked"; the full upstream multiplication theorem is "Assumed"; and "Independent review /
formalisation: Not supplied". So "checked in Lean's kernel" means the finite certificates and arithmetic, not the
algorithm or OpenAI's theorem.

### Reaction
Outside X the effort drew little attention by the evening of 8 Oct: one 3-point Hacker News story by a contributor, and no
press or named expert assessment that we found. The model attributions in posts (e.g. @bubbleboi's "6.1 Astra running in
a loop") are third-party claims unless a repository's disclosure confirms them.

#### Why it matters
It is an early example of what the release unlocked: outsiders with AI coding agents building on AI-written manuscripts
within hours, and within a day a crowdsourced, AI-assisted "citizen science" effort with dozens of contributors and its
own review process. The improvements are only as good as OpenAI's unverified theorem, and a smaller κ changes nothing in practice.
Still, finding a roughly 2^168-fold larger exponent saving in two days suggests the original constants were far from tight, and that
AI agents can now do this kind of parameter and construction search quickly.

Sources: [@RohanArun on X: most of the free model's ideas failed verification; Astra-coordinated agent fleet (18 Oct)](https://x.com/RohanArun/status/2111940317662363937); [PR #3959: Conditional κ = 0.001736738… (Rohan Arun, open, 19 Oct)](https://github.com/CrocSwap/integer-mult-bounds/pull/3959); [@RohanArun on X: a free model (OpenCode Space Bunny) proposes new conditional κ bounds (17 Oct)](https://x.com/RohanArun/status/2111493670318809303); [PR #3190: Conditional κ = 0.001733427… (Rohan Arun, open)](https://github.com/CrocSwap/integer-mult-bounds/pull/3190); [Community repo README, 12 Oct checkpoint (κ ≈ 1.518×10^-3)](https://github.com/CrocSwap/integer-mult-bounds); [PR #527 (Rohan Arun): κ ≈ 1.467×10^-3, rank-sector factory](https://github.com/CrocSwap/integer-mult-bounds/pull/527); [Experiment log of tested ideas](https://github.com/CrocSwap/integer-mult-bounds/tree/main/research/experiment-log); [@RohanArun on X: hit a wall, call for geometry experts (12 Oct)](https://x.com/RohanArun/status/2109629408076083414); [@RohanArun on X: new conditional κ = 0.001467 (12 Oct)](https://x.com/RohanArun/status/2109466530219999637); [GitHub: CrocSwap/integer-mult-bounds (Colkitt, community repo)](https://github.com/CrocSwap/integer-mult-bounds); [Community release notes: κ > 2^-15 (PR #39)](https://github.com/CrocSwap/integer-mult-bounds/blob/main/docs/releases/community-kappa-15.md); [Community round-two review and contribution ledger](https://github.com/CrocSwap/integer-mult-bounds/blob/main/docs/research/community-round2-review.md); [Proof note for the 2^-59 bound (PDF)](https://github.com/CrocSwap/integer-mult-bounds/blob/main/artifacts/paired-note.pdf); [GitHub: Swapnil-jain/integer-mult-kappa (Jain)](https://github.com/Swapnil-jain/integer-mult-kappa); [OpenAI family #109: Integer multiplication below n log n (pinned commit)](https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integer-multiplication-below-n-log-n-September-23-2026); [@0xdoug on X: κ = 5.8×10^-33 (first result)](https://x.com/0xdoug/status/2107658319632433524); [@0xdoug on X: κ = 2^-78](https://x.com/0xdoug/status/2107832360527913445); [@0xdoug on X: κ = 2^-59](https://x.com/0xdoug/status/2107917690568327225); [@0xdoug on X: κ = 2^-34 (8 Oct)](https://x.com/0xdoug/status/2107994110644588859); [@0xdoug on X: 'a real community effort'](https://x.com/0xdoug/status/2108171317321125959); [@0xdoug on X: merged Rohan Arun's PR, κ = 2^-15](https://x.com/0xdoug/status/2108218367962124423); [@SJ_Swapnil_Jain on X: second update, κ = 2^-27](https://x.com/SJ_Swapnil_Jain/status/2108109123774796281); [@SJ_Swapnil_Jain on X: third update, κ > 2^-18](https://x.com/SJ_Swapnil_Jain/status/2108136796886548867); [@SJ_Swapnil_Jain on X: sixth update, κ > 2^-15, Lean kernel checks](https://x.com/SJ_Swapnil_Jain/status/2108196538568851574); [@SJ_Swapnil_Jain on X: seventh update, κ > 2^-14](https://x.com/SJ_Swapnil_Jain/status/2108300632738386419); [@RohanArun on X: 'Mostly used Codex here but Claude collaborating…'](https://x.com/RohanArun/status/2108190499941065193); [@aurel_pr on X: live tracker of the community effort](https://x.com/aurel_pr/status/2108214135179944096); [@bubbleboi on X: '6.1 Astra running in a loop'](https://x.com/bubbleboi/status/2107934679294935317); [HN: 'They just merged my PR for the bound on the integer multiplication problem'](https://news.ycombinator.com/item?id=50007425); [README: current reviewed result #4610, κ = 0.0030217758 (23 Oct)](https://github.com/CrocSwap/integer-mult-bounds/blob/main/README.md); [PR #5413: complete coupled-output caller, conditional κ = 0.003027288235 (Clear1nsight, 24 Oct)](https://github.com/CrocSwap/integer-mult-bounds/pull/5413); [Selected reviewed record (certificates/selected-result.json)](https://github.com/CrocSwap/integer-mult-bounds/blob/main/certificates/selected-result.json); [PR #4610: Conditional κ = 0.0030217758, joint helper interfaces and Lean compiler tightening (Cao Yifan / intqwq)](https://github.com/CrocSwap/integer-mult-bounds/pull/4610); [PR #5241: independent verification of #4610 (Clear1nsight)](https://github.com/CrocSwap/integer-mult-bounds/pull/5241); [@RohanArun on X: 'We broke the wall', conditional κ = 0.00302 (+80%) (22 Oct, ~96k views)](https://x.com/RohanArun/status/2113091328913404342); [@RohanArun on X: paths to a conditional κ = 0.01 (23 Oct)](https://x.com/RohanArun/status/2113677165866766619); [PR #1094: Conditional kappa 0.001672660197076918, rare-pattern T16 exchanges (Rohan Arun, merged 13 Oct)](https://github.com/CrocSwap/integer-mult-bounds/pull/1094); [arXiv 2610.14263: Dağlı, Dağlı & Dağlı, Bipartite Carrier Compression and Machine-Certified Bounds in Schönhage Trilinear Multiplication Tensors](https://arxiv.org/abs/2610.14263)

### 2026-10-07: Exact overlaps conjecture for self-similar measures on the line proved: GPT-6 Astra found the first proof (Kittle & Kogler), 47,000-line Lean formalization; OpenAI's catalogue claims it independently

****; University College London, Institute for Advanced Study, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-07-exact-overlaps-conjecture-gpt-6-astra/

On 7 Oct 2026 Samuel Kittle (UCL) and Constantin Kogler (IAS) posted arXiv 2610.10511, a proof of the exact overlaps conjecture for self-similar measures on the real line and of its generalized form, dim ν = min{1, h_μ/|χ_μ|}. Their "Machine Contribution Statement" says GPT-6 Astra found the first proof on 27 Sep 2026 after about two and a half hours, following a two-hour warm-up in which it found new, simpler proofs of Hochman's 2014 and Varjú's 2019 theorems. GPT-6 Astra also formalized everything in Lean (about 47,000 lines, 9.5 hours, four agents). OpenAI's 6 Oct catalogue independently claims the same theorem as family 148, dated 24 Sep. The proof is an unrefereed preprint with a public Lean formalization.

- Theorem 1.1: for any probability measure μ on finitely many contracting similarities of ℝ, the self-similar measure ν has dim ν = min{1, h_μ/|χ_μ|} (h_μ random-walk entropy, χ_μ Lyapunov exponent). This implies the exact overlaps conjecture: a dimension drop below min{1, H(p)/|χ|} happens only with exact overlaps
- History (paper's introduction): a version for self-similar sets goes back to Simon (1996); Hochman (Annals 2014) proved it under exponential separation, hence for algebraic parameters; Varjú (2019) did transcendental Bernoulli convolutions; Rapaport (2022) algebraic contractions; Rapaport–Varjú (2024) and others special three-map systems
- Key new ingredient (Theorem 1.3): an entropy inequality quantifying the information lost when independent random variables are added, in terms of variance averaged across scales
- Machine Contribution Statement: 'On September 27, 2026, GPT-6 Astra found the first proof of Theorem 1.1 presented to the second-named author (C.K.)'. 'Motivated by a suggestion of Elon Lindenstrauss, GPT-6 Astra was first asked to propose new approaches to simpler proofs of the main results of [Hoc14] and [Var19b], which was successfully achieved after around two hours. Encouraged by the model having apparently rethought the area in two hours, the exact overlaps conjecture was tried and the model succeeded after around two and a half hours.'
- Division of labour: Kittle found a new deduction of Theorem 1.1 from Theorem 1.3 via Hochman's theorem and the variance summation method; 'the introduction and Section 3 are due to the authors, while Section 2 is due to GPT-6 Astra and was rewritten by the authors'; the authors 'take full responsibility for the correctness'
- Lean: 'GPT-6 Astra formalized all results from this paper and all necessary results from previous work including [Hoc14, Theorem 1.4]. The formalization is around 47,000 lines of code and took around 9.5 hours with four agents working in parallel' (github.com/ckkogler/exact-overlaps-one-dim-lean)
- Supplementary repo (all documents 'written by GPT-6 Astra'): the conversation transcript, self-contained AI proofs of Hochman's Theorem 1.1 (8 pp.) and Varjú's Theorem 3 (10 pp.), a 16-page first proof, an extension to arbitrary dimension when the rotations generate a virtually solvable group, and a sharpened Theorem 1.3
- Priority: OpenAI's catalogue (family 148, 'The entropy-rate dimension formula for self-similar measures on the line', dated 24 Sep, with a Lean scope note) claims the same theorem; the authors note OpenAI dates its proof 24 Sep, theirs 27 Sep, and that Theorem 1.3 is a quantitative strengthening of OpenAI's Lemma 3.2
- Kogler previously co-authored the GPT-6 Astra proof of the Odlyzko–Poonen conjecture (2026-09-22-odlyzko-poonen-conjecture-proved)

#### What happened
Self-similar measures are the natural measures on fractals built from finitely many contracting maps x ↦ r_i x + b_i. The
exact overlaps conjecture predicts that such a measure has the "expected" dimension, min{1, entropy/Lyapunov exponent},
unless two different compositions of maps coincide exactly. Michael Hochman's 2014 Annals paper proved it under an exponential
separation condition, which covers all algebraic parameters. The transcendental case stayed open apart from special
families (Varjú's Bernoulli convolutions, Rapaport's algebraic contractions, Rapaport–Varjú's three-map systems).

On 7 Oct 2026 Samuel Kittle (UCL) and Constantin Kogler (IAS) posted a proof of the full one-dimensional conjecture, and of
the generalized formula with random-walk entropy. Their machine contribution statement describes how it was found. Following
a suggestion from Elon Lindenstrauss, Kogler first asked GPT-6 Astra for new, simpler proofs of Hochman's and Varjú's
theorems, which it produced in about two hours. "Encouraged by the model having apparently rethought the area in two hours,"
they then posed the conjecture itself, and the model had a proof after about two and a half hours on 27 Sep. Kittle found a
different route from the key entropy inequality to the main theorem, and the authors rewrote the paper. Section 2 is
GPT-6 Astra's argument, rewritten by the authors. GPT-6 Astra then formalized the whole paper and the required parts of
Hochman's work in Lean: about 47,000 lines in 9.5 hours with four agents in parallel. The authors published the conversation
transcript and the model's original drafts.

OpenAI's 6 Oct release had already listed the same theorem as family 148, with a manuscript dated 24 Sep and a Lean scope
note. Kittle and Kogler state both dates and describe their Theorem 1.3 as a quantitative strengthening of OpenAI's key lemma.

#### Why it matters
This is a central problem in fractal geometry, and the record of how it fell is unusually complete: the transcript, the
model's drafts, a Lean proof and a clear statement of which sections are machine work. It is also one of the first cases of
two independent AI systems settling the same well-known conjecture within days of each other.

Sources: [Kittle & Kogler: The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line (arXiv 2610.10511)](https://arxiv.org/abs/2610.10511); [Lean formalization (ckkogler/exact-overlaps-one-dim-lean)](https://github.com/ckkogler/exact-overlaps-one-dim-lean); [Supplementary material incl. GPT-6 Astra transcript (ckkogler/kk26-supplementary-material)](https://github.com/ckkogler/kk26-supplementary-material); [OpenAI math catalogue (CONTENTS.md, family 148)](https://github.com/openai/math/blob/main/CONTENTS.md); [Hochman (2014): On self-similar sets with overlaps and inverse theorems for entropy (Annals)](https://doi.org/10.4007/annals.2014.180.2.7)

### 2026-10-07: Two claimed proofs of the LeBrun–Salamon conjecture: Hu, Liu & Wan (arXiv, ChatGPT as 'auxiliary tool') and OpenAI's catalogue (family 062)

****; Chongqing University of Technology, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-07-lebrun-salamon-conjecture-proof/

On 7 Oct 2026 Zhengyu Hu, Kefeng Liu and Xueyuan Wan (Mathematical Sciences Research Center, Chongqing University of Technology) posted arXiv 2610.10410, claiming a proof of the LeBrun–Salamon conjecture: every compact connected positive quaternion-Kähler manifold of real dimension 4n, n ≥ 2, is a Wolf space after rescaling. Their AI declaration says only that "the authors used ChatGPT as an auxiliary tool". They state that their argument was "obtained well before" OpenAI's 6 Oct catalogue, which also claims the conjecture (family 062, manuscript dated 23 Sep) by a different route. Both proofs are unrefereed, and neither has a Lean formalization.

- Conjecture: LeBrun & Salamon, 'Strong rigidity of positive quaternionic–Kähler manifolds' (Invent. Math. 118, 1994); equivalent, via twistor spaces, to the homogeneity of contact Fano manifolds (every contact Fano manifold is the adjoint variety of a simple Lie group). Previously known in low dimensions and under large-symmetry assumptions (Buczyński–Wiśniewski–Weber; Occhetta–Romano–Solá Conde–Wiśniewski)
- Hu–Liu–Wan method: transverse rational curves (stable domains of contact degree two) on the twistor space, an intrinsic two-point tensor extended across the nodal boundary, an algebraic additive action from a nilpotent endomorphism, first-jet interpolation, then Hwang–Mok, Beauville's theorem and LeBrun's uniqueness theorem
- AI declaration (verbatim): 'The authors used ChatGPT as an auxiliary tool in this work. The authors verified and completed all mathematical arguments and take full responsibility for the content of this paper.'
- Priority remark (verbatim): 'This project was developed over several months, and its main result and core argument were obtained well before the appearance of the OpenAI manuscript [21]. Our proof was developed independently and follows a different approach.' They describe OpenAI's proof as smoothing a pair of contact lines into very free conics and using a diagonal-jet construction
- OpenAI family 062, 'Projective contact classification and the LeBrun–Salamon conjecture' (manuscript 'Contact Fano manifolds and the LeBrun–Salamon conjecture', 23 Sep 2026): also claims contact-Fano homogeneity and a classification of projective contact manifolds of dimension ≥ 3 (b₂ = 1: adjoint varieties; b₂ ≥ 2: ℙ(T*Z)). CONTENTS.md shows no Lean link for this family

#### What happened
Positive quaternion-Kähler manifolds are Einstein manifolds with holonomy in Sp(n)Sp(1). LeBrun and Salamon conjectured in
1994 that the only compact ones are the symmetric Wolf spaces, such as quaternionic projective space and Gr₂(ℂⁿ⁺²). Through
the twistor construction the problem is equivalent to showing that every contact Fano manifold is homogeneous, an adjoint
variety. Earlier work settled low dimensions and cases with large torus symmetry.

On 7 Oct 2026 Hu, Liu and Wan posted a proof in all dimensions. They study rational curves on the twistor space that are
transverse to the contact distribution, build an additive group action from an intrinsic tensor, and then invoke Hwang–Mok,
Beauville and LeBrun to identify the twistor space and the metric. The paper declares ChatGPT use as an "auxiliary tool"
without details. It also includes a priority remark about OpenAI's catalogue, released the day before, which claims the same
conjecture in family 062 by a different argument.

#### Why it matters
If either proof holds, it closes a well-known classification problem in Riemannian and contact geometry. Like the exact
overlaps conjecture the same day, it shows a new pattern: human groups posting their own proofs with priority notes as soon as
OpenAI's catalogue makes a problem look "taken". The AI's share in the human paper is undisclosed beyond one sentence, so this
entry counts it as human-led.

Sources: [Hu, Liu & Wan: Transverse rational curves and the LeBrun–Salamon conjecture (arXiv 2610.10410)](https://arxiv.org/abs/2610.10410); [OpenAI math catalogue (CONTENTS.md, family 062)](https://github.com/openai/math/blob/main/CONTENTS.md); [LeBrun & Salamon (1994): Strong rigidity of positive quaternion-Kähler manifolds](https://doi.org/10.1007/BF01231528); [Buczyński, Wiśniewski & Weber: Algebraic torus actions on contact manifolds (J. Differential Geom.)](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-121/issue-2/Algebraic-torus-actions-on-contact-manifolds/10.4310/jdg/1659987892.full)

### 2026-10-07: Braverman & He refute the 2004 undirected multiple-unicast (network coding) conjecture with a GPT-6-found counterexample on PG(2, 9)

****; Princeton University, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-07-multiple-unicast-conjecture-false-gpt-6/

On 7 Oct 2026 Mark Braverman and Zhongtian He (Princeton) posted arXiv 2610.10108, "The Multiple Unicast Conjecture is False". The conjecture (Li & Li and Harvey–Kleinberg–Lehman, 2004) says network coding gives no throughput advantage over fractional multicommodity flow in undirected networks. They give a linear network code over 𝔽₉ on a 182-vertex subgraph of the point–line incidence graph of PG(2, 9), carrying 157 unicast sessions at rate ≥ 1 where every flow has rate ≤ 147/157. By earlier amplification results this gives coding gaps of Ω((log n)^ε). They write that they "use GPT-6 to find a nondeterministic counterexample" and thank ChatGPT "for coming up with the counterexample and drafting the proofs".

- Result: a deterministic linear network code over 𝔽₉ on a 182-vertex bipartite subgraph of the incidence graph of PG(2, 9): 157 independent unicast sessions at common rate ≥ 1, while every fractional multicommodity flow has common rate ≤ 147/157; amplification (Braverman–Garg–Schvartzman 2017) gives a family with coding gap Ω((log n)^ε)
- Method: the authors first posed a stronger 'nondeterministic' version of the conjecture (edges carry certificate bits checked locally); GPT-6 found a counterexample using a new choice of Reed–Solomon local codes in the high-girth-graph framework of Braverman–He 2025; an edge orientation plus local search turned it into a causal code
- AI use statement (verbatim): 'We thank ChatGPT for coming up with the counterexample and drafting the proofs. … The authors checked the proof and take responsibility for the correctness of the results.'
- Verification: 'The accompanying repository provides complete data, a standalone symbolic verifier, and an executable simulator with automated tests'; no Lean proof
- Stakes: a positive answer would have implied lower bounds for external-memory and oblivious computation and an Ω(n log n) lower bound for constant-degree Boolean circuits for integer multiplication (Afshani et al. 2019)
- Concurrent work: the authors note that OpenAI's sub-(n log n) exact Fourier circuits in its 6 Oct catalogue also imply a refutation, via the Afshani et al. cyclic-shift argument; their construction is an explicit 182-vertex example
- Second, independent disproof the same day, by two of the conjecture's authors: Xindan Zhang, Baochun Li and Zongpeng Li, 'Network Coding Can Beat Routing in Undirected Multiple-Unicast Networks' (arXiv 2610.09367, 7 Oct 2026). They build a finite, connected, simple network of maximum degree three on which a binary linear block code beats the best fractional routing rate, and a family where coding approaches rate one while routing is at most K/(log n)^c. 'The finite counterexample and the family theorem are formalized in Lean.' Disclosure (verbatim): 'OpenAI's GPT-6 Astra assisted in developing the proofs and the Lean code.'

#### What happened
In directed networks, coding at intermediate nodes is known to beat routing (the classic butterfly example). For undirected
networks with many source–sink pairs, Li and Li and, independently, Harvey, Kleinberg and Lehman conjectured in 2004 that
coding gives no advantage over fractional multicommodity flow. The conjecture mattered beyond networking. A proof would have
given unconditional lower bounds for external-memory algorithms and an Ω(n log n) circuit lower bound for multiplying
integers.

Mark Braverman and Zhongtian He had refuted stronger variants of the conjecture in 2025. This time they posed an even stronger
"nondeterministic" version to GPT-6, in which bits simply appear on edges and each vertex accepts or rejects by local checks.
GPT-6 found a counterexample using Reed–Solomon codes on the projective plane over 𝔽₉. The authors then oriented the edges
and ran a local search, and the example became a genuine causal code, which disproves the conjecture itself. The paper ships
a standalone verifier and simulator.

The authors also note that OpenAI's catalogue, released the day before, implies the same refutation indirectly: its claimed
sub-n log n Fourier circuits, combined with the Afshani et al. argument, would yield a coding advantage.

### Follow-up (14 Oct): from refutation to a faster sort
Baochun Li, Zongpeng Li and Sirui Liu used a new coding counterexample to give a deterministic external-memory integer sort that beats the comparison bound in a range of parameters. This answers Aggarwal and Vitter's 1988 question, and the algorithm is checked in Lean ([arXiv 2610.20079](https://arxiv.org/abs/2610.20079)). ChatGPT and Codex "materially affected the constructions and algorithmic analyses". Own entry: [2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding](/e/2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding/).

#### Why it matters
This is a clean, checkable AI-found counterexample to a 22-year-old conjecture in information theory, by a leading
theoretical computer scientist who describes exactly how the model was steered. Together with OpenAI's claimed sub-n log n
multiplication, it shifts a whole cluster of beliefs about lower bounds in the same week.

Sources: [Braverman & He: The Multiple Unicast Conjecture is False (arXiv 2610.10108)](https://arxiv.org/abs/2610.10108); [OpenAI math catalogue (CONTENTS.md: Fourier circuits and integer multiplication)](https://github.com/openai/math/blob/main/CONTENTS.md); [On the Multiple-Unicast Conjecture: Beyond Cut Metrics (arXiv 2608.06070, background)](https://arxiv.org/abs/2608.06070); [Zhang, Li & Li: Network Coding Can Beat Routing in Undirected Multiple-Unicast Networks (arXiv 2610.09367, Lean-formalized, GPT-6 Astra assisted)](https://arxiv.org/abs/2610.09367); [Lean proofs for the Zhang–Li–Li counterexample (GitHub)](https://github.com/iqua/li-li-conjecture)

### 2026-10-07: Toms–Winter conjecture on C*-algebras completed by two preprints in one week; both credit AI with a key lemma (Thiel; Evington, Hua, Schafhauser, Seth, Tikuisis, White)

****; University of Oxford, University of Ottawa, Chalmers University of Technology, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-07-toms-winter-conjecture-completed-chatgpt-astra-lemma/

On 7 Oct 2026 Samuel Evington, Shanshan Hua, Christopher Schafhauser, Apurva Seth, Aaron Tikuisis and Stuart White posted arXiv 2610.09390, "Regular simple nuclear C*-algebras" (70 pages): every simple, separable, pure, nuclear C*-algebra is Z-stable. With Hannes Thiel's preprint of 5 Oct (arXiv 2610.06761), this "completes the proof of the Toms–Winter conjecture", the statement that finite nuclear dimension, Z-stability and almost unperforation of the Cuntz semigroup are equivalent for simple separable nuclear non-elementary C*-algebras. The abstract ends: "An initial version of an essential lemma in the Bauer trace simplex case was obtained from ChatGPT Astra." The paper documents the single prompt of 8 Sept 2026 and says no further generative AI was used. Thiel's paper acknowledges that "interactive collaboration with AI" led to the key idea of its two opening lemmas. Both are unrefereed preprints.

- Paper: 'Regular simple nuclear C*-algebras', arXiv 2610.09390 [math.OA], 7 Oct 2026, 70 pages; Evington, Hua, Schafhauser, Seth, Tikuisis, White
- Theorem A ('The structure theorem for simple separable nuclear C*-algebras'): for a simple separable nuclear non-elementary C*-algebra A the following are equivalent: (i) finite nuclear dimension; (i') nuclear dimension at most one; (ii) Z-stability; (iii) almost unperforated Cuntz semigroup; (iii') A is pure
- Paper: 'Earlier this week, Hannes Thiel gave a proof that (iii)⇒(iii'). Here, we provide the final step in the puzzle by proving (iii')⇒(ii), completing the Toms–Winter conjecture which is now a unified structural theorem for simple nuclear C*-algebras.'
- Origin (paper): the conjecture 'stems from Toms and Winter's detailed analysis … of Villadsen's first family of examples in 2007' and 'became a central objective for the field in the 2010s'; the equivalence of (i), (i') and (ii) was already known
- AI statement: 'On 8 September 2026 … SW asked ChatGPT 6 Astra using codex whether a ‖·‖_{2,u}-separable W*-bundle M over a compact metrisable space with classification of projections … and real rank zero such that all fibres of M are isomorphic to the hyperfinite II_1 factor, must be trivial. After 27 minutes of computation, and no further human input, it produced a short 4 page proof of this statement'
- Footnote: 'Initially, the separability condition was omitted from the prompt; this was swiftly added when ChatGPT started to produce non-separable counterexamples.' The paper adds that this was 'the first test prompt' White sent after setting the tool up
- Limits of the AI role (paper): 'Beyond the two prompts on 8 September described above, the authors have made no further active use of generative AI to produce this version of the paper (for developing the mathematical content, proofreading, or any other purpose). Instead we chose to investigate the consequences of these ideas ourselves, leading to this paper.'
- Human work after the prompt, as dated in the paper: extending the lemma from W*-bundles to tracially complete C*-algebras (strategy found 19 Sept), the lifting arguments (2 Oct), a rewrite towards a Popa-style proof of hyperfiniteness (3–7 Oct). The authors also report that an unpublished 'proof' of theirs that the tracial completion has real rank zero 'was based on a misunderstanding … and flawed'
- Some authors 'had also considered various potential statements in the spirit of the small projections lemma … at different times over many years but had never been able to make a result of this nature work'
- Thiel, 'Ranks of operators and pureness of C*-algebras', arXiv 2610.06761, 5 Oct 2026: strict comparison by traces forces almost divisibility, hence pureness. Acknowledgments: 'At an initial stage of this project, I used interactive collaboration with AI to investigate the rank problem for simple ASH algebras. This led to the key idea contained in Lemma 2.1 and Lemma 2.2. The extension to the setting of the Global Glimm Property was developed by the author.' No model is named
- The six authors say they had no prior sight of Thiel's paper, 'though some of us verified [it] on 6 October'
- Other results of the paper: simple pure C*-algebras whose quasitraces are traces are K-stable; a local quantisation result 'à la Popa' for amenable tracially complete C*-algebras. White thanks 'OpenAI for providing access to ChatGPT through their ChatGPT for Academic Researchers programme'

#### What happened
C*-algebras are algebras of operators on Hilbert space. The Elliott programme classifies the simple, separable, nuclear ones by
K-theory and traces, in analogy with the Murray–von Neumann and Connes classification of von Neumann algebras. Examples of
Villadsen, Rørdam and Toms showed that classification fails without an extra regularity hypothesis. Toms and Winter conjectured
that three candidate hypotheses of very different kinds coincide: a topological one (finite nuclear dimension), an algebraic one
(absorbing the Jiang–Su algebra Z tensorially) and an order-theoretic one (strict comparison, i.e. an almost unperforated Cuntz
semigroup). Most implications were proved in the 2010s. The open one, from strict comparison to Z-stability, was known only under
restrictions on the trace space.

Two preprints in the first week of October 2026 close it. Hannes Thiel showed that strict comparison implies pureness. Two days
later the six-author paper showed that pure algebras are Z-stable, using the theory of tracially complete C*-algebras that several
of the authors had built over five years.

The paper has a section titled "AI statement, author contribution statement and description of our route to the results". On
8 Sept 2026 Stuart White asked ChatGPT 6 Astra (through Codex) whether a certain kind of W*-bundle with hyperfinite fibres must
be trivial. The model answered with a four-page proof after 27 minutes. A second prompt asked it to check its maximality
argument. The authors then stopped using AI and worked out the consequences themselves over four weeks. They say they will
publish the model's output and plan to test the model on the paper's questions before a later revision.

#### Why it matters
The conjecture was the main structural question in the classification of nuclear C*-algebras for about fifteen years. This is
not an AI-generated proof: the 70-page argument is human. It is a documented case where one short model answer supplied the
idea experts had been missing, and where the authors recorded exactly what was asked, when, and what they did afterwards. Thiel's
independent acknowledgment means both final steps of the conjecture involved AI at the idea stage.

#### Not verified here
The ChatGPT output had not been located on White's page when this entry was written. No expert commentary, press coverage or
Hacker News discussion was found on 13 Oct 2026. "Open since 2009" is the publication year of the Toms–Winter paper that the preprint cites as the
conjecture's source (J. Funct. Anal. 256, 2009); the preprint dates the analysis itself to 2007.

Sources: [arXiv 2610.09390: Regular simple nuclear C*-algebras](https://arxiv.org/abs/2610.09390); [arXiv 2610.06761: Thiel, Ranks of operators and pureness of C*-algebras](https://arxiv.org/abs/2610.06761); [Stuart White's publications page (the paper says the ChatGPT output will be posted in its supplementary materials)](https://sites.google.com/view/stuartwhite/research/publications)

### 2026-10-07: Bolzano (Charles University): open-source multi-agent system run unattended on about 3,800 open problems from research papers; about 200 reported solved, four STOC 2026 questions confirmed by the papers' authors

***; Charles University, Czech Academy of Sciences; Science & math; confidence medium; https://postcutoff.com/e/2026-10-07-bolzano-automated-open-problem-solving-charles-university/

On 7 Oct 2026 Adrián Zámečník, Matěj Kripner, Martin Koutecký, Martin Balko, Jan Grebík, Pavel Hubáček, Robert Šámal and Václav Rozhoň (Charles University, Prague) posted arXiv 2610.09769, accepted at the MATH-AI workshop of NeurIPS 2026. Bolzano is "an open-source orchestration layer around LLMs": parallel prover agents, a verifier agent and a shared research record. After eight expert-checked results in an April 2026 report, the team ran it "without problem-specific human guidance on about 3,800 open problems extracted from four sets of papers, solving about 200 open problems". For papers accepted to STOC 2026 "we answered four questions raised in the papers, as confirmed by their authors". The solved counts outside STOC are the team's estimates, partly based on LLM assessments.

- Paper: 'Bolzano: From Expert-Guided Proof Search to Automated Open-Problem Solving', arXiv 2610.09769 [cs.AI], 7 Oct 2026 (v2 9 Oct); 'Accepted at the 6th Workshop on Mathematical Reasoning and AI (MATH-AI), NeurIPS 2026'
- Table 1 (problems / model / rounds / promising / solved): arXiv math.CO and cs.DS 1,600 / GPT-5.5 / 4 / 250 / 90; STOC 2026 420 / GPT-5.5 / 4 / 41 / 4; earlier FOCS, SODA and STOC 900 / GPT-5.6 Sol / 2 / 57 / 40; Midsummer Combinatorial Workshop 880 / GPT-5.6 Sol / 2 / 137 / 80; total 3,800 / 485 / 214
- Caveat (Table 1 footnote): 'Except for STOC 2026, the solved counts are estimates. These estimates are partially informed by LLM-based assessments of whether the output addresses the source question and whether its argument is substantially correct.' For STOC 2026 only the four results checked by the team and verified by the source papers' authors are counted
- Setup: 'the prover, verifier, and summarizer operated without web search, code execution, direct filesystem access, or any tools'; all runs used one prover at the maximum reasoning effort; open problems were extracted from the papers by a separate LLM workflow with literature search
- STOC 2026 results, confirmed 'in private correspondence in June and July 2026': a one-pass O(log n)-memory algorithm detecting a semi-random planted biclique with sides at least Cn^(2/3) (question from Garg, Hastings, Pabbaraju and Sharan); an exponential separation between monotone rank programs and monotone span programs (Cavalar et al.); a strongly convex rational quartic in two variables whose zero sublevel set is a single irrational point (Slot, Steurer and Wiedmer); an Ω(log n) circuit-depth lower bound for joining and splitting biased CAT states (Parham)
- Footnote: for the convex-quartic and CAT-state results 'the authors reviewed only the main idea, not the full proof'
- Paper: 'Bolzano found a few hundred solutions, which we sent to authors of the source papers, some of whom plan to include these results in revised versions of their papers.'
- Limits stated by the authors: 'The verifier is another LLM, not a formal proof checker. An accepted but incorrect lemma may propagate through persistent memory'; observed failures 'include missing assumptions, proofs addressing weaker versions of the original questions, invalid arguments and rediscoveries of known results'; there is 'no comparison with direct use of the underlying models'
- Earlier report: 'Bolzano: Case studies in LLM-assisted mathematical research', arXiv 2604.16989 (eight problems checked by domain experts; six classed as publishable research and five as essentially autonomous). Project site: bolzano.app

#### What happened
Bolzano is an open-source system from Charles University in Prague. In each round several prover agents attack a problem in
parallel, a verifier agent criticises their output and updates three shared documents (notes, candidate proofs, current outcome),
and a summarizer records the round. A first report in April 2026 described eight results obtained with expert guidance and checked
by specialists.

The new paper asks whether the same loop works with nobody steering. An LLM workflow read four collections of papers (new arXiv
submissions in combinatorics and algorithms, papers accepted to STOC 2026, older FOCS/SODA/STOC papers, and work around the
Midsummer Combinatorial Workshop), extracted every explicitly stated open question, checked that it was still open, and wrote a
self-contained task. Each task got a fixed budget of two or four rounds with GPT-5.5 or GPT-5.6 Sol and no tools. The team then
screened the outputs, compared promising ones with the original question and sent a few hundred to the authors of the source papers.

The headline number, "about 200 open problems", is an estimate. The firmly checked part is the STOC 2026 experiment: four answers
that the papers' own authors confirmed, among them an algorithm matching a known memory lower bound up to logarithmic factors and
an exponential separation between two models of monotone computation.

#### Why it matters
OpenAI's October release of 722 manuscripts came from an internal model and its own problem list. Bolzano is a public,
open-source pipeline built on commercially available models by a university group, and it took its problems from other people's
papers, whose authors could check the answers. The paper is also frank about the bottleneck it exposes: producing candidate
solutions is cheap, and judging their correctness, scope and novelty is now the expensive step.

#### Not verified here
The four STOC results were not checked against the source papers, and the confirmations are private correspondence reported by
the authors. bolzano.app and the code repository were not inspected. No press coverage or Hacker News discussion was found on
13 Oct 2026.

Sources: [arXiv 2610.09769: Bolzano: From Expert-Guided Proof Search to Automated Open-Problem Solving](https://arxiv.org/abs/2610.09769); [arXiv 2604.16989: Bolzano: Case studies in LLM-assisted mathematical research](https://arxiv.org/abs/2604.16989); [Bolzano project site](https://bolzano.app/)

### 2026-10-07: Claimed proof of the cyclicity conjecture of Perron–Frobenius theory for positive operators (Dobrick, GPT-5.6 used to simplify arguments)

***; Alexander Dobrick; Science & math; confidence medium; https://postcutoff.com/e/2026-10-07-cyclicity-conjecture-positive-operators-dobrick/

On 7 Oct 2026 Alexander Dobrick posted arXiv 2610.10093, "On the Cyclicity Conjecture". It claims that the peripheral spectrum of every positive operator on a complex Banach lattice is cyclic, with no growth assumptions. The paper calls this a central conjecture of infinite-dimensional Perron–Frobenius theory that had stayed open since the 1960s. AI played a supporting role: GPT-5.6 models in an OpenCode harness were used for editing and "to simplify some arguments", and an AI-assisted literature review found two of the propositions used. The proof is an unrefereed preprint by one author, and we found no expert reaction as of 10 Oct.

- Conjecture (Open Problem 1.1 in the paper): if T is a positive operator on a complex Banach lattice with spectral radius r(T) = 1 and λ is in the peripheral spectrum, then λ^k is in the spectrum of T for every integer k
- History, per the paper: studied in H. H. Schaefer's Tübingen school in the 1960s–70s. Lotz and Krieger proved it for Abel-bounded operators, and later results added other growth conditions. All earlier results needed extra control of the operator's or resolvent's asymptotics
- Method: a spectral-domination theorem from Vesentini's subharmonicity of the spectral radius (extending Räbiger–Wolff); torsion-operator averaging gives positive minorants that are asymptotically rotationally self-similar; an ultrapower argument makes this exact. The paper also gives a new proof of Lotz's theorem for Abel-solvable operators
- AI disclosure (verbatim): 'During the preparation of this work, the author used an OpenCode harness, custom-written auxiliary skills and large language models from OpenAI's GPT-5.6 family for editorial assistance and to simplify some arguments. The statements of Proposition 2.1 and Proposition 2.2 were found by an AI-assisted literature review. In an earlier version of the paper, the notion of rotational self-similarity had a different AI-inspired name. The author reviewed and verified all AI-assisted material and takes full responsibility for the content of the manuscript. All new contributions are due to the author.'
- Not among the 722 manuscripts in OpenAI's 6 Oct math catalogue (no match for cyclicity/peripheral/Perron in CONTENTS.md, checked 10 Oct)

#### What happened
Alexander Dobrick posted "On the Cyclicity Conjecture" ([arXiv 2610.10093](https://arxiv.org/abs/2610.10093); the paper is dated 7 Sep 2026 and was posted 7 Oct). It claims a full answer to a question from the Perron–Frobenius theory of positive operators. For positive matrices it has been known since the early 20th century that the peripheral spectrum is cyclic. In infinite-dimensional Banach lattices it was proved only under extra growth conditions, such as Lotz–Krieger for Abel-bounded operators. The paper notes that the conjecture itself has no such condition.

The AI role is stated in the paper's disclosure (quoted in full under key facts). GPT-5.6 models, run through an OpenCode harness with custom skills, were used for editing and to simplify some arguments, and an AI-assisted literature search turned up Propositions 2.1–2.2. The author says all new contributions are his own.

#### Why it matters
If the proof holds, a long-standing central problem in operator theory has been settled. It is also typical of autumn 2026 papers, in which a human-led result openly credits a coding-agent harness and frontier models for parts of the work. The claim is unverified: it is a fresh single-author preprint with no refereeing or formalisation yet. Treat the conjecture as "claimed resolved, pending verification".

Sources: [Dobrick: On the Cyclicity Conjecture (arXiv 2610.10093)](https://arxiv.org/abs/2610.10093); [Glück: Growth rates and the peripheral spectrum of positive operators (arXiv 1512.07483), background on the open problem](https://arxiv.org/abs/1512.07483)

### 2026-10-07: Biohub's Virtual Biology Initiative grows to $1.8B: DOE, NIH, Google DeepMind, Isomorphic Labs and Meta join the push for open data to build AI 'virtual cells'

***; Biohub, US Department of Energy, National Institutes of Health, Google DeepMind, Isomorphic Labs, Meta; Science & math; confidence high; https://postcutoff.com/e/2026-10-07-virtual-biology-initiative-1-8b-doe-nih-deepmind-meta/

On Oct 7, 2026 Biohub (the Chan Zuckerberg science organization), the US Department of Energy and the NIH expanded the Virtual Biology Initiative to $1.8B in money, data, compute and measurement technology. Google DeepMind, Isomorphic Labs and Meta together commit $300M. The aim is open, AI-ready cell data for "virtual cell" models that predict how cells respond to drugs, mutations and disease.

- Commitments: Biohub $500M (its April 2026 founding pledge: $400M measurement technology, $100M external research); DOE $500M+ over five years; NIH datasets and resources from $500M+ of prior federal investment; Google DeepMind, Isomorphic Labs and Meta $300M combined (biohub.org)
- DOE contributes exascale computing, X-ray/neutron scattering, cryo-EM and autonomous labs under the Genesis Mission; NIH coordinates datasets and standards through a 'Bio Genesis Mission'
- Supporting organizations: Allen Institute, Broad Institute, Gladstone Institutes, Human Cell Atlas, Human Protein Atlas, Wellcome Sanger Institute, NVIDIA, Renaissance Philanthropy
- Biohub calls it the largest coordinated commitment to AI-ready biological data so far; the data are to be open
- Quotes: Alex Rives (Biohub head of science): 'A virtual cell is one of the most important challenges for the next era of science'; Max Jaderberg (Isomorphic Labs president) on scaling past single-organization limits
- The initiative was launched in April 2026 at $500M (Biohub alone)
- The White House listed the $1.8B initiative in its Oct 8 science-summit fact sheet (see 2026-10-08-white-house-golden-age-science-summit-genesis-pledges)

#### What happened
The expansion adds the US government and three AI companies to Biohub's April 2026 program. Most of the "$1.8B" is in-kind (government
data, compute, instruments) rather than new cash; the company share is $300M combined, with no per-company split published.

#### Why it matters
Virtual-cell models need far more standardized perturbation data than exists. Pooling federal labs, Biohub and frontier AI labs behind open
data is the biggest bet yet that biology can be modelled the way AlphaFold modelled protein structure.

Sources: [Biohub: Open data for predictive AI models of biology: $1.8 billion committed](https://biohub.org/news/virtual-biology-initiative-expansion/); [Reuters: US government, Google join Zuckerberg-backed Biohub's $1.8 billion push into AI biology](https://www.reuters.com/business/healthcare-pharmaceuticals/us-government-google-join-zuckerberg-backed-biohub-18-billion-push-ai-biology-2026-10-07/)

### 2026-10-07: AI for Math Fund (Renaissance Philanthropy, XTX Markets) names 22 new grants; total commitment reported at $35.1M

**; Renaissance Philanthropy, XTX Markets; Science & math; confidence medium; https://postcutoff.com/e/2026-10-07-ai-for-math-fund-2026-grants/

The AI for Math Fund, run by Renaissance Philanthropy with XTX Markets as founding donor, announced 22 new grants in early October 2026. They fund open Lean infrastructure, benchmarks, autoformalization checking, a Simons Institute research pod and tools for understanding AI-generated proofs. EdTech Innovation Hub reports that the round adds $17.1M, for a $35.1M total. The round came the same week as OpenAI's 722-manuscript release, and several projects target the problem that release exposed: checking and understanding machine-made mathematics.

- 22 grant awards in the fund's second round, listed on Renaissance Philanthropy's 'AI for Math Fund: 2026 Winners' page; founding donor XTX Markets
- Per EdTech Innovation Hub: $17.1M added, total commitment $35.1M (an earlier figure was $31.5M), 22 projects involving 30 organizations chosen from 470+ applications, grants of $100k–$1M for 12–24 months (not stated on the official page)
- Projects include: a Simons Institute research pod on AI for math and TCS; an INI programme for AI in maths; 'A Visual Proof Environment for Understanding AI-Generated Mathematics'; 'Reliable Equivalence Checking for Autoformalization'; a modular human-AI approach to formalizing the classification of finite simple groups in Lean; BELLE (Lean elegance benchmarks); LeanExplore; TorchLean; PySR as a conjecture engine; 'Constitutional AI for Proof Feedback at Scale'; a mechanistic study of how language models do mathematics
- Renaissance Philanthropy president Kumar Garg (via EdTech Innovation Hub): 'The math field is undergoing a transformation in the age of AI.'

#### What happened
Renaissance Philanthropy published the 22 winners of the AI for Math Fund's 2026 round. Its founding donor is the trading firm
XTX Markets. Most grants fund open infrastructure for formal mathematics in Lean, such as search, datasets, verified compilation,
tactic synthesis and diagram understanding. Others fund benchmarks and two programmes, at the Simons Institute and the Isaac Newton
Institute (INI). Two projects target checking AI output: "Reliable Equivalence Checking for Autoformalization" and "A Visual Proof
Environment for Understanding AI-Generated Mathematics". The dollar totals come from EdTech Innovation Hub's report, not from
the official page, and the exact announcement date is unclear (reported 7–8 Oct).

#### Why it matters
As labs release AI-generated proofs faster than mathematicians can read them, philanthropic money is flowing into the
verification and understanding layer: open tools to check that a formal statement matches the intended problem and to make
machine proofs readable. That is the gap highlighted by the OpenAI release and the summer 2026 Lean soundness bugs.

Sources: [Renaissance Philanthropy: AI for Math Fund 2026 Winners](https://renaissancephilanthropy.org/ai-for-math-fund-2026-projects); [EdTech Innovation Hub: AI for Math Fund expands to $35.1m as new grants back open tools, theorem proving and AI-generated proofs](https://www.edtechinnovationhub.com/news/ai-for-math-fund-expands-to-351m-as-new-grants-back-open-tools-theorem-proving-and-ai-generated-proofs)

### 2026-10-08: Google DeepMind's AlphaProof Nexus paper published in Science: LLM + Lean agent resolved 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures

****; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/

On 8 Oct 2026 Science published Google DeepMind's AlphaProof Nexus paper ("Advancing mathematics research with AI-driven formal proof search", doi 10.1126/science.aej2213), first posted on arXiv on 21 May 2026. Agents built on Gemini 3.1 Pro search for Lean proofs with compiler feedback. The full agent autonomously resolved 9 of 353 open Erdős problems (two questions open since 1970) at a few hundred dollars per problem and proved 44 of 492 open OEIS conjectures, all Lean-verified. A basic generate-and-check loop solved the same 9 Erdős problems at higher cost.

- Science, 8 Oct 2026, doi 10.1126/science.aej2213 (received 27 May, accepted 17 Aug 2026, per press reports); preprint arXiv 2605.22763 (v1 21 May 2026); 21 authors led by George Tsoukalas, Anton Kovsharov, Sergey Shirobokov; correspondence Pushmeet Kohli and Swarat Chaudhuri
- Erdős problems: 9 of 353 attempted (~2.5%), counted by formal sub-question: #12(i), #12(ii), #125 (variant), #138 (variant), #152, #26 (Tenenbaum variant), #741(i), #741(ii), #846 (Lean files in github.com/google-deepmind/alphaproof-nexus-results)
- The 'two questions open for 56 years' are Erdős #12(i) and #12(ii), posed by Erdős and Sárközy in 1970 (paper, Sec. 3)
- OEIS: Gemini autoformalized 492 open OEIS conjectures; the agent proved 44 that manual review found correctly formalized and previously unproven. Test lemmas checking the first terms of each sequence guarded against misformalization
- Comparison: the later OEIS Open benchmark (arXiv 2608.11941) reports 147 of 492 formalised open OEIS conjectures resolved by newer models at $50/attempt; OpenAI's 6 Oct 2026 release lists 722 manuscripts, mostly not formally verified (see related entries)
- Deployment: proved an exact O(1/t) last-iterate rate for Anchored Gradient Descent-Ascent in min-max convex-concave optimization while also discovering a new step-size schedule; solved 2 of 4 open Hilbert-function problems with Gergely Bérczi, incl. Zanello's log-concavity conjecture for pure O-sequences of codimension 3 and type 2; helped on a Ben Green list problem; quantum optics problems with Mario Krenn
- Agents: a 'basic' agent (prover subagents + Lean compiler feedback) and a 'full-featured' agent with evolutionary coordination, Elo-rated sketches (Gemini 3.0 Flash raters) and AlphaProof as a tool; the basic agent solved all 9 Erdős problems but cost more on the hardest
- Science vol. 394, issue 6820, pp. 234-239 (Research Article). Same issue: Perspective 'AI for research mathematics has arrived' by Jeremy Avigad (CMU) and Matthew Ballard (U. South Carolina), pp. 166-167 (per EurekAlert: even unsuccessful AI proof attempts can help mathematicians understand problems), and an 'Expert Voices' piece by Emily Riehl, 'AI has solved many math problems, but it has not solved math'
- erdosproblems.com credits DeepMind on #12, #26, #125, #138, #152, #741, #846; #741 and #846 were found independently by an internal OpenAI model too; for #846 the page notes a counterexample also follows from earlier work of Reiher, Rödl et al.
- Viral 2nd wave: @Dr_Singularity post on 10 Oct 2026 (~45.6k views at fetch time) presented the result as new

#### What happened
On 8 October 2026 *Science* published Google DeepMind's paper on **AlphaProof Nexus**, a framework of LLM agents that search
for formal proofs in Lean. The preprint (arXiv 2605.22763) had appeared on 21 May 2026 and was covered then by the
tech press. Publication in Science, after peer review, brought a second wave: an EurekAlert release, expert reactions
from the Spanish Science Media Centre, a Science Perspective by Jeremy Avigad and Matthew Ballard, an "Expert Voices"
piece by Emily Riehl in the same issue, a Pushmeet Kohli
thread on X, and on 10 October a viral summary by @Dr_Singularity.

The system pairs Gemini 3.1 Pro prover subagents with the Lean compiler. Every proposed proof is machine-checked, and
failed attempts feed back into the next try. The paper compares a **basic agent** (independent subagents plus compiler
feedback) with a **full-featured agent**, which coordinates subagents with an evolutionary algorithm, ranks proof
sketches with Gemini 3.0 Flash raters and can call AlphaProof (the 2024 RL prover) as a tool.

Results (from the paper):
- **Erdős problems:** 9 of 353 attempted, autonomously, at a few hundred dollars of inference per problem. The 9 are
  formal sub-questions: #12(i), #12(ii), #125, #138, #152, #26, #741(i), #741(ii) and #846, several in variant forms.
  The two "open for 56 years" are #12(i) and (ii), posed by Erdős and Sárközy in 1970. In a post-hoc ablation, the basic
  agent also solved all 9, at higher cost on the hardest ones.
- **OEIS:** Gemini autoformalized 492 open conjectures from the Online Encyclopedia of Integer Sequences. The agent
  proved 44 that manual review judged correctly formalized and previously unproven.
- **Research deployments:** an exact O(1/t) last-iterate rate for Anchored Gradient Descent-Ascent in min-max
  convex-concave optimization, found together with a new step-size schedule. With Gergely Bérczi (Aarhus), it solved 2 of 4
  open problems on Hilbert functions, including Zanello's log-concavity conjecture for pure O-sequences in the
  (codimension 3, type 2) case. It also helped resolve a problem from Ben Green's list, found misformalizations in
  the literature and supported quantum optics work with Mario Krenn.

#### Verification and reactions
- **Lean-verified and peer-reviewed.** All proofs are public Lean files (github.com/google-deepmind/alphaproof-nexus-results),
  and erdosproblems.com credits DeepMind on the pages of the solved problems. Science's article pages were not readable
  for this entry (HTTP 403). Volume and pages come from Science's RSS table of contents; the received and accepted dates come from press summaries.
- **Not all fully new.** On erdosproblems.com, #741 and #846 are credited to DeepMind *and* an internal OpenAI model,
  found independently. The #846 page notes that a counterexample also follows from a theorem of Reiher, Rödl et al.
  Several solutions settle variants or single parts of a problem.
- **Critics.** Anatol Wegner's blog (31 May, written with Gemini assistance; not peer-reviewed) argues that some
  "open" problems had prior literature and that the agent exploited loose formalizations. These are his claims and
  were not checked here. The Perspective and Riehl's piece were seen only as titles in Science's table of contents
  (the pages return HTTP 403 to us); their text is not summarised here. In the SMC Spain reactions, Josep Curto (UOC) called the system "a severe case of
  over-engineering", since the basic agent matched it. Javier Aramayona (ICMAT-CSIC) warned of results "known to be true"
  that "no one fully understands". Pablo Haya Coll (UAM) and Teodoro Calonge (Valladolid) were positive.
- In May, WION reported Demis Hassabis saying AGI is still far away despite the result, and a video posted to Hacker News was titled "Hassabis said solving Erdős isn't real invention" (not checked against the video).

#### Comparison
Measured by count, AlphaProof Nexus is now well behind later systems. The OEIS Open benchmark (arXiv 2608.11941) reports
147 of the same 492 formalised OEIS conjectures resolved at $50 per attempt. OpenAI's 6 Oct release lists 722
manuscripts, but only 162 are Lean-formalized and their review is marked "unchecked". The 44 OEIS and 9 Erdős results
were obtained with a May 2026 model. What sets this paper apart is that every result is formally verified and the
work passed peer review in a top general-science journal.

#### Why it matters
This is a peer-reviewed, fully formal account of LLM agents solving open research problems at scale and at low cost.
Its main finding, that a simple generate-and-check loop does about as well as a complex agent, supports the view that
model capability, not agent engineering, now drives AI mathematical discovery.

Sources: [Science: Advancing mathematics research with AI-driven formal proof search (doi 10.1126/science.aej2213)](https://www.science.org/doi/10.1126/science.aej2213); [Science Perspective: AI for research mathematics has arrived (Avigad & Ballard)](https://www.science.org/doi/10.1126/science.ael6628); [Science Expert Voices: AI has solved many math problems, but it has not solved math (Emily Riehl)](https://www.science.org/doi/10.1126/science.ael8339); [arXiv 2605.22763: Advancing Mathematics Research with AI-Driven Formal Proof Search](https://arxiv.org/abs/2605.22763); [GitHub: google-deepmind/alphaproof-nexus-results (Lean proofs + prose proofs)](https://github.com/google-deepmind/alphaproof-nexus-results); [EurekAlert (AAAS): Introducing AlphaProof Nexus (8 Oct 2026)](https://www.eurekalert.org/news-releases/1146436); [Science Media Centre Spain: expert reactions (8 Oct 2026)](https://sciencemediacentre.es/en/alphaproof-nexus-new-ai-tool-designed-tackle-mathematical-proofs); [Pushmeet Kohli on X: thread on DeepMind and mathematics, paper in Science today (8 Oct 2026)](https://x.com/pushmeet/status/2108265638435119494); [Pushmeet Kohli on X: 'Our technical paper on AlphaProof Nexus appears in Science today'](https://x.com/pushmeet/status/2108265646890815730); [Dr Singularity on X: viral summary (10 Oct 2026)](https://x.com/Dr_Singularity/status/2108936442671992990); [The Decoder: AlphaProof Nexus solves decades-old math problems for a few hundred dollars (May 2026)](https://the-decoder.com/google-deepminds-alphaproof-nexus-solves-decades-old-math-problems-for-a-few-hundred-dollars/); [erdosproblems.com #12 (DeepMind construction)](https://www.erdosproblems.com/12); [erdosproblems.com #846 (DeepMind and OpenAI, independently)](https://www.erdosproblems.com/846); [Anatol Wegner: critical analysis of the AlphaProof Nexus paper (31 May 2026)](https://buttondown.com/anatol/archive/deepminds-alphaproof-nexus/); [WION: Google AI solves decades-old maths problems but DeepMind CEO says AGI is still far away (May 2026)](https://embed.wionews.com/technology/google-ai-solves-decades-old-maths-problems-but-deepmind-ceo-says-agi-is-still-far-away-1779695717629); [Hacker News discussion (May 2026)](https://news.ycombinator.com/item?id=48248173)

### 2026-10-08: GPT-6 Astra finds a 3SAT reduction showing monotone Dualization (minimal hypergraph transversals) has no polynomial algorithm under ETH

****; Hokkaido University, Okayama University, Hosei University, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-08-dualization-eth-lower-bound-gpt-6-astra/

On 8 Oct 2026 Yasuaki Kobayashi, Kazuhiro Kurita and Kunihiro Wasa posted arXiv 2610.11473: a deterministic reduction from 3SAT with n variables to the complement of Dual, producing hypergraphs of size 2^O(n^(2/3) (log n)^(1/3)). Under the Exponential Time Hypothesis, neither Dual nor Dualization has an N^o(√(log N / log log N)) algorithm, so Dualization has no output-polynomial algorithm. The best known algorithm, Fredman–Khachiyan (1996), runs in quasipolynomial time N^O(log N / log log N). Disclosure: "The proof of Theorem 1 was originally discovered by GPT-6 Astra (OpenAI). The authors subsequently reconstructed and revised the argument and rewrote its exposition."

- Problem: Dual asks whether two monotone CNFs are mutually dual (equivalently whether L = Tr(H) for hypergraphs H, L); Dualization asks to enumerate all minimal transversals of H. Their output-polynomial solvability is a long-standing open problem; Fredman–Khachiyan's 1996 algorithm runs in N^O(log N / log log N)
- Theorem 1: a deterministic algorithm maps a 3CNF with n variables to hypergraphs H, L with L ⊆ Tr(H), of size 2^O(n^(2/3)(log n)^(1/3)), such that the formula is satisfiable iff L ≠ Tr(H)
- Consequence under ETH: no N^o(√(log N / log log N)) algorithm for Dual or Dualization, hence no polynomial-time algorithm for Dual and no output-polynomial enumeration for Dualization. It also rules out output-polynomial algorithms for the many problems known to be Dualization-hard (minimal dominating sets, which are 'polynomially equivalent', and others)
- AI disclosure (verbatim): 'The proof of Theorem 1 was originally discovered by GPT-6 Astra (OpenAI). The authors subsequently reconstructed and revised the argument and rewrote its exposition. The same AI model was also used to prepare drafts of several parts of the manuscript. The authors have verified the correctness of the results and take full responsibility'
- The paper is short (main text ends on page 9); unrefereed; no Lean formalization; no expert reactions found on X, HN or blogs as of 9 Oct
- Not in OpenAI's 6 Oct catalogue (no Dualization or transversal family in CONTENTS.md)

#### What happened
Dualization of monotone Boolean functions appears in databases, data mining, AI and convex geometry, and many enumeration
problems are known to be exactly as hard. Fredman and Khachiyan showed in 1996 that it can be done in quasipolynomial time,
and since then the question has been whether it can be done in output-polynomial time. Most work has found tractable
special cases.

Kobayashi, Kurita and Wasa give a subexponential reduction from 3SAT: partial assignments to blocks of variables become
vertices, and the formula is satisfiable exactly when a candidate list of minimal transversals is incomplete. With the
Exponential Time Hypothesis this rules out polynomial-time duality testing. The authors say GPT-6 Astra discovered the
proof of the main theorem.

#### Why it matters
If it holds, it closes the main question about Dualization conditionally. The quasipolynomial algorithm is close to optimal
and a whole family of enumeration problems inherits the lower bound. Because the argument is short and the claim is strong,
expert checking matters. Status: unrefereed.

Sources: [Kobayashi, Kurita & Wasa: An ETH-based quasipolynomial lower bound for Dualization (arXiv 2610.11473)](https://arxiv.org/abs/2610.11473)

### 2026-10-08: Engel & Mauri post a 7-page proof of the hyperkähler SYZ conjecture, two days after OpenAI's catalogue claimed it; AI used only to proofread

****; University of Illinois Chicago, Université Paris Cité, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-08-hyperkahler-syz-conjecture-engel-mauri/

On 8 Oct 2026 Philip Engel (UIC) and Mirko Mauri (Université Paris Cité) posted arXiv 2610.12277, "Hyperkähler SYZ conjecture": for a compact irreducible hyperkähler manifold X, every nef line bundle L is semiample, so a nontrivial nef isotropic line bundle defines a Lagrangian fibration. A corollary: compact hyperkähler manifolds of fixed dimension with b2 ≥ 5 fall into finitely many deformation classes. The key new input is the metric SYZ conjecture (Li; Blum–Liu). Their AI disclosure says a "largely complete proof was human written" as a rough draft and that after OpenAI's 6 Oct release they used ChatGPT Astra only to proofread; "The structure of the proofs appears similar, but we have not yet had time to examine the OpenAI proof carefully." OpenAI's family 041 ("The strong hyperkähler SYZ conjecture", dated 23 Sep) claims the same theorem without a Lean formalization.

- Theorem 1.1 (generalized abundance for compact hyperkähler manifolds): every nef line bundle L on a compact irreducible hyperkähler manifold is semiample; the case q(L) > 0 was known, so the content is the isotropic case q(L) = 0, which is the hyperkähler SYZ conjecture
- Corollary 1.2: the number of deformation classes of compact hyperkähler manifolds of fixed dimension with b2 ≥ 5 is finite (combining with earlier work, [5, Thm. B])
- Method: the strategy 'appears essentially in' Huybrechts–Mauri (2022); the new ingredient is the metric SYZ conjecture (Y. Li, arXiv 2605.00516; Blum–Liu), whose special Lagrangian tori become holomorphic after a hyperkähler rotation; 7 pages
- AI disclosure (verbatim): 'A largely complete proof was human written and in the form of a rough draft. After the October 6, 2026 posting of manuscripts by OpenAI, the authors used ChatGPT Astra to proofread the existing draft. The structure of the proofs appears similar, but we have not yet had time to examine the OpenAI proof carefully.'
- OpenAI family 041 'Hyperkähler SYZ and projective-space bases' (manuscripts dated 23 Sep 2026) claims the strong hyperkähler SYZ conjecture and that the normal projective base of any projective Lagrangian fibration is projective space; no Lean scope note
- No expert reactions found on X, HN or blogs as of 9 Oct

#### What happened
The hyperkähler SYZ conjecture predicts that a nontrivial nef line bundle of square zero on a compact hyperkähler manifold
comes from a Lagrangian fibration. Philip Engel and Mirko Mauri post a short proof. Mauri, with Huybrechts, had earlier
outlined the strategy, and the missing piece was the metric SYZ conjecture, proved in 2026 by Yang Li and by Blum and Liu.
A consequence is a finiteness theorem for deformation classes of hyperkähler manifolds with b2 ≥ 5.

The paper is a priority marker in the race set off by OpenAI's catalogue. Its AI section says the human proof existed as a
rough draft before 6 October and that ChatGPT Astra was used only to proofread it afterwards. It is one of several
human-led papers posted within days of the catalogue on problems it claims (others: LeBrun–Salamon, exact overlaps,
Erdős–Gallai, the Hellinger conjecture, 4-to-1 games).

#### Why it matters
If correct, it settles one of the main structural conjectures about hyperkähler manifolds, and it gives experts a short
human proof to compare with OpenAI's machine-written one.

Sources: [Engel & Mauri: Hyperkähler SYZ conjecture (arXiv 2610.12277)](https://arxiv.org/abs/2610.12277); [OpenAI math catalogue (CONTENTS.md, family 041)](https://github.com/openai/math/blob/main/CONTENTS.md); [Huybrechts & Mauri: Lagrangian fibrations (Milan J. Math. 2022; arXiv 2108.10193)](https://arxiv.org/abs/2108.10193)

### 2026-10-08: Erdős Problem #522 (1961) resolved with a Lean-verified proof: half the zeros of random ±1 polynomials lie in the unit disk, almost surely (Kawada, MIT CSAIL)

***; MIT CSAIL, Google DeepMind; Science & math; confidence medium; https://postcutoff.com/e/2026-10-08-erdos-522-littlewood-polynomial-zeros-lean-kawada/

On 8 Oct 2026 Sebastien Kawada (MIT CSAIL) posted arXiv 2610.11324, which proves that for random Littlewood polynomials (all coefficients ±1, chosen by fair coin flips) the proportion of zeros in the closed unit disk converges to 1/2 almost surely. Erdős asked this in 1961 (Problem #522 on erdosproblems.com); Yakir had proved convergence in probability. The paper goes further: a universal radial law at scale 1/n and non-universal fluctuations. All main results are verified in Lean 4, and the proof "has been accepted as a solution to its formal statement in Google DeepMind's Formal Conjectures project". AI disclosure: the Lean formalization and the checking of several derivations "were aided by internal models developed in the Kellis Lab at MIT CSAIL"; "The author originated the mathematical approach".

- Paper: 'Universal Radial Laws for Random Polynomials: Erdős Problem #522 and Non-Universal Fluctuations', arXiv 2610.11324 [math.PR], 8 Oct 2026; Sebastien Kawada, MIT; 82 pages
- Problem (paper): 'In 1961 Erdős asked whether ν_n(1)/n → 1/2 almost surely', where ν_n(1) counts the zeros of f_n(z) = ε_0 + ε_1 z + … + ε_n z^n in the closed unit disk and the ε_k are independent random signs
- Earlier: Yakir proved the statement in probability (|ν_n(1) − n/2| < n^(9/10) with probability tending to one). Paper: 'In April 2026, Chojecki and Kwon and Zou each posted an informal argument for the almost-sure statement for random signs.'
- Radial law: for bounded symmetric, real Gaussian or circular complex Gaussian coefficients, the proportion of zeros in the disk of radius 1 + x/n converges almost surely to ½(1 + coth x − 1/x), uniformly in x; so almost all zeros lie within O(1/n) of the unit circle
- Rate for random signs: almost-sure convergence at the unit circle at rate O(n^(−1/4) √(log n)), even with arbitrary dependence between degrees
- Fluctuations: the variance of the zero count is (c_G + κ_4/12)·n + O(n^(399/400)), depending on the coefficient law through its fourth cumulant; for random signs the constant is c_G − 1/6
- Verification: 'Ours is the first formally verified proof of Erdős Problem #522. Its Lean formalization was completed on September 22, 2026.' The repository lists 173 main theorems depending only on the axioms propext, Classical.choice and Quot.sound, and a single-file proof checkable in the Lean web editor
- Formal statement: the theorem proves Erdos522.erdos_522 as written in Google DeepMind's Formal Conjectures project, whose maintainers reviewed the formal proof (acknowledgments)
- AI disclosure (acknowledgments): 'The Lean formalization and the checking and refinement of several derivations were aided by internal models developed in the Kellis Lab at MIT CSAIL, including language models, latent representation models, and graph neural networks acting on the dependency structure of the proof. The author originated the mathematical approach and established the results presented here.'

#### What happened
A Littlewood polynomial has every coefficient equal to +1 or −1. Pick the signs by coin flips and ask where the zeros fall: they
cluster near the unit circle, and Erdős asked in 1961 whether exactly half of them, in the limit, lie inside or on it for almost
every infinite sequence of flips. Convergence in probability was known. The almost-sure version needs control over all degrees at
once, because each new coefficient moves every zero.

Kawada's paper proves the almost-sure statement and describes the zeros at finer scales. Magnified by the degree n, their distance
from the circle follows one fixed law for a wide class of coefficient distributions. The fluctuations of the count do depend on
the distribution, through its fourth cumulant.

The whole development is formalized in Lean 4. The theorem matches the statement of Problem #522 that Google DeepMind's Formal
Conjectures project had written down in advance, which removes the usual doubt about whether a formal proof proves the intended
claim. The paper says in-house models of the Kellis Lab at MIT helped with the formalization and with checking derivations, and
that the mathematical approach is the author's.

#### Why it matters
It is a named Erdős problem closed with a machine-checked proof against an independently written formal statement, days after
erdosproblems.com froze proof claims and removed its open/solved labels (6 Oct). It also shows a different pattern from the autonomous
results of autumn 2026: a human-led proof where AI did the formalization work, using a lab's own models that the paper does not
name or describe further.

#### Not verified here
The Lean project was not rebuilt for this entry, and erdosproblems.com/522 was not read (the site removed its open/solved status labels on 6 Oct). No expert reaction was found on 12 Oct 2026.

Sources: [arXiv 2610.11324: Universal Radial Laws for Random Polynomials: Erdős Problem #522 and Non-Universal Fluctuations](https://arxiv.org/abs/2610.11324); [Lean formalization and paper repository (chreia/erdos-522)](https://github.com/chreia/erdos-522); [Formal Conjectures: Erdos522.erdos_522](https://google-deepmind.github.io/formal-conjectures/theorem/?name=Erdos522.erdos_522); [Erdős Problem #522 (erdosproblems.com)](https://www.erdosproblems.com/522)

### 2026-10-08: Hilbert's 16th problem: Qin & Sun prove H(d) = Ω(d² ln² d), the first log-factor gain since 1995, with GPT-5.6 Sol developing the proof and a Lean 4 certificate

***; Chinese Academy of Sciences, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-08-hilbert-number-lower-bound-gpt-5-6-sol-lean/

On 8 Oct 2026 Chaoyang Qin and Xiaoming Sun (Institute of Computing Technology, CAS) posted arXiv 2610.12218, proving that the Hilbert number satisfies H(d) = Ω(d² ln² d), a factor ln d above the d² ln d bound that follows from Christopher–Lloyd (1995). The construction ("Hierarchical Melnikov Realization") perturbs an anisotropic Chebyshev Hamiltonian so that the first-order Melnikov function has simple zeros in many disjoint period annuli. "GPT-5.6 Sol was used to develop the proof strategy and arguments throughout this paper, assist with Lean formalization, and edit the manuscript." The authors say the Lean 4 development has no unproved placeholders or project-specific axioms. They note that OpenAI's 6 Oct catalogue (family 143) claims the finiteness of H(d), the opposite, upper-bound side of the problem.

- Main theorem: H(d) = Ω(d² ln² d) as d → ∞ (lower bound on the maximum number of limit cycles of real planar polynomial vector fields of degree d)
- Prior bounds: Otrokov 1954 (quadratic); Christopher & Lloyd 1995, order d² ln d along d = 2^k − 1, giving Ω(d² ln d) for all d; Li, Chan & Chung 2002 refined constants
- Method: anisotropic Chebyshev Hamiltonian, hierarchically chosen perturbation coefficients, two 3-adic filtrations for the log factors, and a Borel–Gauss analysis for the local rank condition
- Lean: the construction 'is fully formalized in Lean 4'; the formal statements assert a degree-bounded polynomial vector field with an injective finite family of hyperbolic limit cycles; 'no unproved placeholders or project-specific axioms'; code archived at Zenodo (doi:10.5281/zenodo.23219558)
- AI disclosure (verbatim): 'GPT-5.6 Sol was used to develop the proof strategy and arguments throughout this paper, assist with Lean formalization, and edit the manuscript. The authors take full responsibility for the mathematical content and the final text.'
- Priority remark: 'The main results were obtained by the end of August, at which time the finiteness of the Hilbert number H(d) remained an open problem'; OpenAI's family 143 (dated 24 Sep, released 6 Oct) claims uniform finiteness, with only its quintic Liénard case in Lean

#### What happened
The second part of Hilbert's 16th problem asks how many limit cycles a planar polynomial vector field of degree d can have.
Upper bounds are out of reach, so most work builds systems with many cycles. The best growth rate, d² ln d, dates to
Christopher and Lloyd in 1995. Qin and Sun gain another factor of ln d by placing simple zeros of the Melnikov function across
many period annuli at once. They formalize the full construction in Lean 4.

#### Why it matters
It is a quantitative improvement on a 30-year-old bound for one of Hilbert's problems, with a machine-checked construction and
an AI model credited with developing the argument. It does not address finiteness, which OpenAI's catalogue separately claims
without a full formalization.

Sources: [Qin & Sun: Hierarchical Melnikov Realization: Logarithmic Factor Improvement to Hilbert Number Lower Bounds (arXiv 2610.12218)](https://arxiv.org/abs/2610.12218); [Lean formalization archive (Zenodo, doi:10.5281/zenodo.23219558)](https://doi.org/10.5281/zenodo.23219558); [OpenAI math catalogue (CONTENTS.md, family 143)](https://github.com/openai/math/blob/main/CONTENTS.md)

### 2026-10-08: Holomorphic Cancellation Problem (1998) answered in the negative, conditional on a newly announced exotic affine space; proofs developed with generative AI (Gaofeng Huang, Harvard)

***; Harvard University; Science & math; confidence medium; https://postcutoff.com/e/2026-10-08-holomorphic-cancellation-problem-counterexample/

On 8 Oct 2026 Gaofeng Huang (Harvard) posted arXiv 2610.12581. It proves that a contractible Stein manifold X with the density property that embeds properly in C^k satisfies X × C^k ≅ C^(n+k). Combined with a result announced by Auroux, Bai and Pomerleano on 28 Sept 2026, this gives, for some n between 4 and 8, a complex affine variety X with X × C biholomorphic to C^(n+1) although X is not biholomorphic to C^n: a negative answer to the Holomorphic Cancellation Problem of Derksen and Kutzschebauch (1998). The paper's AI disclosure says it "was prepared with generative AI assistance to develop and write the proofs". It also cites an OpenAI math-release preprint claiming a counterexample to Zariski cancellation for n = 4.

- Paper: 'A Counterexample to the Holomorphic Cancellation Problem', Gaofeng Huang (Department of Mathematics, Harvard; funded by a Swiss National Science Foundation Postdoc.Mobility grant), arXiv 2610.12581 [math.CV], 8 pages, 8 Oct 2026
- Theorem 1.1: 'There exist an integer n ∈ {4, …, 8} and a contractible affine variety X of dimension n such that X × C is biholomorphic to C^(n+1) but X is not biholomorphic to C^n.' The exact n is not determined
- Theorem 1.2 (the paper's own tool): if X is a contractible Stein manifold of dimension n with the density property and X embeds properly and holomorphically in C^k, then X × C^k is biholomorphic to C^(n+k). The method is Andersén–Lempert theory
- Conditional input: Auroux, Bai and Pomerleano 'recently announced' that a specific exotic affine space is not biholomorphic to C⁴; the only reference is Pomerleano's lecture 'Exotic affine spaces and Floer theory' at the CMSA Harvard workshop on Lagrangian Floer theory, 28 Sept 2026. No written proof is cited
- AI disclosure: 'This work was prepared with generative AI assistance to develop and write the proofs, to search the literature, and to edit the language. The author checked the mathematical content and references and takes responsibility for the content of the paper.' The model is not named
- Link to OpenAI's release: 'The batch of preprints released by OpenAI on October 6 2026 contains a counterexample to the Zariski cancellation problem for n = 4', citing 'An explicit failure of complex affine-space cancellation' (openai/math, dated 23 Sept 2026). Huang notes consequences such as a non-linearizable algebraic C*-action on C⁵ from that preprint
- Not found as of 12 Oct 2026: expert reactions; the Auroux–Bai–Pomerleano paper

#### What happened
The cancellation problem asks whether a space X must be ordinary n-dimensional space when X times a line is ordinary
(n+1)-dimensional space. The algebraic version (Zariski) is open in characteristic zero for n = 3 and n ≥ 5; the holomorphic
version was posed by Derksen and Kutzschebauch in 1998. Huang shows that any contractible Stein manifold with the density
property becomes ordinary complex space after multiplying by enough copies of C. If one such manifold is not itself ordinary
space, which is what Auroux, Bai and Pomerleano announced in a September lecture, then going up one dimension at a time must
produce a counterexample somewhere between dimensions 4 and 8.

#### Why it matters
A problem open for 28 years gets a negative answer in eight pages, but with two caveats: the key input is an announcement without
a paper, and the proofs were developed with an unnamed AI system and checked by one author. The paper also shows OpenAI's 6 Oct
release being used and cited by mathematicians within two days: its n = 4 counterexample to Zariski cancellation is the algebraic
sibling of this result.

Sources: [arXiv 2610.12581: A Counterexample to the Holomorphic Cancellation Problem](https://arxiv.org/abs/2610.12581); [OpenAI math release: An explicit failure of complex affine-space cancellation (23 Sept 2026)](https://github.com/openai/math/blob/main/preprints/An-explicit-failure-of-complex-affine-space-cancellation-September-23-2026/paper.pdf)

### 2026-10-08: Set theorist Asaf Karagila refuses to review OpenAI's claimed solution of the Partition Principle problem (does PP imply AC?): 'It sucked'; calls the 722-paper release a 'Denial of Service'

***; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-08-karagila-openai-partition-principle/

OpenAI's 6 Oct 2026 math catalogue includes family 244, "The Partition Principle does not imply Choice": if ZF is consistent, so is ZF + PP + choice for ordinal-indexed families + ¬AC (manuscript dated 24 Sep; Lean formalization of the relative-consistency statement). On 8 Oct Asaf Karagila, who works on exactly this problem and lists it on his Problems page with a whisky prize, wrote that the preprint "sucked. It is unclear, muddled, and has a strange structure", that a journal should desk-reject it for quality, and that he will not spend time reading it or send Sam Altman the whisky. He called the mass release "the equivalent of a Denial of Service" for mathematicians. The post reached 172 points on Hacker News.

- Claim (OpenAI family 244): assuming Con(ZF), there is a model of ZF in which every surjective image of a set injects into it (Partition Principle) while AC fails; choice for ordinal-indexed families holds. A second construction gives a transitive symmetric extension with no new countable sequences of ground-model elements
- Lean scope (lean/docs/244.md): formalizes the relative-consistency implication and a model construction from a countable transitive ground model with an internal strongly inaccessible cardinal; 'The paper's stronger transitive-model preservation assertions are outside that selected construction'
- History: Russell (1906) conjectured PP is equivalent to AC; AC ⇒ PP is easy; the converse is often called the oldest open problem in set theory. Frank Gilson's 2026 claim (arXiv 2601.01855) was withdrawn in v8 (10 Sep 2026): 'the previously claimed final PP+not-AC construction is withdrawn'
- Karagila (8 Oct): read the preprint, not the Lean code ('that code was enormous'); found odd lemmas (7.4, 8.1), 'terminology a bit off', and references to unpublished lecture notes, including his own; 'if this was an academic paper submitted to a journal, it should be issued a desk rejection for the quality'
- Karagila on the release: 'OpenAI drops some hundreds of "solutions", incomprehensibly written "solutions", and we are all expected to jump on them'; 'It seems to me, that the AI tech companies would like us to conform to their standards, rather than spend the time and energy to conform to ours'
- Karagila is not against AI as a tool and uses LLMs for minor tasks, but avoids them for mathematics until the community has a framework (e.g. whether chat logs should go to referees)
- Correctness of OpenAI's construction: not assessed by any expert as of 9 Oct

#### What happened
Within a day of OpenAI's release, people kept asking Asaf Karagila whether he had seen the claimed solution and whether he
would send Sam Altman the whisky promised on his Problems page. His answer, in a blog post on 8 October, was no on both
counts. He judged the writing unfit for a journal and said that checking it would mean dropping his own research and
students for a paper whose authors "don't bother to communicate better". He argued that the release lets OpenAI suggest
"progress" while the public hears "solutions", and that funders and policymakers may conclude mathematicians can be
replaced.

The construction itself has a Lean formalization of its main relative-consistency statement, which Karagila did not
examine. A human claim to the same result (Gilson, January 2026) had been withdrawn in September.

#### Why it matters
It shows the gap between a machine-checkable claim and acceptance by the field. The problem's main expert refuses the cost
of reading AI output written below the field's standards, so a possibly correct proof of a century-old problem may stay in
limbo until someone checks the Lean statement against the intended theorem or rewrites the argument.

Sources: [Asaf Karagila: OpenAI, the Partition Principle, and mathematics (8 Oct 2026)](https://karagila.org/2026/openai-pp/); [Hacker News discussion (172 points)](https://news.ycombinator.com/item?id=50013902); [OpenAI math catalogue (CONTENTS.md, family 244)](https://github.com/openai/math/blob/main/CONTENTS.md); [OpenAI math: Lean scope note for family 244](https://github.com/openai/math/blob/main/lean/docs/244.md); [Gilson: A countable-support symmetric iteration separating PP from AC (arXiv 2601.01855, claim withdrawn in v8)](https://arxiv.org/abs/2601.01855)

### 2026-10-08: Full Lyapunov spectrum of the Sompolinsky–Crisanti–Sommers random neural network derived, a problem named open in 1988; GPT-6 Astra produced the first derivation (David G. Clark, Flatiron)

***; Flatiron Institute, OpenAI, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-10-08-lyapunov-spectrum-random-neural-networks-gpt-6-astra/

On 8 Oct 2026 David G. Clark (Center for Computational Neuroscience, Flatiron Institute) posted arXiv 2610.12426, which computes the full Lyapunov spectrum of the random recurrent neural network of Sompolinsky, Crisanti and Sommers in the N → ∞ limit. The 1988 paper had named "evaluation of the whole distribution of Lyapunov exponents" as an open problem. The paper says the work "was done in collaboration with the AI models GPT-6 Astra and Claude Opus 5.5": GPT-6 Astra produced the initial derivation after working alone for 100 minutes, and Claude Opus 5.5 wrote all code and figures. It is a physics-style (non-rigorous) derivation checked against large-N simulations; an unrefereed preprint.

- Paper: 'Lyapunov spectrum of random neural networks', David G. Clark (Flatiron Institute), arXiv 2610.12426 [cond-mat.dis-nn], 8 Oct 2026; code and data at github.com/davidclark1/rnn-lyapunov
- Claim (introduction): Sompolinsky et al. 'named the "evaluation of the whole distribution of Lyapunov exponents" of this network as an open problem nearly four decades ago. The present work solves this problem.'
- Method: a finite-N identity expresses the cumulative distribution of Lyapunov exponents as a response function of the tangent-space dynamics (tangent trajectory chosen by a minimum-norm condition); a cavity method then reduces it at large N to a self-consistent single-site problem. Result: the chaos is extensive
- Use of AI: 'GPT-6 Astra produced an initial derivation. Working with both models, the author then arrived at the more physically transparent derivation presented in Sec. 2. Claude Opus 5.5 wrote all code and generated all figures, under the direction of the author, and assisted in writing and revision of the text.'
- AI methodology appendix: 'The author had previously assumed that the present problem was analytically intractable.' GPT-6 Astra (Max thinking), in a session where it had already derived other results on random recurrent networks, 'worked independently for 100 minutes' after the prompt 'Derive the full N → ∞ Lyapunov spectrum of the SCS model (see Engelken for numerics)… Do not stop until you have nailed the answer.' Its derivation used a theorem of Deninger equating the sum of positive Lyapunov exponents with a Fuglede–Kadison log-determinant
- Rigour: the exchange of the large-N limit with the long-time and zero-regularizer limits 'was assumed' (in both the AI and the final derivation); the theory is checked against known results (maximum exponent, Sompolinsky et al.'s δ → 0 result) and against numerics of Engelken, Wolf and Abbott
- Acknowledgments: Ashok Litwin-Kumar proposed 'this problem as a target for AI-assisted solution'
- Not found: expert reactions or press coverage as of 11 Oct 2026

#### What happened
In 1988 Haim Sompolinsky, Andrea Crisanti and Hans-Jürgen Sommers showed that a large network of randomly coupled neurons becomes
chaotic, which became a standard model for chaos in neural circuits. Their paper listed the full distribution of Lyapunov exponents
(how fast nearby trajectories separate in every direction) as an open problem. Engelken, Wolf and Abbott later computed it
numerically, but no analytic theory existed.

David G. Clark's preprint gives a large-N theory for the whole spectrum. Its "Use of AI" section and appendix describe the process.
Ashok Litwin-Kumar suggested the problem as a target for AI. GPT-6 Astra, prompted inside a session where it had already derived
other results on random networks, worked alone for 100 minutes and produced a derivation that matched simulations, using a theorem
of Deninger on Fuglede–Kadison determinants. Working with GPT-6 Astra and Claude Opus 5.5, Clark then found the more physically
transparent cavity-method derivation presented in the paper. Claude Opus 5.5 wrote all code and figures.


### A parallel paper with ChatGPT 5.6 Sol (10 Oct)
Haoyu Albert Wang and Wei P. Dai posted [arXiv 2610.14922](https://arxiv.org/abs/2610.14922), "Full Lyapunov Spectra of Random Neural Networks, Part I: Finite-Time and Asymptotic Theory": a dynamical mean-field framework for the finite-time and stationary Lyapunov spectra, checked against singular-value and QR numerics. The abstract says "This work was carried out in collaboration with ChatGPT 5.6 Sol", which "assisted with theoretical analysis, numerical implementation and checking, and manuscript drafting". The authors write that toward the end they "became aware of David G. Clark's contemporaneous work … which reports overlapping mean-field results". So two groups, each working with a different lab's models, reached the same 1988 open problem within days.

#### Why it matters
It is a long-standing problem in theoretical neuroscience and statistical physics, not pure mathematics, settled with an AI-first
derivation and an unusually explicit methodology appendix, including the prompt. It extends the 2026 wave of AI-assisted results
beyond mathematics. Like most physics derivations, it is not a rigorous proof: limit exchanges are assumed.

Sources: [Wang & Dai: Full Lyapunov Spectra of Random Neural Networks, Part I (arXiv 2610.14922)](https://arxiv.org/abs/2610.14922); [Clark: Lyapunov spectrum of random neural networks (arXiv 2610.12426)](https://arxiv.org/abs/2610.12426); [Code and data (davidclark1/rnn-lyapunov)](https://github.com/davidclark1/rnn-lyapunov); [Sompolinsky, Crisanti & Sommers (1988): Chaos in random neural networks (Phys. Rev. Lett. 61, 259)](https://doi.org/10.1103/PhysRevLett.61.259)

### 2026-10-08: PPT-distillable entanglement can fall strictly below the regularised Rains bound: results "derived almost entirely by ChatGPT Astra" (Lami)

***; Scuola Normale Superiore, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-08-ppt-entanglement-distillation-rains-bound-chatgpt-astra/

On 8 Oct 2026 Ludovico Lami (Scuola Normale Superiore, Pisa) posted arXiv 2610.12454. It gives a regularised formula for the entanglement distillable with PPT operations and two new single-letter upper bounds, and uses them to show that the PPT distillable entanglement can be strictly smaller than the regularised Rains bound, "resolving an open problem in the theory of entanglement manipulation" (Regula et al., 2019). For a 3×3 Werner state a certified upper bound of 0.62107 ebits lies below the Rains value of about 0.64766. The introduction begins: "The main results and techniques presented here have been derived almost entirely by ChatGPT Astra." Unrefereed preprint, checked by the author.

- Paper: 'On PPT entanglement distillation', arXiv 2610.12454 [quant-ph], 8 Oct 2026; Ludovico Lami, Scuola Normale Superiore, Pisa (single author); 39 pages
- Results (abstract): '(a) a regularised formula for the PPT distillable entanglement in terms of a measured-relative-entropy-like quantity; and (b) two different single-letter converses, one based on operator quadratic forms and the other on Hirschman's strengthening of the Hadamard three-line theorem'
- Open problem resolved (abstract): 'we show that the PPT distillable entanglement can be strictly smaller than the regularised Rains bound, thereby resolving an open problem in the theory of entanglement manipulation [Regula et al., NJP 21:103017, 2019]'
- Explicit gap: 3×3 Werner state with antisymmetric weight 25/26: certified upper bound 0.62107 ebits, against a (regularised) Rains bound of (25/26)·log2 5 − log2 3 ≈ 0.64766 ebits
- Also: a faithful lower bound on PPT distillable entanglement in terms of the entanglement negativity, a quantitative version of the known fact that every NPT state is PPT-distillable
- AI disclosure (note opening the introduction): 'The main results and techniques presented here have been derived almost entirely by ChatGPT Astra. The human author (yours truly) gave many suggestions on how to extend and generalise the initial findings, which applied only to Werner states, and spent many hours (but an order of magnitude fewer than would have been needed a year ago) validating and improving the proof techniques, as well as adapting the presentation to his style.'
- The numerical separation is backed by companion certificate files (appendices on certified parameters and quadrature)

#### What happened
Entanglement distillation turns many copies of a noisy entangled state into fewer, nearly perfect entangled pairs ("ebits"). The
best achievable rate with local operations is very hard to compute, so theorists study a larger, more tractable class of
operations, the PPT channels, and upper bounds such as the Rains bound. Whether the regularised Rains bound is always attained by
PPT operations was listed as an open problem in 2019.

Lami's paper answers no. It gives a formula for the PPT distillable entanglement, derives two new upper bounds, and exhibits a
simple family of states (Werner states on two three-level systems) where the new bound is provably below the Rains bound.

The paper's first note attributes the work to the model, named there as "ChatGPT Astra" (OpenAI's GPT-6 Astra). The author
describes his own part as suggesting generalisations, spending many hours on validation, and adapting the text to his style.

#### Why it matters
The disclosure is unusually direct for a single-author theory paper. His
remark about time is a data point on how fast the checking burden is changing: validation took an order of magnitude less time
than it would have a year before, by his estimate.

#### Not verified here
The certificate files were not re-run. No independent expert reaction was found on 12 Oct 2026.

Sources: [arXiv 2610.12454: On PPT entanglement distillation](https://arxiv.org/abs/2610.12454)

### 2026-10-08: Laderman's 23 multiplications are optimal for 3×3 matrices over F_2 and over the integers: AI-found proof checked in Lean (Tomar)

***; independent; Science & math; confidence medium; https://postcutoff.com/e/2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar/

On 8 Oct 2026 Tejasvi Singh Tomar (independent researcher) posted arXiv 2610.11823: the rank of the 3×3 matrix multiplication tensor over the two-element field is exactly 23. Laderman's 1976 algorithm gives the upper bound; Rudich and Rousseau had proved a lower bound of 22 a week earlier. As a corollary, any bilinear algorithm with integer coefficients needs 23 multiplications. The proof is a computer search whose certificates are checked in Lean 4 (the largest checks as compiled code). The paper's "Use of AI tools" note says: "Generative AI tools were used to find and prove the results, to write the programs and the Lean formalization, and to draft the text." Over the rationals and reals the rank is still only known to lie between 19 and 23.

- Paper: 'The rank of 3×3 matrix multiplication over F_2 is 23', arXiv 2610.11823 [math.RA], 8 Oct 2026; Tejasvi Singh Tomar, 'Independent researcher'
- Theorem 1.1: R_{F_2}(⟨3,3,3⟩) = 23. Corollary 1.2: 'A bilinear algorithm over F_2 or with integer coefficients that multiplies two 3×3 matrices uses at least 23 multiplications. In particular R_Z(⟨3,3,3⟩) = 23, and no recursive 3×3 algorithm with integer constants has an exponent below log_3 23 ≈ 2.854.'
- Upper bound: Laderman's algorithm (1976), 23 multiplications, coefficients 0 and ±1. Paper: 'whether Laderman's 23 multiplications are necessary has been open since 1976'
- Lower bounds over F_2 in 2026, as listed in the paper: 19 over all fields (Bläser 2003); 20 (C. Wang, arXiv 2603.07280; certificate re-checked by Beuchert); 21 (Wang, arXiv 2609.06725, and independently Yang et al., arXiv 2609.18722, formalized in Lean); 22 (Rudich and Rousseau, arXiv 2610.01639, in Lean); 23 (this paper)
- Method: the substitution method over Wang's 496 symmetry classes of subspaces (bounds raised for 111 of them), a 27×27 flattening, then a computer search listing all candidate sets of 22 first factors and an exact completion search showing none works
- Verification: 'The computation emits certificates, which are checked in the Lean 4 proof assistant by checkers whose soundness is proved in Lean. The largest checks are evaluated as compiled code, so the proof relies on the Lean compiler in addition to its kernel.'
- AI disclosure: 'Generative AI tools were used to find and prove the results, to write the programs and the Lean formalization, and to draft the text.' No model is named
- Code and certificates: github.com/tejasvi/rank333-f2, archived at doi.org/10.5281/zenodo.23236154
- Still open: the rank over fields of characteristic zero (between 19 and 23), and whether non-integer coefficients can beat 23
- 12 Oct 2026: Tomar published a second repository, github.com/tejasvi/rank333-f2-kernel, 'Kernel-only Lean proof': 'Every certificate in it is checked by the Lean kernel. The proof uses no native_decide, no compiled code and no axioms beyond propext, Classical.choice and Quot.sound.' Its verification notes count 150,678 packages with 14,185,917 Lean modules. If it holds up, this removes the paper's stated reliance on the Lean compiler
- AI disclosures of the earlier steps of the race (PDFs read 13 Oct): Wang, 20 (arXiv 2603.07280): 'Generative AI tools materially contributed to this work'; the search code was written by hand, Claude Code generalized it, Claude and ChatGPT checked derivations and helped draft. Wang, 21 (arXiv 2609.06725): no AI statement found. Yang et al., 21 (arXiv 2609.18722, Zhejiang University): 'The structural proof, independently developed by Qiushi Engine', the team's autonomous research system, with 'additional Codex assistance' for the Lean development. Rudich and Rousseau, 22 (arXiv 2610.01639, Polytechnique Montréal): 'AI agents (Claude, by Anthropic) wrote the search code and the Lean proofs that do not require a human auditor, under our direction. The audit-surface was human designed.'
- Related exact rank: Ian D'Ambrosio, 'The Exact F2-Rank of 2x3 by 3x4 Matrix Multiplication' (Zenodo 21895176, 11 Aug 2026): the rank is 20, confirming over GF(2) the Hopcroft–Kerr conjecture; 'produced with material assistance from Beyond, an autonomous research system operated by Nth Research Collective'
- Author: Tomar's GitHub profile (tejasvi) lists '@Google' and Mountain View, CA; the papers say 'Independent researcher'. He has posted three AI-generated, Lean-checked preprints in eleven days (28 Sept, 8 Oct, 9 Oct)

#### What happened
Strassen showed in 1969 that two 2×2 matrices can be multiplied with 7 multiplications instead of 8, and by 1971 it was proved that
7 is the minimum. For 3×3 matrices the best algorithm, by Julian Laderman (1976), uses 23 multiplications, and the best lower bound
valid over every field has been 19 since 2003.

Over the two-element field F_2 the gap closed quickly in 2026. Automated and formally verified proofs gave 20, then 21 (two
independent groups in September), then 22 (Rudich and Rousseau, 1 Oct, in Lean). Tomar's preprint claims the last step: no
algorithm with 22 multiplications exists over F_2. Because any algorithm with integer coefficients reduces to one over F_2, the
same bound holds over the integers.

The proof is a large case analysis by computer. Lean checks the certificates the search produces. The author is explicit that the
biggest checks run as compiled code, which means trusting the Lean compiler and not only its small kernel.

#### Why it matters
The result says that the textbook-style schemes found by hand, by SAT solvers and by systems such as AlphaTensor and AlphaEvolve
cannot improve on 23 for 3×3 matrices unless they use non-integer coefficients. It is also an example of a research line where
several authors, working with AI tools and proof assistants, pushed a bound that had not moved for 23 years to its exact value
within months.

#### Not verified here
The certificates were not re-run for this entry, and the kernel-only repository was not rebuilt. The paper does not name the AI
models used. No independent review, and no reaction from the authors of the earlier bounds (Wang, Yang et al., Rudich and
Rousseau), was found on 12 or 13 Oct 2026 (web and Hacker News; X not searched). Tahir's conditional bound of 21, cited in the
paper, was not located. The Google affiliation comes only from a self-edited GitHub profile field.

Sources: [arXiv 2610.11823: The rank of 3×3 matrix multiplication over F_2 is 23](https://arxiv.org/abs/2610.11823); [Code, certificates and Lean checkers (tejasvi/rank333-f2)](https://github.com/tejasvi/rank333-f2); [Rudich & Rousseau, Lower bound of 22 for 3×3 matrix multiplication over Z (arXiv 2610.01639)](https://arxiv.org/abs/2610.01639); [Yang et al., A structural proof of the lower bound 21 for 3×3 matrix multiplication over F_2 (arXiv 2609.18722)](https://arxiv.org/abs/2609.18722); [Wang, A lower bound of 21 for 3×3 matrix multiplication over F_2 (arXiv 2609.06725)](https://arxiv.org/abs/2609.06725); [Kernel-only Lean 4 proof (tejasvi/rank333-f2-kernel, 12 Oct 2026)](https://github.com/tejasvi/rank333-f2-kernel); [Wang, Automated Lower Bounds for Bilinear Complexity over Finite Fields (arXiv 2603.07280; lower bound 20)](https://arxiv.org/abs/2603.07280); [D'Ambrosio, The Exact F2-Rank of 2x3 by 3x4 Matrix Multiplication (Zenodo)](https://zenodo.org/records/21895176)

### 2026-10-08: Developer's Claude-and-Jev pipeline over 5.7M archival passages turns up an unrecorded 1812 Indian meteorite fall, lost VOC rhinoceros shipments and three uncatalogued eruptions (unreviewed candidates)

**; Anthropic, TypeSafe AI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-08-antiquity-ai-archive-search-meteorite-rhinos/

On Oct 8, 2026 developer Jesse Waites described "Antiquity", an open-source pipeline built with Claude Code that used TypeSafe's Jev to filter and Claude Haiku to analyse 5.7 million passages from Dutch East India Company (VOC) records, Dutch and American newspapers and ship logs. It surfaced an 1812 meteorite fall in Maharashtra (26 years before the earliest recorded one there), 1738–1740 Javan rhinoceros shipments and three volcanic eruptions missing from the Smithsonian catalogue. The finds are candidates that specialists had not yet reviewed. The post reached 151 points on Hacker News.

- Corpus: GLOBALISE's digitised VOC archive (4.35M pages, 1600s–1790s), Delpher Dutch newspapers, two centuries of American newspapers and ship logbooks; 5.7M passages in total
- Tools (author's account): Claude Code to build the pipeline, Jev as a fast filter, Claude Haiku for detailed reading, a custom GPU semantic search; code at github.com/jessewaites/antiquity. The whole VOC archive was processed in a 12-hour overnight run; filtering 59,000 elephant mentions with Jev cost about $3
- Meteorite candidate: an iron-rich stone that fell near 'Cokurrgaum', India on Aug 6, 1812, reported in the Bombay Gazette, which would be 'the earliest recorded meteorite fall in Maharashtra, predating the current earliest record by 26 years'
- Rhinoceros candidate: three Javan rhinoceroses shipped in 1738–1740 for the King of Kandy, one reaching Colombo before dying, documented in sworn crew statements and official correspondence
- Eruption candidates: Gamkonora (Feb 3, 1722), Ciremai (early 1712) and Slamet (Feb 1780), absent from the Smithsonian Global Volcanism Program; 769 eruption reports were found in the corpus overall
- Verification: 'checked against the original pages and the catalogues I could access, but not yet reviewed by the specialists who maintain those catalogues'; the author had contacted meteorite curators, rhino historian Kees Rookmaaker and volcanologists

#### What happened
Jesse Waites, a software developer, published a long post on Oct 8, 2026 about "Antiquity", an open-source toolkit he built with Claude Code to search digitised historical archives. A cheap decision model (Jev) filtered millions of passages for relevant mentions, and Claude Haiku read the survivors in detail. He reports three kinds of finds: a meteorite fall, rhinoceros shipments and volcanic eruptions that are missing from the standard catalogues.

All findings come from the author's own post. He checked them against the original pages but says specialists have not reviewed them yet.

#### Why it matters
It shows how cheap models make it practical for one person to read whole archives (one filtering pass cost about $3), and it produces the kind of result catalogue maintainers can check against the cited pages.

Sources: [Jesse Waites: I Pointed AI at 400 Years of Historical Archives. It Found a Forgotten Meteorite, Lost Rhinos, and Unrecorded Volcanic Eruptions](https://jessewaites.com/blog/post/i-pointed-ai-at-400-years-of-archives/); [GitHub: jessewaites/antiquity](https://github.com/jessewaites/antiquity); [Hacker News discussion (151 points)](https://news.ycombinator.com/item?id=50019056)

### 2026-10-08: Agents in Claude Science build what Anthropic calls the first complete map of the sky in ultraviolet light, filling GALEX's gaps from Gaia and Planck data

**; Anthropic, Johns Hopkins University; Science & math; confidence high; https://postcutoff.com/e/2026-10-08-claude-science-complete-uv-sky-map/

On Oct 8, 2026 Anthropic published a post by Johns Hopkins astrophysicist Brice Ménard (also an Anthropic researcher) describing how many Claude agents working in Claude Science assembled a full-sky ultraviolet map. They combined NASA's GALEX observations (about two-thirds of the sky) with Swift, FIMS/SPEAR and TD-1 data and predicted the missing third from Gaia stars and Planck dust maps. On held-out sky regions the predictions came within about 10% of the real UV measurements. The interactive map is public; no paper or peer review is mentioned.

- Published Oct 8, 2026 on Anthropic's research blog; author Brice Ménard, 'an astrophysicist at Johns Hopkins University and a researcher at Anthropic'
- Inputs: NASA GALEX (about 38,000 observations covering roughly two-thirds of the sky), NASA Swift, the Korean FIMS/SPEAR instrument, the European TD-1 satellite, and ESA's Gaia and Planck catalogs and maps
- Output: an all-sky map in the far-UV (~154 nm) and near-UV (~232 nm) bands; about a third of the sky is 'inpainted' (predicted), using more than 100 million Gaia stars and Planck dust data
- Check: in deliberate occlusion tests on regions with real data, the predicted UV was within about 10% of the measurements; Ménard also inspected the outputs by eye and caught artifacts such as atmospheric (airglow) contamination
- Process: 'many agents working in Claude Science' over several days of iterations, individual runs lasting hours; Ménard gave high-level instructions and reviewed the results. The post names no Claude model version
- Ménard: 'For once, I was able to pursue a project like this without sacrificing time I would otherwise have devoted to research.' Before, 'the only UV map I could show was full of holes'
- The map is published as an interactive viewer at menard.pha.jhu.edu/uvmap; no paper, preprint or peer review is mentioned

#### What happened
Anthropic's research blog published "The missing map of the sky" on October 8, 2026
([Anthropic](https://www.anthropic.com/research/the-missing-map-of-the-sky)). Brice Ménard, an astrophysicist at Johns Hopkins and an
Anthropic researcher, describes a long-wanted project: an ultraviolet map of the whole sky. GALEX, the main UV survey, left about a third
of the sky unobserved. Claude agents in Claude Science merged GALEX with other UV missions and trained a model that predicts UV brightness
from Gaia's catalog of more than 100 million stars and Planck's dust maps, then used it to fill the gaps. When regions with real data were
hidden on purpose, the predictions matched the measurements to within about 10%.

The work took several days of iteration. Ménard gave the direction, checked the outputs and caught problems such as atmospheric glow in
the data; the agents did the rest. The map is online as an interactive viewer.

#### Why it matters
It is a small but concrete example of agentic AI doing the data-heavy "infrastructure" work that scientists often postpone for years.
It is not a discovery in itself, and it has not been through peer review. It is one of a series of Anthropic posts in autumn 2026
showcasing Claude Science on real research problems.

Sources: [Anthropic: The missing map of the sky (Using Claude Science to produce the first complete map of the sky in UV light)](https://www.anthropic.com/research/the-missing-map-of-the-sky); [Interactive UV sky map (Brice Ménard, Johns Hopkins)](https://menard.pha.jhu.edu/uvmap/); [GALEX mission data (MAST archive)](https://archive.stsci.edu/missions-and-data/galex)

### 2026-10-09: Thurston's geometrization conjecture formalised in Lean: the Hamilton–Perelman proof machine-checked with GPT-6, Claude and DeepSeek agents (Chow's group)

****; UC San Diego, Cornell University, Princeton University, Tsinghua University, Amazon AGI, NVIDIA Research, University of Oregon; Science & math; confidence high; https://postcutoff.com/e/2026-10-09-geometrization-conjecture-lean-formalization/

On 9 Oct 2026 Yuan Liao, Ziyang Qin, Jui-Hui Chung, Ayush Khaitan, Zhengyang Zhang, Hongzhou Lin, Zijian Zhang, Ligeng Zhu, Peng Lu and Bennett Chow posted arXiv 2610.13897, a Lean 4 formalization of the Hamilton–Perelman Ricci-flow proof of Thurston's geometrization conjecture (posed 1982, proved by Perelman in 2003). The Lean endpoint `GC.Endpoint.geometrization` depends only on Lean's three standard axioms, with no `sorry`. The formalization was written by AI agents in the "Humanize" builder–reviewer workflow: "We use GPT-6-Astra as the reviewer", and builders were chosen among GPT-6 Luna, GPT-6 Sol, GPT-6 Astra, Claude Opus 5.5 and DeepSeek V4.1 Flash. It extends the group's Poincaré formalization of 27 Sept 2026.

- Paper: 'A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture', arXiv 2610.13897 [math.DG], 9 Oct 2026; Yuan Liao and Ziyang Qin contributed equally. Affiliations: UC San Diego, Cornell, Princeton AI (Princeton University), Tsinghua University, Amazon AGI, NVIDIA Research, University of Oregon
- Theorem formalized: every connected, closed, oriented smooth three-manifold is a connected sum of prime manifolds; each prime factor contains finitely many disjoint, smoothly embedded incompressible tori, and the interior of every piece cut along them carries a complete metric locally isometric to one of the eight Thurston geometries (finite volume in the hyperbolic case)
- Scope stated by the authors: the statement 'does not require the cutting tori to form the canonical JSJ family', does not require the family to be minimal and makes 'no uniqueness assertion'; the prime convention allows S³ as a factor
- Verification (paper): 'successful endpoint and complete-library builds, a separate consumer of the expanded public statement, and transitive axiom checks'; the four public declarations yield the same axiom set '[propext, Classical.choice, Quot.sound]' and 'contain neither sorryAx nor a project-specific axiom'. The authors add that the 'mathematical interpretation also requires review of the definitions and hypotheses'
- AI role (Section 3, 'Agent-assisted formalization with Humanize'): the Humanize RLAR workflow ('Ralph Loop with Agentic Review') separates a builder agent from an independent reviewer. 'We use GPT-6-Astra as the reviewer. For the builder, we choose among GPT-6-Luna, GPT-6-Sol, GPT-6-Astra, Claude-Opus-5.5, and DeepSeek-V4.1-Flash based on their observed generation speed in tokens per second (TPS).' 'Human authors remain responsible for the intended theorem statements, the mathematical interpretation of the formalization, and the final verification claims.'
- Mathematical route: Hamilton's Ricci-flow program and Perelman's completion, following Morgan–Tian and Kleiner–Lott; long-time surgery flow, thick-region hyperbolic pieces, the Kleiner–Lott local-collapse argument for graph-manifold pieces, and Hamilton's incompressibility argument (with Mostow–Prasad rigidity). 'A fuller account of the analytic and topological constructions will appear elsewhere.'
- Code: github.com/qinz1yang/differential-geometry, release v0.1.5 (preview), published 9 Oct 2026, Lean and Mathlib v4.35.0-rc3, Apache-2.0; the release 'adds Thurston's geometrization conjecture for closed, connected, oriented smooth three-manifolds, and three-dimensional Mostow–Prasad rigidity'. The repository had 74 stars on 12 Oct 2026
- Reused sources (paper): the group's Poincaré development, parts of Armstrong and Kempe's DeGiorgi library, Jordan–Schönflies, Classification of Compact Surfaces, Tau Ceti and a canonical-topology development
- Funding and access: Khaitan acknowledges the DARPA expMath program; Chow thanks OpenAI 'for providing access to OpenAI models through the ChatGPT for Academic Researchers program'
- Not stated in the paper: line counts, cost or time taken. Not found as of 12 Oct 2026: independent re-checks, expert reactions, press or a Hacker News thread

#### What happened
Thurston's geometrization conjecture says every closed three-manifold can be cut into pieces that each carry one of eight
standard geometries. It contains the Poincaré conjecture as a special case. Grigori Perelman proved it in 2002–03 by completing
Richard Hamilton's Ricci-flow program, and it took several teams years to write out and check the details.

On 27 Sept 2026 Bennett Chow's group published a Lean formalization of the Poincaré case. On 9 Oct the group, now ten authors
including researchers at Amazon AGI and NVIDIA Research, posted the full geometrization theorem. The Poincaré case needs only
finite-time extinction of the flow. Geometrization needs the flow with surgery for all time, the analysis of the thick
(hyperbolic) and thin (collapsed) parts, and a proof that the cutting tori are incompressible. The short paper reproduces the
Lean statement, explains its exact scope and lists the checks performed.

The Lean code was produced with Humanize, a workflow in which a builder agent writes code in rounds and a separate reviewer agent
checks progress against the plan. The paper names the models, which the Poincaré paper had not done: GPT-6 Astra reviewed, and the
builder was chosen among GPT-6 Luna, GPT-6 Sol, GPT-6 Astra, Claude Opus 5.5 and DeepSeek V4.1 Flash by generation speed.

#### Why it matters
It is one of the deepest theorems of modern geometry, and the proof is far longer than anything formalized before 2026. The paper
is also a data point on method: the proof ideas are human and decades old, but the formal text was written by commercial models
in a builder–reviewer loop. The usual caveat applies. The Lean kernel guarantees the proof of the stated theorem; people still
have to confirm that the definitions and the statement say what mathematicians mean. The authors say so themselves, and their
statement is weaker than the textbook one in stated ways (no canonical JSJ family, no uniqueness).

Sources: [arXiv 2610.13897: A Lean Formalization of the Hamilton–Perelman Proof of Thurston's Geometrization Conjecture](https://arxiv.org/abs/2610.13897); [GitHub: qinz1yang/differential-geometry (release v0.1.5 preview)](https://github.com/qinz1yang/differential-geometry/releases/tag/v0.1.5); [Humanize RLAR workflow (humanfia.ai)](https://humanfia.ai/flows/rlar); [arXiv 2609.33842: the group's Poincaré conjecture formalization](https://arxiv.org/abs/2609.33842)

### 2026-10-09: "We need ABP": six PDE specialists publish GPT-6 Astra's proofs of the kinetic Krylov–Safonov theorem and of Pucci's 1966 conjecture, with full chats (Loher, Mooney, Mouhot, Niebel, Silvestre, Zhu)

****; OpenAI, University of Cambridge, University of Chicago, UC Irvine; Science & math; confidence medium; https://postcutoff.com/e/2026-10-09-kinetic-krylov-safonov-pucci-conjecture-gpt-6-astra/

On 9 Oct 2026 Amélie Loher, Connor Mooney, Clément Mouhot, Lukas Niebel, Luis Silvestre and Yuzhe Zhu posted arXiv 2610.13921, "an exposition of a proof, discovered by Astra, of an Aleksandrov estimate for kinetic equations with rough coefficients". The estimate was the missing step for a Krylov–Safonov Harnack inequality for kinetic (Kolmogorov-type) equations in nondivergence form, which the paper calls "a challenging open problem for a long time". Their site weneedabp.github.io lists six AI source manuscripts with prompts and chats, each marked "Human contribution: None": the kinetic Krylov–Safonov theorem (GPT-6 Astra Ultra, 6 Sept), Pucci's 1966 conjecture on the optimal exponent in the elliptic ABP estimate (22 Sept and 1 Oct) and its parabolic version (5 Oct). The AI manuscripts are unrefereed; the human exposition covers the Aleksandrov estimate.

- Paper: 'An Aleksandrov estimate for kinetic equations with rough coefficients', arXiv 2610.13921 [math.AP], 9 Oct 2026; Amélie Loher, Connor Mooney, Clément Mouhot, Lukas Niebel, Luis Silvestre, Yuzhe Zhu
- Abstract: 'We give an exposition of a proof, discovered by Astra, of an Aleksandrov estimate for kinetic equations with rough coefficients.'
- Paper: 'The proof of the kinetic Krylov–Safonov theorem was found by OpenAI's GPT-6 Astra Ultra on September 6, 2026.'
- Theorem 1.1: for the Kolmogorov operator Ku = ∂_t u + v·∇_x u − tr(A D²_v u) with λI ≤ A ≤ ΛI and merely measurable A, Ku = f ≥ 0 in Q_1 and u = 0 on the incoming boundary imply sup u ≤ C‖f‖_{L^p(Q_1)} for some p(n, λ, Λ) < ∞
- Why it was missing (paper): the expansion-of-positivity and covering steps of Krylov–Safonov were known to extend to kinetic equations; 'the main obstacle to extending the argument has been the lack of an ABP inequality, or a suitable substitute, for equations of Kolmogorov type or equations in Carnot groups'
- Method: by duality the estimate becomes an L log L (entropy) bound for the adjoint equation, obtained 'by regularising along an auxiliary flow and exploiting entropy dissipation' (site); the paper explains it in the elliptic, parabolic and kinetic settings
- Site 'Source material' page: 'Here we collect AI-generated results. We do not claim credit for these results. Anyone could have obtained them by prompting the AI about it.'
- Source 01, kinetic Krylov–Safonov theorem: 6 Sept 2026, OpenAI GPT-6 Astra Ultra, 'Approx. 150% of a Pro 20x plan', 'Human contribution: None'; prompt, chat and manuscript published
- Pucci's conjecture (1966): the ABP estimate sup|u| ≤ C‖L_A u‖_{L^p(B_1)} for trace-one matrices with ellipticity δ holds if and only if p > max{n/2, n[1 − (n−1)δ]}. Site: 'This conjecture was solved by Astra'. Sources 03–04: dimensions d ≠ 3 (22 Sept, GPT-6 Astra Ultra) and d = 3 (1 Oct, GPT-6 Astra); Source 05: parabolic Pucci conjecture (5 Oct)
- Other sources: adjoint L log L estimate on the Heisenberg group and step-2 Carnot groups (10 Sept, Astra Ultra); generalisation to Carnot groups (7 Oct, GPT-6 Pro)
- Remark 1.3 of the paper: 'A similar proof of the Aleksandrov estimate was made public in [8] on September 14, 2026. The precise involvement of AI in their proof was not made public.' ([8] = Z. Hao and X. Zhang, arXiv 2609.15534)
- Open: the optimal exponent in the kinetic case ('kinetic Pucci conjecture'); the authors say they are preparing lecture-note proofs of the full kinetic Krylov–Safonov theorem and an exposition of the Pucci proof
- Hao and Zhang's own paper (arXiv 2609.15534, read 13 Oct) does state an AI role: 'We also acknowledge the use of AI tools (ChatGPT 5.6 Pro) for assistance in collecting relevant literature and for inspiring discussions, in particular for suggesting the use of (Y_t, V_t) in proving Krylov's estimate. In addition, AI tools were used to improve the language and to assist in preparing the two figures.' Their abstract calls the results 'the first Harnack and Hölder regularity theory for kinetic equations with merely measurable coefficients'
- Site as of 13 Oct 2026: the elliptic page carries an 'AI-assisted exposition' of the elliptic Pucci conjecture (added 9 Oct per the site's GitHub history) and one of an ABP estimate for nonlocal equations ('The supplied AI manuscript proves an Aleksandrov estimate for a specific class of even kernels'); the kinetic page says 'We are preparing a complete proof of the kinetic Krylov–Safonov theorem in the style of lecture notes.' No human-written exposition of the Pucci proof and no lecture notes had been posted

#### What happened
The Krylov–Safonov theorem (1979–80) says that solutions of elliptic and parabolic equations in nondivergence form are Hölder
continuous and satisfy a Harnack inequality even when the coefficients are only bounded and measurable. Its proof starts from the
Aleksandrov–Bakelman–Pucci (ABP) maximum estimate. For kinetic equations, where diffusion acts only in the velocity variable and
transport spreads it to space, no ABP estimate was known, and the nondivergence theory was stuck there.

The six authors write that the proof of the kinetic theorem "was found by OpenAI's GPT-6 Astra Ultra on September 6, 2026". The new
idea is to work with the adjoint equation and prove an entropy-type (L log L) bound by regularising along an auxiliary flow. Their
paper explains this argument in the elliptic and parabolic cases first, then for the Kolmogorov equation. The same approach, pushed
further by the model, is reported on their site to settle Pucci's conjecture from 1966 on the best exponent p in the elliptic ABP
estimate, and its parabolic analogue.

The project site is called "We need ABP". It publishes each AI manuscript with its prompt, research instructions, the full chat, the
model, the date and the share of a ChatGPT Pro plan used, and invites others to contribute expositions. A paper by Hao and Zhang
(arXiv 2609.15534, 14 Sept) made a similar proof of the Aleksandrov estimate public; the six authors note that the role of AI in that
proof was not made public. Hao and Zhang's acknowledgments do credit ChatGPT 5.6 Pro with literature search and with suggesting the
process used in their proof of Krylov's estimate, so two groups reached the kinetic estimate in September 2026 with model input.

#### Why it matters
It is a case where recognised specialists (Mouhot is at the University of Cambridge, Silvestre at the University of Chicago, Mooney at UC Irvine) attribute a result in their
own field wholly to a model and refuse credit for it. It also shows a new division of labour: the model produces the proof, and
the experts' paper is an exposition that makes the idea usable by others. The statements about Pucci's conjecture rest on
AI-written manuscripts that the group has not yet published a human account of, so they should be read as claims under review.

#### Not verified here
No independent expert reaction, press coverage or Hacker News discussion was found on 12 or 13 Oct 2026 (web and HN searches; X not searched). The AI manuscripts and chats
were not read for this entry; dates, models and the "Human contribution: None" labels are quoted from the site's source list.

Sources: [arXiv 2610.13921: An Aleksandrov estimate for kinetic equations with rough coefficients](https://arxiv.org/abs/2610.13921); [We need ABP: an open mathematical collaboration (project site)](https://weneedabp.github.io/); [We need ABP: source material (AI manuscripts, prompts, chats, models and dates)](https://weneedabp.github.io/ai.html); [We need ABP: elliptic page (statement of Pucci's conjecture)](https://weneedabp.github.io/elliptic.html); [Hao & Zhang, Harnack estimates for nondivergence kinetic equations (arXiv 2609.15534)](https://arxiv.org/abs/2609.15534); [We need ABP: kinetic page (status of the lecture notes)](https://weneedabp.github.io/kinetic.html)

### 2026-10-09: Smale's sixth problem claimed for five bodies in the plane: finitely many central configurations for all positive masses, AI-found and proved in Lean (Tomar)

****; independent; Science & math; confidence medium; https://postcutoff.com/e/2026-10-09-smale-sixth-problem-five-bodies-lean-tomar/

On 9 Oct 2026 Tejasvi Singh Tomar (independent researcher) posted arXiv 2610.13100: for every choice of five positive masses the planar five-body problem has finitely many central configurations up to similarity. This is Smale's sixth problem for n = 5 in the plane; Albouy and Kaloshin (Annals of Mathematics, 2012) had proved it outside a codimension-two set of masses. The paper says the theorem "is proved in Lean 4 with Mathlib" and is accepted by two independent kernel implementations, and its "Use of AI tools" note reads: "Generative AI tools were used to find and prove the results, to write the programs and the Lean formalization, and to draft the text." Unrefereed; no expert reaction found yet.

- Paper: 'Finiteness of planar central configurations of five bodies for all positive masses', arXiv 2610.13100 [math.DS], 9 Oct 2026; Tejasvi Singh Tomar, 'Independent researcher'; 91 pages
- Theorem 1.1: 'For every m ∈ (R>0)^5 the planar five-body problem with masses m has finitely many central configurations up to similarity.'
- Smale's sixth problem (1998) asks whether the number of central configurations is finite for every choice of positive masses. Known before: n = 3 (five configurations, Euler and Lagrange), n = 4 (Hampton–Moeckel 2006), n = 5 for masses outside a closed algebraic subset of codimension two (Albouy–Kaloshin 2012)
- Method: follows Albouy and Kaloshin's singular-sequence diagrams; determines the possible valuation vectors at all positive masses, excludes diagrams by expanding the potential at a place, then an exact computer-algebra analysis with 'exact rational certificates'
- Special masses: on the rays through (1,4,4,4,4) and (1,1,4,4,4) the complex equations can have a continuum of solutions, so the real configurations are counted directly: 306 and 294 up to orientation-preserving similarity
- Formal proof: 'Theorem 1.1 is proved in Lean 4 with Mathlib … It is also accepted by nanoda and lean4lean, two independent implementations of the Lean kernel.' The paper says re-running the computations from a fresh copy takes about 860 CPU-hours
- AI disclosure ('Use of AI tools'): 'Generative AI tools were used to find and prove the results, to write the programs and the Lean formalization, and to draft the text.' No model is named
- Certificates, checking programs and the Lean project are archived at doi.org/10.5281/zenodo.23202861
- The same author posted a second AI-generated, Lean-checked result the day before: the rank of 3×3 matrix multiplication over F_2 is 23 (arXiv 2610.11823)
- Scope: the plane and five bodies only; Smale's problem for n ≥ 6 stays open
- Earlier paper by the same author: 'Uniqueness of four-body convex central configurations for all masses' (arXiv 2609.35632, 28 Sept 2026, 34 pages, Lean formalization): exactly one strictly convex planar central configuration for each ordering of four positive masses. Abstract: 'This answers Problem 10 in the list of Albouy, Cabral and Santos, which Santoprete calls the Simó-Yoccoz conjecture.' Same disclosure: 'Generative AI tools were used to find and prove the results, to write both programs of Section 5 and the Lean formalization, and to draft the text.'
- Author: the papers say 'Independent researcher'; Tomar's GitHub profile (tejasvi) lists '@Google' and Mountain View, CA (self-edited field, not otherwise confirmed)

#### What happened
A central configuration is an arrangement of point masses in which the gravitational pull on each body points to the common centre
of mass and is proportional to the distance from it. Such arrangements are the shapes that can rotate rigidly or collapse
homothetically. Whether there are only finitely many of them for given masses is the sixth problem on Steve Smale's 1998 list of
problems for the 21st century. Four bodies were settled in 2006. For five bodies in the plane, Alain Albouy and Vadim Kaloshin
proved finiteness in 2012 except for masses in an explicit exceptional set.

Tomar's paper claims to remove that exceptional set. It keeps the Albouy–Kaloshin framework, adds expansions of the potential that
rule out more limiting diagrams, and settles what remains by exact computer algebra with rational certificates. For two special
families of masses, where the complexified equations genuinely have infinitely many solutions, it counts the real configurations
directly.

The author states that generative AI tools found and proved the results and wrote the Lean formalization. The Lean proof of the
main theorem is reported to pass the standard kernel and two independent re-implementations.


### A second paper with the same result (12 Oct)
Zhe Chen and Zhicheng Tong posted [arXiv 2610.16107](https://arxiv.org/abs/2610.16107), "Finiteness of planar central configurations of five bodies for all positive masses: On Smale's sixth problem", with the same theorem. Its comments say the first author submitted an initial version on 6 Oct that "remained on hold", and that "Shortly thereafter, arXiv:2610.13100 appeared reporting identical results". The method keeps Albouy–Kaloshin's classification of singular sequences and replaces their exceptional-set construction by a direct exclusion (finite boundary charts for seven of eight diagrams, an exhaustive order census for Diagram 5). AI statement: "This manuscript was prepared with AI assistance and preliminary automated verification. Rigorous manual verification by the second author is currently underway and will be incorporated into a forthcoming version, alongside a complete formal proof in Lean." Two independent claims of the same result raise the chance it is right, but neither is yet checked by specialists.

#### Why it matters
If it holds, it closes a case of a famous problem with a machine-checked proof produced largely by AI, by someone outside a
university. A Lean proof shifts the question from "is the argument right" to "does the formal statement say what the theorem
says". The paper points to the file with the formal statement so readers can compare, but nobody else is known to have done that
yet.

#### Not verified here
The Lean project was not rebuilt for this entry. The paper does not say which models were used. No reaction from specialists in
celestial mechanics (Albouy, Kaloshin, Moeckel, Hampton were searched by name) was found on 12 or 13 Oct 2026 on the web or
Hacker News; X was not searched. The arXiv paper is still at v1 and the announced companion paper on complex solution sets has not appeared.

Sources: [Chen & Tong: Finiteness of planar central configurations of five bodies (arXiv 2610.16107)](https://arxiv.org/abs/2610.16107); [arXiv 2610.13100: Finiteness of planar central configurations of five bodies for all positive masses](https://arxiv.org/abs/2610.13100); [Zenodo archive: certificates, checkers and Lean project](https://doi.org/10.5281/zenodo.23202861); [Albouy & Kaloshin, Finiteness of central configurations of five bodies in the plane (Annals of Mathematics, 2012)](https://doi.org/10.4007/annals.2012.176.1.10); [arXiv 2609.35632: Uniqueness of four-body convex central configurations for all masses (Tomar)](https://arxiv.org/abs/2609.35632)

### 2026-10-09: Terence Tao's Caltech lecture 'Math 2.0': AI has ended 'proof scarcity', and blind problem-solving is now 'actively harmful' to mathematics

****; UCLA, Caltech; Science & math; confidence high; https://postcutoff.com/e/2026-10-09-tao-math-2-0-caltech-lecture/

On the evening of 9 Oct 2026 Terence Tao gave the public lecture "Math 2.0" at Caltech (AIM Public Lecture Series, Beckman Auditorium) and posted the slides on his blog on 10 Oct. It was his first full talk after OpenAI's 722-manuscript release. He argues that frontier labs have moved mathematics from an era of "proof scarcity" to one of "proof abundance". Problem solving was a good proxy for the field's real goals under scarcity, he says, but "further blind optimization of problem-solving alone is now actively harmful to the long-term health of mathematics". He proposes a "Math 2.0" in which AI strengthens human understanding: open exposition problems, living networks of results, Mathlib-scale formal libraries, ablation studies, and open, interpretable maths models, which he says will be released "very soon".

- Venue: AIM Public Lecture Series, Beckman Auditorium, Caltech, Friday 9 Oct 2026, 7:00–8:30 pm. Announced over a year earlier as 'Machine Assisted Proof' for May; postponed to October and, 'with recent events', substantially rewritten. Caltech's page still carries the old title and abstract
- Slides posted 10 Oct on Tao's blog (math.GM, talk) as math-2-0-caltech-2026.pdf; the content was also added to Tao's 'AI-maintained' living summary of his views on AI
- Core thesis (slides): proofs are objectively verifiable, digitizable and backed by large, high-quality data, so frontier labs 'have thus prioritized advances in mathematical problem-solving capability'; 'we are now entering an era of proof abundance in mathematics'
- 'Increasingly often, no special expertise from the user prompting the AI is required to obtain such solutions.' AI performance is 'extremely jagged: astounding in some directions, while inadequate in others'
- 'Despite the impression given by press releases and social media, most of the problems at the current frontier of mathematics remain unsolved, even with massive computational expenditure.' He calls for assessments that report 'resource consumption and negative results' and cites First Proof (1stproof.org) as one such effort
- Open problems as 'lighthouses': 'Reaching these lighthouses prematurely by automated tools can disrupt the exploration of the paths not taken, and sterilize the surrounding field.' 'Indiscriminate use of AI to solve problems in a non-renewable fashion damages the long-term health and progress of the field'
- Alignment thought experiment: an AI finds a cancer cocktail whose model prediction is 'confirmed in Lean' and that passes a stage 3 trial; would you inject it without 'at least one human cancer expert who understands the mechanism'?
- Examples given of 'Math 2.0' uses: 'open exposition problems' (the revised Caltech event, Nov 13–15); the Integrated Explicit Analytic Number Theory Network (a living 'spreadsheet' of results); Mathlib; ablation studies; exploring 'the neighborhood of a theorem'; 'tailored, open-source math models' ('Some open math models to be released very soon - stay tuned!')
- New experimental mathematics: the Equational Theories Project (2024–25, 22M+ universal-algebra statements settled) and the 2026 SAIR/LMFDB Inverse Galois challenge IGP24 (degree-24 polynomials for all 25,000 possible Galois groups; stage two will extend to degree 31)
- Closing line: 'The primary bottleneck to achieving "Math 2.0" is not technology, or even institutional culture. It is imagination.' Acknowledgements: Nestor Guillen, Bryna Kra, Rachel Ward, Po-Shen Loh; 'AI was used to generate images and diagrams for this talk'
- Coverage: OfficeChai ran 'Terence Tao Says Math 2.0 Must Move Past Problem-Solving in the Age of AI' (10 Oct, per Google News; not read) after an earlier piece on his Mastodon post; The Independent and the Daily Mail (9 Oct) had quoted the Mastodon version (6 Oct), which introduced the Math 1.0 / 2.0 terms

#### What happened
Tao's Caltech public lecture had been booked more than a year earlier under the title "Machine Assisted Proof". It slipped from
May to October and, in his words, "with recent events the talk changed significantly". He gave it on Friday evening, 9 October,
three days after OpenAI posted 722 AI-written manuscripts on GitHub, and posted the slides the next morning. The short blog post
sums it up as three parts: the historical "Math 1.0", "the current state induced by the excessive emphasis on automated solving of
open problems by AI", and a "Math 2.0" in which AI tools "sustain both the internal mathematical community, and the role that
mathematical understanding plays in advancing and aligning the applications of mathematics to the real world".

The slides give the argument in steps. Mathematics was built around proof scarcity. Proofs are objectively checkable (even by Lean),
fully digital and backed by a huge digitized literature, which makes them ideal for machine learning. Frontier labs therefore
invested heavily in maths, and the field is now in an era of proof abundance. Many hard open problems can be solved "in relatively
short amounts of time" given compute and frontier models, often by users with no special expertise. Tao separates this from the
public impression. A slide titled "Reporting bias" mocks the feeling that every problem on social media is being solved by AI, and
he insists most frontier problems remain open "even with massive computational expenditure", asking for reports that include
compute used and failed attempts.

His central claim is about alignment within mathematics. Solving open problems was always a proxy for longer-term goals, mainly
human understanding that can then be carried safely into messier applied problems. The proxy worked under scarcity and "has become
misaligned in the era of proof abundance". Open problems are "lighthouses", and reaching them early with automated tools can
"sterilize the surrounding field". A diagram shows AI short-circuiting the cycle from pure problems to understanding to aligned
applied solutions. A medical thought experiment (a Lean-confirmed, trial-passing cancer cocktail that nobody understands) makes the
point that verification is not the same as understanding.

The constructive half lists "Math 2.0" uses of AI: open exposition problems instead of open research problems (the revised Caltech
event, renamed "Old Problems, New Proofs", Nov 13–15), the Integrated Explicit Analytic Number Theory Network, large formal libraries
such as Mathlib, ablation studies (re-proving a theorem without a key input), mapping the "neighborhood of a theorem", interactive
syntheses of whole theories, and open, interpretable maths models. On the last he adds: "Some open math models to be released very
soon - stay tuned!" He also points to large-scale "experimental mathematics" (the Equational Theories Project, the IGP24 Inverse
Galois challenge). He ends: "The primary bottleneck to achieving 'Math 2.0' is not technology, or even institutional culture. It is
imagination."

#### Why it matters
Tao is the mathematician most often cited on AI and maths, and this is his fullest public statement after OpenAI's release. It turns
his 6 October Mastodon post into a framework that other commentators already use, "proof scarcity" giving way to "proof abundance".
It is neither a boycott call nor a celebration. He accepts the capability jump and calls further problem-solving races harmful, and
he makes specific proposals that institutions can adopt. The "open math models" remark suggests a non-lab, open-source tool release
is coming (no date or details given).

Sources: [Terence Tao: Math 2.0 (blog post, 10 Oct 2026)](https://terrytao.wordpress.com/2026/10/10/math-2-0/); [Slides: Math 2.0 (Caltech, Oct 2026, PDF)](https://teorth.github.io/tao-web/slides/math-2-0-caltech-2026.pdf); [Caltech: AIM Public Lecture Series, Terence Tao (event page, old title 'Machine Assisted Proof')](https://www.caltech.edu/campus-life-events/calendar/aim-public-lecture-series-terence-tao-1); [Terence Tao: living summary of his views on AI (AI-maintained)](https://teorth.github.io/tao-web/ai-views.html); [Terence Tao on Mathstodon (6 Oct): Math 1.0 vs Math 2.0](https://mathstodon.xyz/@tao/117395267721642920); [OfficeChai: Tao hints OpenAI wasn't behaving responsibly, says it is making math less fertile (on the Mastodon post)](https://officechai.com/ai/terence-tao-hints-openai-wasnt-behaving-responsibly-by-simultaneously-releasing-300-math-proofs-says-its-making-math-less-fertile/); [The Independent: Mathematicians are very, very angry at OpenAI. Here's why (9 Oct)](https://www.independent.co.uk/tech/mathematicians-are-very-very-angry-at-openai-here-s-why-b3064273.html)

### 2026-10-09: arXiv, 5–9 October 2026: more conjectures settled with disclosed AI help (Quillen's local parameter conjecture in dimension 4, Laughlin gap, poset reconstruction), next to human-only results

***; OpenAI, Anthropic, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-09-arxiv-ai-disclosed-results-early-october-2026/

A catalogue of smaller arXiv preprints from 5–9 Oct 2026 that claim to settle a named conjecture or open problem, with the AI role as the authors state it. AI-developed proofs: Quillen's local parameter conjecture in dimension four ("ChatGPT and Codex (GPT-6) were used to develop the proof"), reconstruction of finite posets of width three ("discovered by ChatGPT 6 Pro"), and a much better lower bound on the Laughlin gap that improves an OpenAI release result ("completed by the AI model Claude Opus 5.5"). Others used AI for literature search or checking, or state none. All are unrefereed. Larger results of the same week have their own entries.

- Quillen's local parameter conjecture, dimension four (Yunpeng Li, CUHK Shenzhen, arXiv 2610.13469, 72 pages): for a regular local ring R of mixed characteristic and dimension four and a regular parameter r, every finitely generated projective module over R[1/r] is free. Known before when R contains a field (Bhatwadekar–Rao, Popescu) and in dimension ≤ 3 (Gabber). Disclosure: 'OpenAI's ChatGPT and Codex (GPT-6) were used to develop the proof and prepare the manuscript. The human author takes full responsibility for the mathematical arguments and the final manuscript.' Single author, not expert-checked
- Laughlin gap (Qiang Gao, arXiv 2610.12700, 20 pages): raises the constant in OpenAI's Fock-space inequality from 0.0461673… to 0.2190037845…, 'certified by exact rational arithmetic', so the spectral gap above the Laughlin state at filling 1/3 is at least 1/5 instead of 1/25 for large spheres (numerics put the true gap near 0.4). 'The proof of the improvement and the SDP search were completed by the AI model Claude Opus 5.5 (Anthropic) under the author's direction'; the author noticed that OpenAI's certificate 'appeared to be hand-picked rather than obtained by optimization' and asked Claude to set up a semidefinite-programming search. A Lean proof by 'ChatGPT 6.1 Sol' is 'in progress'
- Reconstruction of finite posets of width three (Patrizio Cintioli, University of Camerino, arXiv 2610.12690, 19 pages): every finite ordered set of width at most three and at least four elements is reconstructible from its deck, the width-three part of an open problem of Schröder. 'The main result was discovered by ChatGPT 6 Pro (OpenAI), which was also used in developing the proof and preparing the manuscript. The mathematical proofs were also subjected to an independent adversarial review using Claude Fable 5.1 (Anthropic).'
- Tate conjecture for ordinary Calabi–Yau threefolds (Lukas Brantner, Oxford and CNRS, arXiv 2610.13939, 6 pages): integral geometric Tate conjecture over finite fields via canonical lifts. AI as a literature guide: 'We devised the overall proof strategy. ChatGPT was used to explore the literature and review earlier drafts. The model directed our attention to the work of Patakfalvi and Zdanowicz, which helped us handle the case of divisor classes when b1(X) > 0.'
- Randomized Frank–Wolfe lower bound (Sebastian Pokutta, Zuse Institute Berlin, arXiv 2610.13590): an Ω(L·diam(S)²/T²) lower bound for randomized linear-minimization-oracle methods over strongly convex sets. 'Generative AI tools assisted throughout the research process with formulation and revision of mathematical arguments, implementation of verification and plotting scripts, formalization, and manuscript editing', with 'machine-checked proofs for the results covered by the formalization'
- Light AI use: the biwheel conjecture for braces (Feng and Zhou, Jimei University, arXiv 2610.12955; de Carvalho–Lucchesi–Murty's conjecture that the minimum number of perfect matchings of a brace of order n is (n − 2)²/4; AI tools used 'to assist in discussing proof strategies, checking proofs, and improving the exposition'); counterexamples to Minkowski-type inequalities for modules (Jorge Pérez, arXiv 2610.13888; 'the vast majority of its theoretical results were obtained without the aid of generative AI tools', calculations with GPT-5.6 Sol and Codex)
- A paper about an AI proof: Farzad Aryan's 'On the Imaginary Part of ζ′/ζ' (arXiv 2610.12853) is 'motivated by the recent breakthrough of Anthropic's Claude' on zeta zeros and includes 'An Expository Note on Claude's Argument', explaining how Claude 'replaced the termwise positivity argument by a global linear-algebraic analysis'. Aryan is the author of the 2017 unconditional pair-correlation framework that the Claude result built on
- Same week, no AI disclosure found in the PDF (sweep scan): Pisanski's 1980 conjecture on quadrilateral embeddings of regular bipartite graphs (Metzger and Ulrigg, University of Washington, 2610.13085); counterexamples to the Ambro–Kawamata effective non-vanishing conjecture (Rahul Ajit, 2610.09270); a counterexample to Oxley's circuit–cocircuit intersection conjecture from the extended binary Golay code (Houshan Fu, 2610.06033); the fractional Lane–Emden conjecture (Mei and Wei, 2610.10188); the Mueller–Ho conjecture (2002) on vortex lattices in two-component Bose–Einstein condensates (Luo and Wei, 2610.12203)
- None of these results had press coverage, a Hacker News thread or a public expert assessment that we could find on 12 Oct 2026

#### What happened
The stream of AI-assisted arXiv results described in the summer 2026 catalogue continued in the first full week of October, the
week of OpenAI's 722-manuscript release. This entry lists the smaller items of 5–9 Oct. The larger ones have their own entries:
the Lean formalization of geometrization, the counterexample to Nirenberg's conjecture, near-linear Subset Sum, universality of
random permutations, explicit tensors of border rank 3n and the holomorphic cancellation problem.

Three patterns are visible.

- **Single authors with AI-developed proofs of hard statements.** The dimension-four case of Quillen's local parameter conjecture
  is a 72-page proof in algebraic K-theory by one author from a data-science school, developed with ChatGPT and Codex. Nobody
  else has checked it yet, so confidence is low until specialists comment.
- **Outsiders improving OpenAI's results within days.** Qiang Gao read OpenAI's Laughlin-gap proof, saw that its certificate was
  not optimized, and had Claude Opus 5.5 run a semidefinite-programming search that multiplied the constant by almost five.
- **Honest small roles.** Brantner credits ChatGPT with pointing to one relevant paper. Several authors state that the results
  predate AI tools or used them only for checking and wording.

Human-only papers settling named conjectures appeared in the same week.

#### Why it matters
The catalogue shows what the everyday use of frontier models in research mathematics looked like in October 2026: routine,
disclosed in a sentence or two, and uneven in reliability. Results proved by a model and checked by one person need independent
confirmation before they are cited as theorems.

Sources: [arXiv 2610.13469: Quillen's local parameter conjecture in dimension four](https://arxiv.org/abs/2610.13469); [arXiv 2610.12700: Improving the lower bound on the Laughlin gap: An SDP search](https://arxiv.org/abs/2610.12700); [OpenAI math release: A Fock-space inequality and the Laughlin spectral gap (Lean)](https://github.com/openai/math/tree/main/lean/OAI/Analysis/LaughlinFock); [arXiv 2610.12690: Reconstruction of Finite Posets of Width Three](https://arxiv.org/abs/2610.12690); [arXiv 2610.13939: The Tate Conjecture for Ordinary Calabi-Yau Threefolds](https://arxiv.org/abs/2610.13939); [arXiv 2610.13590: Lower Bounds for Randomized Linear-Oracle Optimization over Strongly Convex Sets](https://arxiv.org/abs/2610.13590); [arXiv 2610.12955: The biwheel conjecture for braces](https://arxiv.org/abs/2610.12955); [arXiv 2610.13888: Counterexamples to Minkowski-type inequalities and the Rees converse for modules](https://arxiv.org/abs/2610.13888); [arXiv 2610.12853: On the Imaginary Part of ζ′/ζ](https://arxiv.org/abs/2610.12853); [arXiv 2610.13085: Resolving Pisanski's Conjecture](https://arxiv.org/abs/2610.13085); [arXiv 2610.09270: Counterexamples to the Ambro–Kawamata effective non-vanishing conjecture](https://arxiv.org/abs/2610.09270); [arXiv 2610.06033: A counterexample to the circuit-cocircuit intersection conjecture](https://arxiv.org/abs/2610.06033); [arXiv 2610.10188: A proof of the fractional Lane–Emden conjecture](https://arxiv.org/abs/2610.10188); [arXiv 2610.12203: A Complete Proof of Mueller–Ho Conjecture](https://arxiv.org/abs/2610.12203)

### 2026-10-09: DeepMind researchers use Gemini Argon to shrink the record Turing machines that encode Goldbach (25→21 states), the Riemann Hypothesis (120→104) and ZF consistency (393→380), with Lean proofs

***; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2026-10-09-deepmind-cryptids-smaller-turing-machines-goldbach-riemann-zf/

A Google DeepMind GitHub repo ("cryptids", created Oct 9, 2026) by Maxence Faldor and Moritz Firsching, "using Gemini Argon", gives smaller 2-symbol Turing machines whose halting is equivalent to a counterexample to Goldbach's conjecture (21 states, from 25), to the falsity of the Riemann Hypothesis (104, from 120) and to an inconsistency of ZF (380, from 393). Each reduction is proved in Lean 4 to halt exactly when the previous record machine does, without sorry, and audited with leanprover/comparator.

- Goldbach: goldbach_21 reduces the 25-state machine (Yao / Leng 2024) by removing 4 states; Lean proof that Halts M ↔ ¬GoldbachConjecture
- Riemann Hypothesis: riemann_104 reduces Joseph Shunia's 120-state machine (Sept 2026, arXiv 2609.30306), itself down from O'Rear's 744 states (2016); Halts M ↔ ¬RiemannHypothesis using Mathlib's definition
- ZF: zf_380 reduces Andrew Wade's 393-state machine (Aug 2026), down from 432 (2025); its halting is equivalent to O'Rear's Hilbert system deriving a contradiction, and the repo proves it never halts in ZFSet
- The repo also has kernel-checked behaviour proofs for some BB(6) holdout machines (potential 'Cryptids', subject to community acceptance)
- All proofs checked by the Lean kernel without sorry, using only the standard axioms; reviewed by Eric Wieser; Lean v4.35.0-rc4
- AI role, as stated in the repo: the machines and Lean semantics were 'developed by Maxence Faldor and Moritz Firsching using Gemini Argon'; no paper or blog post yet
- Context: Yedidia and Aaronson (2016) started this line of work, which bounds where the Busy Beaver function becomes independent of ZF; records are tracked on the bbchallenge wiki

#### What happened
The repo appeared quietly on DeepMind's GitHub on Oct 9 and was found by our sweep's watched-accounts step on Oct 16 (44 stars). It
is a formal-verification package, not an announcement: the AI role is one line in the library README and in the history table, which
credits each new record to "Faldor & Firsching, using Gemini Argon".

#### Why it matters
Smaller machines tighten the known bounds on where the Busy Beaver function stops being computable from standard mathematics. Each
reduction comes with machine-checked proofs, so the new records can be checked without trusting the model or the authors. It is also
one of the first Gemini 4 Argon research results we have seen.

Sources: [GitHub: google-deepmind/cryptids (Some Cryptids and Lean proofs)](https://github.com/google-deepmind/cryptids); [cryptids: Cryptids/README.md (records table and provenance)](https://github.com/google-deepmind/cryptids/blob/main/Cryptids/README.md); [bbchallenge wiki: Cryptids](https://wiki.bbchallenge.org/wiki/Cryptids)

### 2026-10-09: Erdős Problem #32 ($50 part): additive complements to the primes cannot have size O(log N), using a sieve weight from OpenAI's AI-generated prime-gaps proof

***; HSE University, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-09-erdos-32-additive-complements-primes-radomskii/

In an arXiv preprint posted on 9 Oct 2026, Artyom Radomskii (HSE University) proves that every set B for which every large integer is a prime plus an element of B has at least c·log X·log log X elements up to X. This answers negatively the question Erdős attached a $50 prize to (Erdős Problem #32: can O(log N) be achieved?) and proves a quantitative form of Ruzsa's Conjecture 2. The key lemma comes from his 29 Sep paper, which adapts the "truncated divisor-sum" weight construction of OpenAI's AI-generated "Improved Long Gaps Between Primes" preprint. Neither paper has an AI-use statement; the AI link is the borrowed method. Unrefereed; erdosproblems.com had not updated the problem by 15 Oct.

- Paper: A. Radomskii, 'A lower bound for additive complements to the primes', arXiv 2610.14127 [math.NT], 9 Oct 2026; single author, HSE University Basic Research Program
- Result (Corollary 1): every asymptotic additive complement B to the primes satisfies B(X) ≥ c·log X·log log X for an absolute c > 0. 'Corollary 1 gives a negative answer to the O(log X) question in Problem 32'
- Not settled: the paper says its bound 'does not decide the separate question whether a complement can satisfy B(X) = o((log X)^2)', the main question of #32, and does not prove Ruzsa's density-1 Conjecture 1
- Previous knowledge (erdosproblems.com/32): Erdős (1954) built complements with B(N) ≪ (log N)^2; Ruzsa proved liminf B(N)/log N ≥ e^γ ≈ 1.781; the site records that Erdős offered $50 for deciding whether O(log N) is possible
- AI lineage: the analytic input is the 'large-shift lemma' (Lemma 3.1) of Radomskii's 'Large gaps between Romanoff numbers' (arXiv 2609.38408, 29 Sep), whose proof 'adapts the truncated divisor-sum construction of [2]', where [2] is 'OpenAI. Improved Long Gaps Between Primes', github.com/openai/LongGapsBetweenPrimes, the GPT-6 Astra write-up of the GPT-5.6 prime-gap result
- The Romanoff paper itself improves the Kalmynin–Konyagin bound for the longest block of integers in [1, X] not of the form p + 2^n to ≫ log log X, 'using the weight construction in [5, §4]' (the same OpenAI preprint)
- AI disclosure: none in either PDF (acknowledgements thank only the HSE program); this entry does not claim the author used an AI model
- Status: preprint, not refereed; erdosproblems.com/32 still showed 'Proof claims (0)' and a last edit of 23 Jan 2026 on 15 Oct; no Hacker News thread found

#### What happened
Erdős Problem #32 asks how thin a set A of non-negative integers can be if every large integer is a prime plus an element
of A. Erdős showed in 1954 that about (log N)^2 elements up to N are enough. He asked whether o((log N)^2) or even O(log N) is
possible, and offered $50 for the O(log N) question (erdosproblems.com, citing Guy's *Unsolved Problems*). Ruzsa proved
that at least e^γ·log N elements are needed and conjectured that the ratio to log N must tend to infinity.

On 9 Oct 2026 Artyom Radomskii of HSE University posted an eight-page paper proving that at least c·log X·log log X elements are needed.
This answers the O(log N) question negatively and proves Ruzsa's Conjecture 2 in a quantitative form. Given a candidate
set B, he constructs a block of consecutive integers in [X/2, X] that avoids every p + b. The analytic input is a
"large-shift lemma" from his 29 Sep paper on Romanoff numbers (integers of the form p + 2^n), where it improves a bound of
Kalmynin and Konyagin.

That lemma is where the AI connection lies. In the Romanoff paper Radomskii writes that he uses "the weight construction in
[5, §4]", and in the new paper that the lemma's proof "adapts the truncated divisor-sum construction of [2]". Both
references are OpenAI's "Improved Long Gaps Between Primes", the GPT-6 Astra write-up of the prime-gap improvement that a
user generated with GPT-5.6 on erdosproblems.com in August 2026. Neither of Radomskii's papers says that he used an AI model.

#### Why it matters
This is a second-generation effect: a technique that first appeared in an AI-generated proof is picked up by a human
specialist and settles part of a different Erdős problem within weeks. The main question of #32, whether o((log N)^2) is
achievable, is still open. The result is an unrefereed preprint and had not been reviewed on erdosproblems.com by 15 Oct.

Sources: [arXiv 2610.14127: Radomskii, A lower bound for additive complements to the primes](https://arxiv.org/abs/2610.14127); [arXiv 2609.38408: Radomskii, Large gaps between Romanoff numbers](https://arxiv.org/abs/2609.38408); [Erdős Problem #32 (erdosproblems.com)](https://www.erdosproblems.com/32); [OpenAI: Improved Long Gaps Between Primes (GitHub)](https://github.com/openai/LongGapsBetweenPrimes)

### 2026-10-09: Nirenberg's 1962 conjecture on closed asymptotic curves disproved; Claude and ChatGPT found the first, polygonal counterexample (Banks, Ghomi, Qiu, Raffaelli)

***; Georgia Institute of Technology, Anthropic, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-09-nirenberg-conjecture-closed-asymptotic-curves-counterexample/

On 9 Oct 2026 Harrison Banks, Mohammad Ghomi, Hongda Qiu and Matteo Raffaelli posted arXiv 2610.13848. It constructs a smooth embedded annulus in R³ with negative curvature, bounded by two convex curves in tangent planes, that contains two closed asymptotic curves. This contradicts a conjecture Louis Nirenberg made in 1962. The acknowledgement says the work was "partly carried out with AI tools Claude (Anthropic) and ChatGPT (OpenAI)": on a discretized version with six-sided polygonal curves, "the AI models found a counterexample", which the authors smoothed and extended. Unrefereed preprint.

- Paper: 'A counterexample to Nirenberg's conjecture on closed asymptotic curves', arXiv 2610.13848 [math.DG], 10 pages, 3 figures, dated 6 Oct and posted 9 Oct 2026
- Theorem 1.1: 'There exists a C∞ embedded annulus M ⊂ R³ whose interior has negative Gauss curvature and contains two distinct closed asymptotic curves. The boundary components of M are strictly convex curves in parallel planes, each tangent to M along the corresponding boundary component.'
- The conjecture: in his program to extend Alexandrov's 1938 rigidity theorem for analytic tight surfaces to smooth surfaces, Nirenberg 'conjectured that a negatively curved annulus bounded by convex curves in tangent planes cannot contain a closed asymptotic curve'. He had proved rigidity assuming at most one closed asymptotic curve per component
- Method: Lelieuvre's classical formula yields a closed curve, realized as an asymptotic curve on a ruled annulus; the second closed asymptotic curve comes from the periodic solution of a Riccati equation. 'All computations in this paper have been carried out and verified in a Mathematica notebook'
- AI disclosure (Acknowledgement): 'This paper is the culmination of six months of work, partly carried out with AI tools Claude (Anthropic) and ChatGPT (OpenAI). Initially we investigated the weak form of the conjecture … We discretized the problem and considered polygonal curves with 6 sides. The AI models found a counterexample of this form, which we then smoothed and extended to a belt, and after a perturbation to an annulus by numerical means. Later, on a separate track, we investigated the conjecture via Lelieuvre's formula … The authors have checked all AI assisted content and take full responsibility for this manuscript.'
- Model versions are not given. Not found as of 12 Oct 2026: expert reactions or press coverage

#### What happened
Asymptotic curves are the directions along which a saddle-shaped surface does not bend. In 1962 Louis Nirenberg, trying to prove
that smooth "tight" surfaces are rigid, needed to exclude closed asymptotic curves in the negatively curved parts of a surface,
and conjectured that they cannot occur. Banks, Ghomi, Qiu and Raffaelli show that they can: their annulus contains two.

The acknowledgement describes two tracks. First the authors discretized a weaker form of the conjecture into polygons with six
sides, and Claude and ChatGPT found a polygonal counterexample, which the authors smoothed and extended numerically. Later they
found the explicit construction via Lelieuvre's formula that the paper presents.

#### Why it matters
It settles a named conjecture of a major analyst after 64 years, in the negative. The AI role is specific and limited: the models
supplied the first evidence that the conjecture was false, and the published proof is a human, classical construction. The paper
does not say which model versions were used.

Sources: [arXiv 2610.13848: A counterexample to Nirenberg's conjecture on closed asymptotic curves](https://arxiv.org/abs/2610.13848)

### 2026-10-09: Random permutations of quadratic length contain every pattern: Alon's 1000n² problem solved, 'the proof was found by GPT-6 Pro' (Chen, Wang)

***; Beijing Institute of Technology, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-09-random-permutations-universal-alon-gpt-6-pro/

On 9 Oct 2026 Yu Chen and Yiting Wang posted arXiv 2610.13530: a uniformly random permutation of length 32n² contains every permutation of length n as a pattern with probability at least 1 − exp(−cn/log n). This removes the log log n factor from He and Kwan's bound and solves Noga Alon's problem of showing that 1000n² suffices. The abstract ends: "The proof was found by GPT-6 Pro", and the declaration adds "The authors do not claim credit for the proof ideas." Alon's sharper conjecture, with constant 1/4, remains open. Unrefereed preprint.

- Paper: 'Quadratic-size Random Permutations are Universal', arXiv 2610.13530 [math.CO], 9 Oct 2026; Yu Chen (Beijing Institute of Technology) and Yiting Wang (supported by the ERC grant RANDSTRUCT)
- Theorem 1.1: there are absolute constants c > 0 and n₀ such that for n ≥ n₀ a uniformly random permutation of length 32n² fails to be n-universal with probability at most exp(−cn/log n)
- Background (paper): Alon conjectured that length (1 + ε)n²/4 suffices, observed that O(n² log n) does, 'and posed the weaker problem of proving that 1000n² suffices'; He and Kwan proved 2000n² log log n. Length below about n²/4 cannot work, because of the longest increasing subsequence
- What remains open (paper): 'the sharp constant 1/4 predicted by Alon's conjecture remains open'; the authors sketch how to lower 32 to any constant above 1 and why 1 is a barrier for the method
- Declaration on the usage of generative AI: 'The proof was found by GPT-6 Pro. The authors do not claim credit for the proof ideas. They have verified and digested the proof and take full responsibility for the correctness of the manuscript.'
- Not found as of 12 Oct 2026: reactions from Alon, He or Kwan, or other expert comment

#### What happened
A permutation is n-universal if every ordering of n items appears inside it as a pattern. Any universal permutation needs length
of order n², and Noga Alon asked whether a random permutation of length a constant times n² is already universal. Xiaoyu He and
Matthew Kwan came within a log log n factor. Chen and Wang's paper removes that factor, so the order of magnitude is now known.

The authors state plainly that the model did the mathematics: GPT-6 Pro found the proof, and their own contribution was to
verify, digest and write it. The prompt and the time taken are not given.

#### Why it matters
It is a clean case of a model resolving a problem that specialists had posed and partly solved, with authors who attribute the
ideas to the model in the abstract. The problem is of interest to specialists more than a famous conjecture, and the result is
not sharp.

Sources: [arXiv 2610.13530: Quadratic-size Random Permutations are Universal](https://arxiv.org/abs/2610.13530)

### 2026-10-09: Subset Sum in O(n + N^(1+ε)) time claimed: Zhejiang team designed the route, GPT Astra Ultra wrote the complete proofs (Chen, Mao, Zhang)

***; Zhejiang University, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-09-subset-sum-near-linear-time-gpt-astra-ultra/

On 9 Oct 2026 Lin Chen, Yuchen Mao and Guochuan Zhang (Zhejiang University) posted arXiv 2610.12953: Subset Sum on integers of absolute value at most N can be decided deterministically in O_ε(n + N^(1+ε)) time for every fixed ε > 0, and d-dimensional subset sum in O(n + N^(d+ε)), giving a feasibility test for integer programs with d constraints. The paper's "AI disclaimer" says the authors designed the main proof route and "GPT Astra Ultra was used to turn the insights of the authors into complete mathematical proofs"; GPT then generalized the one-dimensional randomized algorithm to the deterministic multidimensional one "without any guidance from the authors". Unrefereed preprint.

- Paper: 'Subset Sum and Integer Programming Feasibility Test in Near-Optimal Time', arXiv 2610.12953 [cs.DS], 9 Oct 2026; Lin Chen, Yuchen Mao, Guochuan Zhang (Zhejiang University); 105 pages
- Abstract: 'A long open problem is whether Subset Sum can be solved in O(N) time. We show that Subset Sum can be solved in O(N^(1+ε)) time where ε > 0 is an arbitrarily small constant.'
- Corollary 1.2: subset sum for an explicitly listed multiset in [−N, N] 'can be decided deterministically in O_ε(n + N^(1+ε)) word-RAM time, with unrestricted multiplicities'. Theorem 1.1 gives O_{d,ε}(n + N^(d+ε)) for vectors in dimension d
- Earlier target-independent bounds in the paper's table: Õ(n + N^(5/3)) (Polak–Rohwedder–Węgrzycki 2021) and Õ(n + N^(3/2)) (Chen–Lian–Mao–Zhang 2024; Jin 2023); Bringmann's 2017 algorithm runs in Õ(n + t), where the target t can be as large as nN
- Idea (abstract): dense inputs with the target inside the zonotope were already known to be easy. 'Our key observation is that a sparse input can exhibit a dense structure when lifted to higher dimension'; the thin region near the zonotope's boundary is split into subproblems and solved recursively
- AI disclaimer: 'The main proof route for 1D subset sum was designed by the authors … GPT Astra Ultra was used to turn the insights of the authors into complete mathematical proofs. During the proof, GPT encounters trouble in proving several lemmas, which was eventually resolved by taking ideas from the authors.'
- AI disclaimer, continued: 'After obtaining the randomized near-linear time algorithm for 1D subset sum, GPT was able to generalize it directly to deterministic algorithms for multidimensional multiset without any guidance from the authors. GPT was used for drafting the paper. GPT also verified this papers main results using the Lean 4 proof.' A research note with the conversations is linked (github.com/chenlin198662/SSP-note)
- Caveats: the Lean proof is described as produced by GPT and is not linked from the abstract page; no independent check or expert reaction was found as of 12 Oct 2026

#### What happened
Subset Sum asks whether some of n given integers add up to a target. It is NP-complete, but it can be solved in time polynomial
in the size of the numbers. A long line of work asked how small the dependence on the largest number N can be, with exponents
falling from 2 to 5/3 to 3/2. Chen, Mao and Zhang, who wrote several of those earlier papers, now claim an exponent arbitrarily
close to 1, which is essentially optimal, and extend it to several dimensions and to integer programs with a fixed number of
constraints.

The paper is explicit about the division of labour. The authors supplied the structure of the argument. GPT Astra Ultra wrote out
the proofs, got stuck on several lemmas that the authors resolved, and then generalized the algorithm to the deterministic,
multidimensional, multiset case on its own. The conversations are published as a research note.

#### Why it matters
It is a central problem of pseudopolynomial algorithms, claimed by a group with a track record on it. It also shows a working
pattern that is becoming common in theoretical computer science: experts provide the plan, a model produces a very long, detailed
proof. A 105-page AI-written proof is hard for referees to check, so the claim should be treated as open until others verify it.

Sources: [arXiv 2610.12953: Subset Sum and Integer Programming Feasibility Test in Near-Optimal Time](https://arxiv.org/abs/2610.12953); [Research note: conversations between GPT and the first author (chenlin198662/SSP-note)](https://github.com/chenlin198662/SSP-note/blob/main/SSP-research%20note.pdf)

### 2026-10-09: AI weather models forecast Dubai's unprecedented 2024 storm by 'translocation', not extrapolation (University of Chicago, Science Advances)

**; University of Chicago; Science & math; confidence high; https://postcutoff.com/e/2026-10-09-ai-weather-models-gray-swan-dubai-2024/

A Science Advances paper (vol. 12, issue 41, published 9 Oct 2026) by Y. Qiang Sun, Pedram Hassanzadeh and colleagues tested GraphCast, AIFS and FuXi on the April 2024 Dubai storm, which had twice the highest rainfall in the training data for that region. GraphCast and AIFS forecast it accurately up to 8 days ahead; FuXi underestimated the rain. The authors attribute the success to "translocation", learning from similar or stronger events elsewhere on the globe, and find no evidence that the models extrapolate from weaker events. Events in the tail of the global distribution are still forecast poorly.

- Paper: 'Predicting regional gray swans via translocation: AI weather models and Dubai's unprecedented 2024 rainfall', Science Advances 12(41), doi 10.1126/sciadv.ady9406; Y. Qiang Sun, Pedram Hassanzadeh, Tiffany Shaw, Hamid A. Pahlavan, Adam Marchakitus (Department of the Geophysical Sciences, University of Chicago)
- Abstract: the storm 'had twice the training set's highest rainfall in that region. GraphCast and AIFS accurately forecast this event up to 8 days ahead. FuXi forecasts the event but underestimates the rainfall.'
- Mechanism (abstract): fine-tuning and receptive-field analyses 'suggest that these models' success stems from "translocation": learning from comparable/stronger dynamically similar events in other regions during training. Evidence of "extrapolation" (learning from weaker events) is not found.'
- Limit (abstract): 'Even events within the global distribution's tail are poorly forecasted, which is not only due to data imbalance (generalization error) but also spectral bias (optimization error).'
- Written from the abstract (Crossref); the full text was not read

#### What happened
A common worry about data-driven weather models is that they cannot predict events more extreme than those they were trained on
("gray swans"). The Chicago group examined a real case: the 2024 Dubai rainfall, far beyond the regional record in the training
set. Two of three leading models forecast it well more than a week ahead. The analysis indicates that the models reused what they
had learned from comparable storms in other parts of the world.

#### Why it matters
It separates two kinds of extremes. Regionally unprecedented events can be forecast if something similar happened elsewhere.
Globally unprecedented ones remain a weakness, and the paper traces part of it to how the networks are optimized, which suggests
where to improve them.

Sources: [Science Advances: Predicting regional gray swans via translocation](https://www.science.org/doi/10.1126/sciadv.ady9406)

### 2026-10-09: Machine-learning-guided evolution makes a plant Rubisco 29% faster (MIT, Stanford, ANU; Science Advances)

**; Massachusetts Institute of Technology, Stanford University, Australian National University; Science & math; confidence high; https://postcutoff.com/e/2026-10-09-ml-assisted-directed-evolution-plant-rubisco/

A paper in Science Advances (vol. 12, issue 41, published 9 Oct 2026) used the structure-informed protein model ESM-IF1 to choose which positions of tobacco Rubisco's large subunit to mutate, then selected variants in Rubisco-dependent E. coli. The best single change, T391I, raised the carboxylation rate by 29% and aerobic carboxylation efficiency by 43%. Several improved variants carry amino acids not seen in plant evolution. Rubisco is the enzyme that fixes CO₂ in photosynthesis and has long resisted engineering.

- Paper: 'Machine learning–assisted directed evolution of plant Rubisco', Science Advances 12(41), doi 10.1126/sciadv.aeh3246; authors Julie L. McDonald, Jiacheng Lin, Yunlong Zhao, Brian L. Hie, Rosemary Birch, Mary Gehring, Bryan D. Bryson, Spencer M. Whitney, Matthew D. Shoulders, Robert H. Wilson (MIT; Stanford; Australian National University)
- Method (abstract): 'We applied the structure-informed machine learning (ML) model ESM-IF1 to identify plausible amino acid sites in the large subunit of Nicotiana tabacum Rubisco to target for directed evolution', followed by selection in Rubisco-dependent Escherichia coli
- Result (abstract): 'the T391I substitution improved carboxylation rate by 29% and aerobic carboxylation efficiency by 43%'
- Abstract: several improved variants 'carried amino acid changes not found in the evolutionary lineage of plants, despite being assembly competent in plant chloroplasts'
- Limits: the abstract reports enzyme kinetics, not crop yield; it speaks of 'potential for enhancing crop productivity'. Written from the abstract (Crossref); the full text was not read

#### What happened
Rubisco catalyses the step of photosynthesis that captures CO₂. It is slow and inefficient and, in the words of the abstract, "has proven challenging to engineer".
The MIT-led team used ESM-IF1, an inverse-folding protein model, to pick positions in the tobacco enzyme
where changes were plausible, built libraries at those positions and let bacteria that depend on Rubisco select the better
variants. Some winning mutations do not occur in any plant lineage.

#### Why it matters
"A better Rubisco" is on the list of grand challenges for biology (see the related entry). This is a peer-reviewed, lab-validated
step on a plant enzyme, using a small specialised model as a guide for experiments. It is an enzyme-level gain, not yet a
demonstrated gain in crop yield.

Sources: [Science Advances: Machine learning–assisted directed evolution of plant Rubisco](https://www.science.org/doi/10.1126/sciadv.aeh3246)

### 2026-10-10: Vaught's conjecture fails for infinitary logic: Ackerman, Freer and Knight post a Lean-verified counterexample, built over 8 months with Claude Code and Codex

****; Harvard University, MIT, University of Oxford, Anthropic, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-10-vaught-conjecture-infinitary-counterexample-lean/

arXiv 2610.15217 (submitted 10 Oct 2026) by Nathanael Ackerman (Harvard), Cameron Freer (MIT) and Robin Knight (Oxford) constructs a sentence of L_ω1,ω with exactly ℵ1 countable models up to isomorphism but no perfect set of non-isomorphic models, which disproves Vaught's conjecture for infinitary logic. The construction follows Knight's framework (his 2002 counterexample claim never won general acceptance). The paper says the full argument is "certified in Lean", checked by Comparator and registered on the Palomar Registry. Freer spent 8 months formalizing it with coding agents (Claude Code and Codex, Opus 4.5–5.5, Fable 5/5.1, GPT 5.2–6). The Lean certificate is the main check so far; the preprint is unrefereed.

- Paper: 'A counterexample to Vaught's Conjecture for L_ω1,ω', Nathanael L. Ackerman, Cameron E. Freer, Robin W. Knight, arXiv 2610.15217 [math.LO], 26 pages, submitted 10 Oct 2026
- Result: a sentence of L_ω1,ω with exactly ℵ1 countable models up to isomorphism and no perfect set of non-isomorphic models, so the number of countable models is neither countable nor 2^ℵ0 in the strong (perfect-set) sense
- Scope: the paper disproves Vaught's conjecture 'for infinitary logic'; it does not claim the original first-order case (Knight's 2002 manuscript claimed a first-order counterexample)
- History (paper, §1.2): Knight announced a first-order counterexample in 2002; despite a Notre Dame conference devoted to it, the argument 'did not achieve general acceptance as a correct proof, although no definitive flaw in the construction had been identified'
- AI use: in Feb 2026 Ackerman and Freer proposed 'to use AI coding agents to help formalize his proof in Lean'. 'Over the following 8 months, Freer worked extensively with coding agents (Claude Code and Codex with models ranging from Opus 4.5 through 5.5, Fable 5 and 5.1, and GPT 5.2 through 6) and occasional help from GPT Pro (5.2 through 6)'
- Not a routine formalization: Knight's object needed 'substantial revisions' to satisfy his conditions; 'Closing this gap involved a great deal of human mathematical direction and AI-assisted exploration'. Finding the template system 'required a great deal of work using coding agents to interact with LLMs, Lean, and SMT and Horn clause solvers'
- Verification: 'a formally verified counterexample … with the entire argument certified in Lean'; one version checked by Comparator and registered as PALOMAR-2026-10-07-000001; a cleaner version on GitHub (cameronfreer/vaught-conjecture-counterexample, created 30 Sep 2026); the paper notes that 'not every line of this exposition has been formally verified'
- Tools: Freer's lean4-skills pack for AI coding agents (Oct 2025), the Lean LSP MCP server, and Freer's infinitary-logic Lean library

#### What happened
On 10 Oct 2026 Nathanael Ackerman, Cameron Freer and Robin Knight submitted [arXiv 2610.15217](https://arxiv.org/abs/2610.15217) (listed 13 Oct). It gives a sentence of the infinitary logic L_ω1,ω with exactly ℵ1 countable models up to isomorphism and no perfect set of non-isomorphic models. Vaught's conjecture (1961) says this cannot happen, so the paper disproves its infinitary version. The original first-order conjecture is not claimed.

The paper's history section explains how it came about. In 2002 Knight announced a first-order counterexample. The community tried hard to check it, including a conference at Notre Dame, but the proof "did not achieve general acceptance", although nobody found a definitive flaw. In February 2026 Ackerman and Freer suggested formalizing Knight's newer, clearer infinitary draft in Lean with AI coding agents. Freer then worked for 8 months with Claude Code and Codex (Opus 4.5 to 5.5, Fable 5 and 5.1, GPT 5.2 to 6, plus occasional GPT Pro). The draft's combinatorial object turned out to need "substantial revisions"; closing the gap took "a great deal of human mathematical direction and AI-assisted exploration", and SMT and Horn-clause solvers were used near the end. The template system in the paper is a simplified version of the first one found.

The formal proof is on the [Palomar Registry](https://palomar-registry.org/entry?id=PALOMAR-2026-10-07-000001&version=1) (Comparator-checked) and in a [GitHub repository](https://github.com/cameronfreer/vaught-conjecture-counterexample) with a cleaner API. The authors note that the written exposition and the Lean development are organized differently in places.

No expert reactions or press coverage were found on 14 Oct (web search; X not searched directly).

#### Why it matters
Vaught's conjecture is one of the best-known open problems in model theory. Here the human idea is two decades old; what changed is that coding agents made a full machine check practical, and the check forced repairs that the informal argument had missed. It is a clear case of AI-assisted formalization settling a disputed human proof rather than finding a new one. Whether the method extends to the first-order case is open.

Sources: [arXiv 2610.15217: A counterexample to Vaught's Conjecture for L_ω1,ω](https://arxiv.org/abs/2610.15217); [Palomar Registry entry PALOMAR-2026-10-07-000001](https://palomar-registry.org/entry?id=PALOMAR-2026-10-07-000001&version=1); [GitHub: cameronfreer/vaught-conjecture-counterexample (Lean 4)](https://github.com/cameronfreer/vaught-conjecture-counterexample); [GitHub: cameronfreer/lean4-skills](https://github.com/cameronfreer/lean4-skills); [Wikipedia: Vaught conjecture (background, Knight's claim)](https://en.wikipedia.org/wiki/Vaught_conjecture)

### 2026-10-10: Erdős Problem #1117 (Hayman's Problem 2.16) answered in the negative with ChatGPT and Codex (GPT-6): maximum-modulus points cannot tend to infinity

***; OpenAI, The Chinese University of Hong Kong, Shenzhen; Science & math; confidence medium; https://postcutoff.com/e/2026-10-10-erdos-problem-1117-maximum-modulus-gpt-6-codex/

In an arXiv preprint (v1 Oct 10, 2026; v2 Oct 14) Yunpeng Li (CUHK Shenzhen) answers the second, still-open question of Erdős Problem #1117, recorded as Problem 2.16 in Hayman's problem book: for a non-monomial entire function, the number of points of maximum modulus on |z| = r cannot tend to infinity. He proves the count exceeds 2k only on a locally finite set of radii (k fixed by the first two nonzero Taylor terms) and that 2k is sharp. The paper states that "OpenAI's ChatGPT and Codex (GPT-6) were used to develop the proof".

- Paper: arXiv 2610.15067 [math.CV], 'Maximum-modulus points on concentric circles', single author Yunpeng Li, School of Data Science, CUHK Shenzhen; v1 Sat 10 Oct 2026 (16 KB), v2 Wed 14 Oct 2026 (14 KB; v2 replaces a 'global generic-degree argument of version 1' with a local one)
- Result: write f(z) = z^m g(z), g(0) ≠ 0, and let k be the first positive exponent with a nonzero Taylor coefficient of g; then {r > 0 : ν_f(r) > 2k} is locally finite in (0, ∞), the constant 2k is sharp, and lim inf ν_f(r) ≤ 2k < ∞ with no growth assumption
- Problem history (paper and erdosproblems.com/1117): Erdős asked around 1964 whether lim sup or lim inf of ν_f(r) can be infinite; Hayman's problem list (Problem 2.16) records it; Herzog and Piranian answered the lim sup question yes in 1968; Glücksam and Pardo-Simón gave an 'approximate' affirmative answer to the lim inf question; the second question is also attributed to Clunie
- AI disclosure (paper, verbatim): 'Assistance disclosure. OpenAI’s ChatGPT and Codex (GPT-6) were used to develop the proof and prepare the manuscript. The human author takes full responsibility for the mathematical arguments and the final manuscript.'
- Status on Oct 17, 2026: erdosproblems.com/1117 (last edited 29 Dec 2025) still lists the second question as open and shows one 'proof claim'; no formalization; unrefereed
- Same author earlier posted a 72-page preprint on Quillen's local parameter conjecture in dimension four (arXiv 2610.13469), listed in the 2026-10-09 catalogue

#### What happened
Yunpeng Li of the School of Data Science at CUHK Shenzhen posted a short paper (v1 on Oct 10, 2026, revised Oct 14) that answers the
second question of Erdős Problem #1117. For an entire function f that is not a monomial, let ν_f(r) count the points on the circle |z| = r
where |f| reaches its maximum. Herzog and Piranian showed in 1968 that ν_f(r) can be unbounded along some radii. Erdős's second question,
whether a single function can have ν_f(r) → ∞ on all large circles, stayed open. Li shows the answer is no: the count is bounded by 2k,
where k is the gap between the first two nonzero Taylor terms of f, except on a locally finite set of radii, and the bound is sharp.

The paper's closing disclosure says ChatGPT and Codex running GPT-6 were used "to develop the proof and prepare the manuscript".
It does not say which steps the models found.

#### Why it matters
It is another decades-old Erdős question (the list carries Hayman's 1967/1974 problem-book provenance) answered in a preprint whose author
credits a GPT-6 model with developing the proof. As with most such claims this month, it is unrefereed and erdosproblems.com has not yet
updated the problem's status.

Sources: [arXiv 2610.15067: Maximum-modulus points on concentric circles (Yunpeng Li)](https://arxiv.org/abs/2610.15067); [Erdős Problems #1117 (T. F. Bloom)](https://www.erdosproblems.com/1117)

### 2026-10-11: William Banks writes out OpenAI's Landau–Siegel proof with explicit constants: (1 − β) log q ≥ 0.003 for every real character, and says he verified it in full

****; OpenAI, Anthropic, University of Missouri; Science & math; confidence medium; https://postcutoff.com/e/2026-10-11-landau-siegel-zeros-banks-explicit-account/

arXiv 2610.15483 (11 Oct 2026) by number theorist William D. Banks gives "a detailed account" of the argument in OpenAI's 1 Oct manuscript claiming that Landau–Siegel zeros do not exist, as a direct proof with explicit constants: for every primitive real character χ of conductor q ≥ 10¹⁰, a real zero β of L(s, χ) satisfies (1 − β) log q ≥ 3.0 × 10⁻³, and with a computation of Lu, Zaman and Zhao this holds for every q ≥ 3. The argument is credited to OpenAI; Claude produced the first report and expanded draft; Banks says he "verified the mathematical arguments in their entirety". It is an unrefereed preprint but the first independent expert write-up of the claim.

- Paper: 'On the OpenAI proof that Landau-Siegel zeros do not exist', William D. Banks, arXiv 2610.15483 [math.NT], 19 pages, submitted 11 Oct 2026 (written at the Max Planck Institute for Mathematics, Bonn)
- Theorem: absolute q₀ = 10¹⁰ and δ₀ = 3.0 × 10⁻³ such that every real zero β ∈ (0,1) of L(s, χ), χ primitive real of conductor q ≥ q₀, satisfies (1 − β) log q ≥ δ₀; with Lu–Zaman–Zhao's computation the bound holds for all q ≥ 3
- Contrast: OpenAI's Lean scope note gave no explicit value of the constant; an effective constant matters because Siegel's 1935 theorem is ineffective (no computable constant), which blocks effective class-number bounds
- Method (abstract): an interpolation-determinant argument in the biquadratic field generated by √2 over the quadratic field of χ; a real zero near 1 forces χ(p) = −1 for most primes p, and the Frobenius congruence at those primes makes a nonzero determinant divisible by more prime powers than its archimedean size allows. The paper does not rely on the 7/8 quasi-Riemann manuscript
- AI use: 'The argument presented in this paper is due to OpenAI [6]. The large language model Claude (Anthropic) was used throughout the preparation of the paper. It produced a report on the manuscript [6] together with an expanded draft of the proof, which is the basis of the present text. It was also used for … working out the explicit constants … and for proof checking. The author has worked through every line of the paper, has verified the mathematical arguments in their entirety, and takes full responsibility for its content.'
- Follow-ups in the 15 Oct listing: Bryce Kerr's 'A simpler proof that there are no Landau–Siegel zeros' (arXiv 2610.21099) replaces the interpolation determinant with the pigeonhole principle and Masser's theorem, thanks Banks for his draft, and says 'ChatGPT assisted with the mathematical development'; Iwao Kimura (arXiv 2610.21511) proves his own explicit constant (1 − β) log q ≥ 5811 × 10⁻⁷ in Lean, does not use Banks's proof, notes Banks's constant is better for log q ≥ 9.88, and determines all imaginary quadratic fields with class number ≤ 2000
- Terence Tao's blog (19 Oct 2026), 'A sketch of the proof of non-existence of Siegel zeroes': says 'various digestions and simplifications of the argument have been produced' by Banks (2610.15483), Kerr (2610.21099) and Ben Green (hexagonmath.org/2610.00311v2), and sketches Green's formulation, which he calls 'particularly elegant'; the strategy is 'the adelic polynomial method', with a quantity 'essentially the covolume of a certain lattice'. He omits the analytic steps and uses 'minor assumptions and cheats' for exposition; he gives no explicit verdict on correctness and notes the quasi-Riemann arguments 'are still in the process of being fully digested'
- Terence Tao's 'experimental AI-math challenge' (blog, 20 Oct 2026): use any of the new effective arguments ('now we have a fair number of new (and effective) ways to rule out Siegel zeroes, either by establishing a quasi-Riemann hypothesis, or by some version of the argument in this previous post') to derive an explicit upper bound on a hypothetical tenth imaginary quadratic discriminant of class number one without Heegner's methods; the explicit work 'seems eminently suitable for AI'. Rules: disclose AI use, check purely AI-generated solutions with an independent AI tool, Lean not required, 'not prioritize speed above quality'. Commenters report using Codex, GPT-6 Astra, Claude Opus and the SAIR open model (self-reported)
- Tao's AI disclosure: 'a chat session with the new SAIR open math model was used to help understand the preprints'. Per Tao, Green reports two further short proofs (Pescador, Barreto) and a planned 'communal writeup'

#### What happened
On 11 Oct 2026 William D. Banks posted [arXiv 2610.15483](https://arxiv.org/abs/2610.15483) (listed 13 Oct). He rearranges the argument of OpenAI's 1 Oct manuscript "Uniform exclusion of Landau–Siegel zeros" (part of [the quasi-Riemann family](/e/2026-10-06-quasi-riemann-hypothesis-openai/)) as a direct proof and makes every constant explicit. The result: for primitive real characters χ of conductor q ≥ 10¹⁰, every real zero β of L(s, χ) satisfies (1 − β) log q ≥ 0.003. A recent computation by Lu, Zaman and Zhao covers smaller conductors, so the bound holds for all q ≥ 3.

The method is algebraic: an interpolation determinant in the biquadratic field generated by √2 over the quadratic field attached to χ. A zero close to 1 forces χ(p) = −1 for most primes, and Frobenius congruences then make a nonzero algebraic integer too divisible for its size. The paper does not use the 7/8 zero-free half-plane.

The acknowledgments are explicit about AI use: the argument "is due to OpenAI"; Claude "produced a report on the manuscript … together with an expanded draft of the proof, which is the basis of the present text", worked out the explicit constants and helped check proofs; Banks "has worked through every line of the paper, has verified the mathematical arguments in their entirety".

No public reactions from other number theorists were found on 14 Oct (web search; X not searched directly).

#### Why it matters
The Siegel-zero claim was the part of OpenAI's release that specialists called the most consequential ("it settles the question", Hector Pasten). Banks's paper is the first written check by an outside number theorist, and it adds what the Lean version lacked: a numerical constant. An explicit lower bound for 1 − β would make classical ineffective results, such as Siegel's class-number bound, effective. The usual caveat applies: one author's check, with heavy Claude assistance, in an unrefereed preprint.

#### Follow-ups
On 15 Oct two more papers built on the same OpenAI argument. Bryce Kerr's four-page [arXiv 2610.21099](https://arxiv.org/abs/2610.21099) gives "a conceptually simpler proof", developed from a draft of Banks's exposition and with ChatGPT's help. Iwao Kimura's [arXiv 2610.21511](https://arxiv.org/abs/2610.21511) makes the argument explicit independently, proves it in Lean with Claude Opus 5.5 and uses it to list all imaginary quadratic fields with class number up to 2000 ([own entry](/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/)).

On 22 Oct Banks extended the argument from ℚ to any number field k ([arXiv 2610.31402](https://arxiv.org/abs/2610.31402)): for every quadratic extension F/k, every real zero β of L(s, χ_{F/k}) satisfies (1 − β) log d_F ≥ c(k) with an effective c(k) > 0, and through a reduction of Stark the Brauer–Siegel theorem becomes effective and uniform over all Galois extensions of k. "The proof is Kerr's form of the OpenAI argument". Disclosure: "The large language model Claude (Anthropic) was used throughout the preparation of this paper. The key ideas came from the author, who prompted Claude with the question considered here and with the papers [1] and [7]; from these inputs Claude produced an initial argument. The author then worked through every line of the mathematics by hand, verifying the arguments in their entirety, and wrote the final version of the paper."

Sources: [arXiv 2610.15483: On the OpenAI proof that Landau-Siegel zeros do not exist](https://arxiv.org/abs/2610.15483); [Terence Tao: An experimental AI-math challenge: tenth discriminant bounds and Siegel zeroes (20 Oct 2026)](https://terrytao.wordpress.com/2026/10/20/an-experimental-ai-math-challenge-using-tenth-discriminant-bounds-to-explore-arguments-ruling-out-siegel-zeroes/); [Terence Tao: A sketch of the proof of non-existence of Siegel zeroes (19 Oct 2026)](https://terrytao.wordpress.com/2026/10/19/a-sketch-of-the-proof-of-non-existence-of-siegel-zeroes/); [OpenAI: Uniform exclusion of Landau–Siegel zeros (1 Oct manuscript)](https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf); [arXiv 2610.31402: Banks, Landau-Siegel zeros over a base field and effective Brauer-Siegel](https://arxiv.org/abs/2610.31402)

### 2026-10-11: Skryabin's 2006 conjecture proved: every noetherian Hopf algebra has a bijective antipode (proof strategy suggested by GPT-6)

***; Université libre de Bruxelles, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-11-skryabin-conjecture-noetherian-hopf-algebra-antipode/

arXiv 2610.15473 (11 Oct 2026) by Andrea Sciandra proves that a Hopf algebra over any field has a bijective antipode as soon as its left or right regular module has finite uniform dimension; in particular every noetherian Hopf algebra has a bijective antipode, which was conjectured by S. Skryabin in 2006. The "Use of AI" note says "The proof strategy was suggested by GPT-6 (OpenAI) through an iterative dialogue with the author". Unrefereed preprint.

- Paper: 'A noetherian Hopf algebra has bijective antipode', Andrea Sciandra (Université Libre de Bruxelles), arXiv 2610.15473 [math.RA], 11 Oct 2026
- Theorem: if either regular module of a Hopf algebra H (over an arbitrary field) has finite uniform dimension, the antipode S is bijective; corollaries: left and right uniform dimensions agree, and H is right noetherian iff left noetherian
- Conjecture: Skryabin (2006), every noetherian Hopf algebra has a bijective antipode; the proof starts from Skryabin's injectivity theorem for weakly finite Hopf algebras
- AI use: 'The proof strategy was suggested by GPT-6 (OpenAI) through an iterative dialogue with the author; intermediate results helped identify the key difficulties and arrive at a simpler argument. All AI-assisted material used in this paper has been checked by the author, who takes full responsibility for the content of the paper.'

#### What happened
On 11 Oct 2026 Andrea Sciandra posted [arXiv 2610.15473](https://arxiv.org/abs/2610.15473). Skryabin had shown that the antipode of a weakly finite Hopf algebra is injective and conjectured in 2006 that noetherian Hopf algebras have bijective antipodes. Sciandra proves more: one-sided finite uniform dimension is enough. The proof builds "essential left ideals of common denominators" for finite-dimensional subcoalgebras and uses an inverse-antipode identity for surjectivity.

The "Use of AI" note credits GPT-6 with the proof strategy, reached "through an iterative dialogue with the author".

No expert reactions were found on 14 Oct (web search; X not searched directly).

#### Why it matters
Bijectivity of the antipode for noetherian Hopf algebras was a standard open question in the area. It is another named conjecture where a frontier model supplied the strategy and a single human author did the rest.

Sources: [arXiv 2610.15473: A noetherian Hopf algebra has bijective antipode](https://arxiv.org/abs/2610.15473)

### 2026-10-12: Bloch constant conjecture (Ahlfors–Grunsky, 1937) disproved: Teng Zhang shows B < 0.47186… − 1.3×10⁻⁹, with a Lean proof generated by Codex

****; Xi'an Jiaotong University, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-12-bloch-constant-conjecture-disproved/

arXiv 2610.16808 (12 Oct 2026) by Teng Zhang (Xi'an Jiaotong University) proves that the classical Bloch constant B is strictly smaller than the Ahlfors–Grunsky value A_AG = 0.4718616534…, disproving the conjecture of Ahlfors and Grunsky (1937) that B = A_AG; an explicit bound B < A_AG − 1.3×10⁻⁹ is given. The author says the proofs were his own, ChatGPT was used for editing and "as an exploratory tool", and the Lean 4 formalization of the main results "was generated using OpenAI's Codex and checked by the Lean kernel". The repository the paper cites was not reachable on 14 Oct.

- Paper: 'The Bloch constant is strictly smaller than the Ahlfors–Grunsky value', Teng Zhang (Xi'an Jiaotong University), arXiv 2610.16808 [math.CV], 30 pages, 12 Oct 2026
- Conjecture: Ahlfors and Grunsky (Math. Z. 1937) built a function whose inverse Riemann surface is tiled by equilateral triangles, giving B ≤ A_AG = Γ(1/3)Γ(11/12)/(√(1+√3) Γ(1/4)) = 0.471861653…, and conjectured equality
- Result: B < A_AG (Corollary 1.2) and B < A_AG − 1.3 × 10⁻⁹ (Theorem 1.3); the paper says the explicit bound 'can certainly be improved'
- Method: starts from a genus-two translation surface tiled by equilateral triangles; its developing map gives an Ahlfors–Grunsky function, which is then deformed to lower the supremum of univalent-disk radii
- Lower bounds for context (paper's Table 1): Ahlfors √3/4; Chen–Gauthier and later improvements up to √3/4 + 0.0153 (Wikström)
- AI use: 'ChatGPT was used for English-language editing and proofreading, and as an exploratory tool for discussing possible approaches to selected parts of the argument. The mathematical arguments and proofs in the original manuscript were independently developed, checked, and written by the author. The Lean 4 formalization in [19] was generated using OpenAI's Codex and checked by the Lean kernel.'
- Lean: cited as github.com/zhangteng2000/Bloch-constant; that URL returned 404 on 14 Oct 2026, so the formalization could not be inspected

#### What happened
On 12 Oct 2026 Teng Zhang posted [arXiv 2610.16808](https://arxiv.org/abs/2610.16808). Bloch's constant B is the largest r such that every holomorphic f on the unit disk with |f′(0)| = 1 maps some subdomain conformally onto a disk of radius r. In 1937 Ahlfors and Grunsky gave the upper bound A_AG ≈ 0.4718617 and conjectured it was sharp. Zhang proves B < A_AG, and B < A_AG − 1.3×10⁻⁹, by deforming an Ahlfors–Grunsky-type function obtained from a genus-two translation surface tiled by equilateral triangles.

The AI statement credits the mathematics to the author. ChatGPT was used for editing and "as an exploratory tool for discussing possible approaches to selected parts of the argument", and the Lean 4 formalization of Theorem 1.1, Corollary 1.2 and Theorem 1.3 "was generated using OpenAI's Codex and checked by the Lean kernel". The GitHub repository the paper cites returned 404 on 14 Oct, so the formal statement could not be compared with the paper.

No expert reactions were found on 14 Oct (web search; X not searched directly).

#### Why it matters
If it holds up, a famous extremal problem in geometric function theory has a negative answer after almost 90 years. The AI role is modest in the mathematics but large in the checking: the Lean proof, once public, would let anyone confirm the inequality without trusting the 30-page argument.

Sources: [arXiv 2610.16808: The Bloch constant is strictly smaller than the Ahlfors–Grunsky value](https://arxiv.org/abs/2610.16808); [Lean formalization cited by the paper (404 on 14 Oct 2026)](https://github.com/zhangteng2000/Bloch-constant)

### 2026-10-12: Cureton and Thorner, with AxiomProver, extend the quasi-Riemann zero-free half-plane to every Dedekind zeta function (Re s > 31/32, Lean-formalized)

****; Axiom Math; Science & math; confidence medium; https://postcutoff.com/e/2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover/

A preprint posted on Hexagon on Oct 12, 2026 by Dean Cureton and Jesse Thorner proves that the Dedekind zeta function of any number field has no zeros with Re s > 31/32, and the same for finite-order Hecke L-functions. It follows the strategy of OpenAI's Oct 6 quasi-Riemann work, which covered ζ, Dirichlet L-functions and special fields. The abstract says the results "were proved and formalized in Lean using AxiomProver".

- Paper: 'A quasi-Riemann hypothesis for Dedekind zeta functions' (Hexagon 2610.00200, math.NT; v1 Oct 12, v2 Oct 14, 2026)
- Theorem 1.1: for every number field K, ζ_K(s) ≠ 0 for Re(s) > 31/32. Corollary 1.2: the same half-plane is zero-free for finite-order Hecke L-functions over K (wherever holomorphic; s = 1 included for nonprincipal characters)
- AI disclosure (abstract): 'These results were proved and formalized in Lean using AxiomProver'; Hexagon page: 'The proofs and Lean formalization were generated by AxiomProver'; 'Human understanding: some parts'
- Repo README (AxiomMath/DedekindZetaZeroFree, created Oct 12): 'The investigators, Dean Cureton and Jesse Thorner, directed the research and made the mathematical judgments. AxiomProver generated the proofs and Lean formalization'
- Abstract: 'the hardest part was the formalization of the work of Kazhdan and Patterson on metaplectic forms in order to accommodate all number fields'
- Lean: formal challenge Solution/DedekindZetaSolution.lean; only the standard axioms propext, Quot.sound, Classical.choice; depends on Mathlib, Tau Ceti and a vendored class-field-theory library, with code adapted from openai/math
- Builds on: 'recent work on Artin's primitive root conjecture' (OpenAI's Oct 6 release, family 029) that gave a zero-free half-plane for finite-order Hecke L-functions over cyclotomic fields
- Already used: Frei, Sofos and Widmer (arXiv 2610.18491) cite 'recent breakthroughs of OpenAI and Cureton–Thorner'

#### What happened
OpenAI's Oct 6 math release claimed a fixed zero-free half-plane for ζ and Dirichlet L-functions (Re s > 7/8), and separately one for Hecke
L-functions over cyclotomic fields in its work on Artin's primitive root conjecture. On Oct 12 Dean Cureton and Jesse Thorner posted a
preprint on Hexagon, the repository for LLM-assisted mathematics, that follows that strategy for every number field. The bound is weaker
(31/32), but it covers all Dedekind zeta functions and all finite-order Hecke L-functions. Axiom Math's AxiomProver wrote the proofs and the
Lean formalization, which is public and uses only Lean's standard axioms. The abstract says the hard step was formalizing Kazhdan and
Patterson's work on metaplectic forms.

The story reached Chinese tech press on Oct 18 (36Kr). That article mixes in other results and loosely calls this a proof of the "Generalized
Riemann Hypothesis". It is not: the result is a zero-free half-plane, and the Riemann hypothesis line Re s = 1/2 remains open. 36Kr also
claimed that Bürgisser had proved VP ≠ VNP; we found no source for that and do not repeat it.

#### Why it matters
It is one of the first results to build directly on OpenAI's claimed quasi-Riemann breakthrough. It comes with a machine-checked proof and with
named mathematicians directing the work, and another paper (Frei–Sofos–Widmer) already uses it. If OpenAI's base results hold up, the
number-field version extends their reach to algebraic number theory. It has not been refereed.

Sources: [Hexagon: A quasi-Riemann hypothesis for Dedekind zeta functions (Cureton, Thorner)](https://hexagonmath.org/2610.00200v1); [GitHub: AxiomMath/DedekindZetaZeroFree (Lean 4 formalization)](https://github.com/AxiomMath/DedekindZetaZeroFree); [Frei, Sofos, Widmer (arXiv 2610.18491), citing the result](https://arxiv.org/abs/2610.18491); [36Kr: 'After OpenAI, AI has once again cracked the Generalized Riemann Hypothesis' (Oct 18; secondary, loose)](https://eu.36kr.com/en/p/4031090627858820)

### 2026-10-12: Erdős Matching Conjecture (1965) proved for every fixed k and all large s, via a sharp tail bound whose k ≥ 6 cases generative AI completed

****; Shanghai Jiao Tong University; Science & math; confidence medium; https://postcutoff.com/e/2026-10-12-erdos-matching-conjecture-large-matchings-ai/

arXiv 2610.16203 (12 Oct 2026) by Cheng Chi, Shuang Sun and Yan Wang (Shanghai Jiao Tong University) proves the conjecture of Łuczak, Mieczkowska and Šileikis on the maximal tail probability of sums of k i.i.d. nonnegative random variables, and uses it to prove the Erdős Matching Conjecture for every fixed k ≥ 3, all sufficiently large s and every n ≥ k(s+1). The authors say they proved the tail theorem for k = 2–5 and k ≥ 10^100 without AI, and that "generative AI developed a unified argument valid for every k ≥ 6"; a gap in the hypergraph argument was closed on an AI suggestion. Unrefereed preprint.

- Paper: 'A Proof of the Erdős Matching Conjecture for Large Matchings via Sharp Tail Probability for Sums of i.i.d. Nonnegative Random Variables', Cheng Chi, Shuang Sun, Yan Wang (SJTU), arXiv 2610.16203 [math.CO], 12 Oct 2026
- Erdős (1965): the maximum number of edges of an n-vertex k-uniform hypergraph with no s+1 pairwise disjoint edges is max{C(k(s+1)−1, k), C(n,k) − C(n−s,k)} (clique vs cover construction)
- Result 1 (probability): for every k ≥ 1 and 0 ≤ x ≤ 1/k, sup over i.i.d. nonnegative X_i with E X_1 ≤ x of P(X_1 + … + X_k ≥ 1) equals max{1 − (1 − x)^k, (kx)^k}, with all maximizing distributions characterized; previously known for x < 1/(2k−1) (k ≥ 5), extended by Frankl–Kupavskii to x ≤ 3/(5k−2)
- Result 2 (hypergraphs): the Erdős Matching Conjecture for every fixed k ≥ 3, all sufficiently large s, and every n ≥ k(s+1), with the extremal families characterized
- AI use: 'For Theorem 1.2, the authors independently developed the proof framework and proofs for k = 2, 3, 4, 5 and k ≥ 10^100 without any AI assistance. Building on the authors' proof framework, generative AI developed a unified argument valid for every k ≥ 6 in early September 2026, thereby resolving the remaining cases'
- AI use (Theorem 1.4): the initial argument 'left a gap of width o(s) in the range of n around n = ρ_k s … This gap was closed following a suggestion of using weak convergence, given by generative AI (see Lemma 8.3 and Section 8.2)'; the authors 'independently checked all AI-generated arguments in detail'. The model is not named

#### What happened
On 12 Oct 2026 Cheng Chi, Shuang Sun and Yan Wang posted [arXiv 2610.16203](https://arxiv.org/abs/2610.16203). Erdős asked in 1965 for the largest k-uniform hypergraph on n vertices without s+1 disjoint edges, and conjectured that one of two constructions (a clique on k(s+1)−1 vertices, or all edges meeting a fixed s-set) is always best. The conjecture had been proved for n large compared with ks and in special cases, but not in general.

The paper first settles a conjecture of Łuczak, Mieczkowska and Šileikis on how likely a sum of k i.i.d. nonnegative random variables with small mean is to reach 1, and then derives the Erdős Matching Conjecture for every fixed k ≥ 3 once s is large enough, across the whole range n ≥ k(s+1).

The disclosure is unusually specific about the division of labour: the humans proved the tail theorem for k = 2–5 and k ≥ 10^100, and "generative AI developed a unified argument valid for every k ≥ 6 in early September 2026, thereby resolving the remaining cases". An AI suggestion (weak convergence) also closed a gap of width o(s) near the crossover point of the two constructions. The model is not named.

No expert reactions were found on 14 Oct (web search; X not searched directly).

#### Why it matters
The Erdős Matching Conjecture is one of the central open problems of extremal set theory. Here AI filled in a precise, bounded part of a human programme, the infinite middle range of k, rather than supplying the strategy. The k ≥ 10^100 boundary of the human proof shows how much that middle range was worth.

Sources: [arXiv 2610.16203: A Proof of the Erdős Matching Conjecture for Large Matchings](https://arxiv.org/abs/2610.16203)

### 2026-10-12: Chow polynomials of matroids are not log-concave: counterexamples found in an LLM search (Claude Opus 5.5, ChatGPT Astra) refute three real-rootedness conjectures

***; Anthropic, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-12-chow-polynomials-matroids-not-log-concave/

arXiv 2610.17135 (12 Oct 2026) by Gergely Bérczi, Basile Coron, Christopher Eur and Shiyue Li constructs explicit infinite families of matroids from finite geometries whose multichain counts fail to be log-concave. This breaks log-concavity of Chow, augmented Chow, chain and h-polynomials and disproves the real-rootedness conjectures of Ferroni–Schröter, Huh–Stevens and Athanasiadis–Kalampogia-Evangelinou from the lowest possible ranks. The authors say "LLMs were substantially used in the searches": Eur found a non-constructive disproof with ChatGPT Astra on 23 Sep, and on 4 Oct a launched LLM search found the explicit families with Claude Opus 5.5 within 5–12 hours.

- Paper: 'Chow polynomials and chain polynomials of matroids are not log-concave', Bérczi, Coron, Eur, Li, arXiv 2610.17135 [math.CO], v1, 12 Oct 2026
- Disproves: real-rootedness conjectures of Ferroni–Schröter (Chow polynomials), Huh–Stevens and Athanasiadis–Kalampogia-Evangelinou; the failures start 'from the lowest possible ranks'
- Construction: 'hyperplane coalescence' plus classical operations of projective geometry and combinatorial designs
- AI timeline (paper): 23 Sep, Eur 'found a non-constructive disproof of the real-rootedness conjecture for Chow polynomials using ChatGPT Astra'; 30 Sep, Coron found with ChatGPT Astra a counterexample to log-concavity of h-polynomials of graph associahedra 'of very high vertex count'; 4 Oct, Bérczi, Coron and Li 'orchestrated and launched an LLM-search'; 'About 5 hours and 12 hours later, SL found using Claude Opus 5.5 three explicit families of matroids as counterexamples'; Bérczi then found another family with ChatGPT Astra
- AI use (general): 'LLMs were substantially used in the searches of these counterexamples … The mathematical content was digested and reshaped from the LLM search reports, checked, and written by the authors'; a detailed AI-methodology analysis is promised for an update
- Origin: the BIRS-Oaxaca workshop 'Algebra and Geometry of Matroids' (27 Sep – 2 Oct 2026), where Eur told Li about the LLM-generated disproof

#### What happened
On 12 Oct 2026 Gergely Bérczi, Basile Coron, Christopher Eur and Shiyue Li posted [arXiv 2610.17135](https://arxiv.org/abs/2610.17135). Chow polynomials (Hilbert–Poincaré series of the Chow rings of matroids) and chain polynomials had been conjectured to have only real zeros, which would imply log-concavity. The paper gives explicit infinite families of matroids, built from finite geometries by "hyperplane coalescence", for which even log-concavity fails.

The AI disclosure gives a dated account. Eur found a non-constructive disproof of real-rootedness with ChatGPT Astra on 23 Sep and mentioned it at a BIRS-Oaxaca matroid workshop. On 30 Sep Coron found a (very large) counterexample for graph associahedra with ChatGPT Astra. On 4 Oct three of the authors launched a targeted LLM search; about 5 and 12 hours later Li "found using Claude Opus 5.5 three explicit families of matroids as counterexamples … of lowest ranks possible". The authors "digested and reshaped" the search reports into the paper.

Because the counterexamples are explicit, they can be checked by computation; the paper is still unrefereed. No outside reactions were found on 14 Oct (web search; X not searched directly).

#### Why it matters
Log-concavity of matroid invariants is the area of June Huh's Fields-Medal work, and real-rootedness conjectures there were widely studied. This is a clear example of a now-common pattern: humans state the target, LLM searches produce explicit counterexamples within hours, and the paper is the human digest.

Sources: [arXiv 2610.17135: Chow polynomials and chain polynomials of matroids are not log-concave](https://arxiv.org/abs/2610.17135)

### 2026-10-12: "The proof presented in this paper is due to Claude Opus 5.5": Alexis Drouot publishes Claude's 40-minute elementary proof of his conjecture on geometric bulk conductances

***; University of Washington, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-10-12-drouot-claude-opus-proof-bulk-conductances/

arXiv 2610.16208 (12 Oct 2026) by Alexis Drouot (University of Washington) gives an elementary proof, using only Stokes' theorem and the Poincaré lemma, that the geometric Kubo formula in any even dimension equals a topological integer times the bulk conductance, extending his 2D result with Xiaowen Zhu. Martinian and Stoiber had recently proved the statement by other means. The comments field says "The proof presented in this paper is due to Claude Opus 5.5": given his paper and the obstruction, Claude "proposed a 5-page solution within 40 minutes". Drouot's Section 1.4 reflects on what this means for training mathematicians.

- Paper: 'Geometric bulk conductances in even dimensions', Alexis Drouot (University of Washington; CPT Aix-Marseille), arXiv 2610.16208 [math-ph], 15 pages, 12 Oct 2026
- Result: for a short-range projector P on ℓ²(ℤ^{2n}, ℂ^m) and a transverse family U of open sets, Tr(P[P,1_{U1}] ∧ … ∧ [P,1_{U2n}]) is a topological integer X_U times the bulk conductance of P; the proof deforms a cochain (an idea due to Kitaev) instead of the sets
- Priority: an elementary proof of a recent result of Martinian–Stoiber, which Drouot learned of only 'at terminal stages'; Claude found the reference when asked for a literature search, and Drouot believes Claude 'did not base its response on it'
- AI role: 'Early October 2026, I asked this question to Claude Opus 5.5 … Claude proposed a 5-page solution within 40 minutes. Even in dimension two, Claude's proof is much simpler and more elegant than the one presented in [DZ26]. The present paper is Claude's solution, together with a few of my own edits.'
- Human role (listed by the author): identify the question and formulate the conjecture; ask Claude with the relevant material; simplify Claude's proof; expand technical details; organize and interpret; track down ownership of ideas
- Reflection: 'While it is disturbing to see an LLM solve what could have been a full thesis in under an hour, I believe I learnt a lot from this project … I prefer to stay out of controversies and prepare for a brutal transformation of mathematics … How should we teach the next generation of researchers, now that LLMs can solve Ph.D.-suited projects in just a few prompts?'

#### What happened
On 12 Oct 2026 Alexis Drouot posted [arXiv 2610.16208](https://arxiv.org/abs/2610.16208). In 2024 he and Xiaowen Zhu proved a bulk-edge result for 2D topological insulators along curved boundaries and he conjectured the even-dimensional analogue. In early October 2026 he gave Claude Opus 5.5 the 2D paper, a project description and the obstruction he saw in higher dimensions; "Claude proposed a 5-page solution within 40 minutes", simpler than the original even in 2D. The paper is "Claude's solution, together with a few of my own edits", and the comments field says "The proof presented in this paper is due to Claude Opus 5.5".

The statement itself had been proved shortly before by Martinian and Stoiber, which Drouot learned only near the end (through a literature search by Claude). The new contribution is the elementary proof.

Section 1.4 is a personal reflection: Drouot finds it "disturbing to see an LLM solve what could have been a full thesis in under an hour", says he will "prepare for a brutal transformation of mathematics", and asks how to train researchers "now that LLMs can solve Ph.D.-suited projects in just a few prompts".

No reactions were found on 14 Oct (web search; X not searched directly).

#### Why it matters
This is a frank first-person record of an AI proof on a research-level question, with a clear credit line to the model and a list of what the human did. Drouot's description of the work as thesis-sized is a data point for the debate on graduate training that Tao and others have opened.

Sources: [arXiv 2610.16208: Geometric bulk conductances in even dimensions](https://arxiv.org/abs/2610.16208)

### 2026-10-13: Eight extremal combinatorialists write up GPT-6 Astra's proof of the Erdős–Simonovits rational exponents conjecture (Conlon, Janzer, Liu et al.)

****; Caltech, EPFL, KAIST, IBS, Emory University, Epoch AI, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-13-rational-exponents-conjecture-proof-paper-conlon/

arXiv 2610.18713 (13 Oct 2026), "A proof of the rational exponents conjecture", by David Conlon, Oliver Janzer, Tao Jiang, Jaehoon Kim, Joonkyung Lee, H. Liu, Sean Longbrake and Liana Yepremyan, presents and explains the proof of the Erdős–Simonovits rational exponents conjecture (Erdős problem #571) that Epoch AI reported GPT-6 Astra found during FrontierMath Erdős benchmarking in September. The authors, who did much of the earlier human work on the problem, write that "this advance draws heavily on ideas and methods developed by many researchers over several years". It is the first human-written paper on the result; the original came as a Lean proof and an informal write-up.

- Paper: arXiv 2610.18713 [math.CO], 13 Oct 2026, 13 pages; authors from Caltech (Conlon), EPFL (Janzer), Miami University (Jiang), KAIST (Kim), KIAS (Lee), IBS Daejeon (Liu) and Emory (Longbrake, Yepremyan)
- Theorem 1.2: for every rational r in [1, 2] there is a graph H with ex(n, H) = Θ(n^r), the Erdős–Simonovits rational exponents conjecture
- Framing: 'we wish to explain and situate a proof of the conjecture for all exponents claimed by Epoch AI [1], obtained during benchmarking for the recently released model GPT-6 Astra.' Reference [1] is Thomas Bloom's erdosproblems.com/571 page
- The authors name four sources the proof builds on: the Bukh–Conlon lower bound, an Erdős–Simonovits result on powers of rooted graphs, the Kang–Kim–Liu subdivision conjecture, and a technique of Conlon, Janzer and Lee
- Key technical result (Theorem 1.3): a variant of the Kang–Kim–Liu subdivision conjecture for the graphs H_k (each edge replaced by a path of length k plus two apex vertices), which lets one 'percolate' from the exponent 1 to all rational exponents
- Before the AI proof, the conjecture was known for exponents 1 + a/b with b > a² and 2 − a/b with b ≥ (a − 1)², among other ranges
- Still open, per the concluding remarks: the Bukh–Conlon conjecture on rooted powers of trees and the Kang–Kim–Liu subdivision conjecture, both of which would imply the result
- AI use in this paper: 'We used ChatGPT Pro to draw the figures in this paper and to proofread an early draft.'

#### What happened
In September, Epoch AI and Thomas Bloom reported that GPT-6 Astra, in larger-budget runs outside the FrontierMath Erdős benchmark, had proved the Erdős–Simonovits rational exponents conjecture (#571) with a Lean proof (see [the FrontierMath Erdős entry](/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/)). The benchmark report called the informal write-ups placeholders until experts prepared proper papers.

On 13 Oct 2026 eight extremal combinatorialists posted that paper, [arXiv 2610.18713](https://arxiv.org/abs/2610.18713). Several of them wrote the earlier partial results the proof builds on, including the Bukh–Conlon lower bound, the Kang–Kim–Liu subdivision conjecture and the Conlon–Janzer–Lee technique. The abstract says: "We present a proof of this conjecture claimed by Epoch AI, explaining how it builds upon ideas and methods developed by many researchers." The introduction adds: "As with many of the recent AI-driven breakthroughs, this advance draws heavily on ideas and methods developed by many researchers over several years."

The core is Theorem 1.3, a variant of the Kang–Kim–Liu subdivision conjecture for graphs H_k (each edge of H replaced by a path of length k, plus two vertices joined to the two sides of the bipartition). Combined with the Erdős–Simonovits result on powers of balanced rooted graphs and the Bukh–Conlon lower bound, it reaches every rational exponent starting from the exponent 1. The two stronger conjectures (Bukh–Conlon on rooted tree powers, Kang–Kim–Liu on 1-subdivisions) remain open.

The paper's own AI statement covers only figures and proofreading by ChatGPT Pro. No Hacker News thread was found on 14 Oct (Algolia search); X was not searched directly.

#### Why it matters
It is the second wave of one of the strongest AI-found results of 2026. Experts in the area accept the proof, rewrite it for humans and place it in the literature, while making the point that the model built on their programme. Its verification moves from a Lean certificate plus placeholder write-up to an expert-written preprint.

Sources: [arXiv 2610.18713: A proof of the rational exponents conjecture (Conlon, Janzer, Jiang, Kim, Lee, Liu, Longbrake, Yepremyan)](https://arxiv.org/abs/2610.18713); [erdosproblems.com: Problem #571](https://www.erdosproblems.com/571); [GitHub: tadamcz/erdos571 (Lean proof from the Epoch AI run)](https://github.com/tadamcz/erdos571)

### 2026-10-13: Thompson's group F is nonamenable: specialists Sang-hyun Kim and Yash Lodha publish a human-written version of OpenAI's proof

****; OpenAI, KIAS, Purdue University; Science & math; confidence medium; https://postcutoff.com/e/2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition/

OpenAI's 6 Oct 2026 math release (family 248, manuscript dated 23 Sep, Lean scope note) claims that Thompson's group F is nonamenable, settling a problem first raised by Richard Thompson in 1973 and central to Geoghegan's conjecture. On 13 Oct Sang-hyun Kim and Yash Lodha (Purdue), specialists in groups of piecewise linear and projective homeomorphisms, posted arXiv 2610.18360, "a human written exposition of OpenAI's recent proof": a streamlined proof plus a generalization, with "no novelty claimed" and "No part of this writing, including the references, was produced by AI." It is the first public write-up of the claim by experts in the field; it is unrefereed.

- Claim: OpenAI family 248, 'Thompson's group F is nonamenable' (manuscript dated 23 Sep 2026): 'This confirms Geoghegan's conjecture and resolves the amenability problem for F'; the catalogue links a Lean scope note (lean/docs/248.md)
- Exposition: Sang-hyun Kim (Korea Institute for Advanced Study) and Yash Lodha (Purdue), 'A note on the proof of nonamenability of F', arXiv 2610.18360 [math.GR], dated 13 Oct 2026
- Quote: 'It has been a long standing open problem whether F is amenable … Recently, OpenAI announced a solution. Using the same idea, we give a streamlined proof, and describe a generalization in Theorem 2.7. All the arguments here are based on simplifications of those in the OpenAI announcement, and no novelty is claimed here. No part of this writing, including the references, was produced by AI.'
- History (the note): first considered by Richard Thompson (letter to George Francis, 26 Sep 1973) and popularized by Ross Geoghegan as part of his conjecture
- Lodha's earlier work includes a nonamenable type-F∞ group of piecewise projective homeomorphisms (J. Topology 2020), so the authors are specialists in exactly this problem area

#### What happened
[OpenAI's 6 Oct release](/e/2026-10-06-openai-math-release-722-manuscripts/) listed, as family 248, a proof that Thompson's group F is nonamenable, with a Lean scope note. F, the group of dyadic piecewise linear homeomorphisms of [0, 1], is a central test case in the theory of amenable groups, and the question had been open since Thompson raised it in 1973 (Kim and Lodha cite his letter to George Francis of 26 Sep 1973).

On 13 Oct Sang-hyun Kim and Yash Lodha posted [arXiv 2610.18360](https://arxiv.org/abs/2610.18360). They give "a streamlined proof" using "the same idea" as OpenAI's manuscript, add a generalization (Theorem 2.7), and state that no part of the note was produced by AI. They claim no novelty.

No further expert reactions were found on 14 Oct (web search; X not searched directly).

#### Why it matters
Most of the release's famous claims still await reading by specialists. This is one of the first cases where experts in the exact area have reconstructed an AI proof of a famous problem by hand, which makes errors much less likely than for a claim checked only by its authors' Lean scope note. Formal publication and wider expert review are still to come.

Sources: [arXiv 2610.18360: Kim & Lodha, A note on the proof of nonamenability of F](https://arxiv.org/abs/2610.18360); [OpenAI: Thompson's group F is nonamenable (23 Sep manuscript)](https://github.com/openai/math/tree/main/preprints/Thompsons-group-F-is-nonamenable-September-23-2026); [openai/math CONTENTS.md (family 248)](https://github.com/openai/math/blob/main/CONTENTS.md)

### 2026-10-13: Ulam's question answered: every Lie group is a group of permutations of a countable set (Monod & Thom; key construction from GPT-6 Astra)

****; EPFL, TU Dresden, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-13-ulam-problem-lie-groups-countably-representable-gpt-6-astra/

arXiv 2610.18464 (13 Oct 2026), "Lie groups are countably representable" by Nicolas Monod and Andreas Thom, proves that every Lie group can be realised by permutations of a countable set, answering a question Ulam asked in 1960 (and hence that every locally compact second countable group is countably representable). The abstract says "the key deformation construction originated in an argument produced by OpenAI's GPT 6 Astra during extended mathematical discussions"; the authors then developed, checked and substantially rewrote it, and found that parts were already in the literature. Thom had disputed OpenAI's attribution of an earlier Astra result in September.

- Paper: arXiv 2610.18464 [math.GR], 13 Oct 2026, by Nicolas Monod and Andreas Thom
- Theorem A: every Lie group (second countable) is countably representable, i.e. embeds in Sym(N). Corollary B: every locally compact second countable group is countably representable
- History in the paper: Schreier and Ulam showed in 1935 that R is countably representable; Ulam asked whether all Lie groups are [Ula60, II.7; Ula64, V.2]. Linear Lie groups were known (Thomas 1999; Kallman 2000; Ershov–Churkin 2004), and Conversano–Monod (J. Algebra 2024) reduced the problem to connected simple Lie groups. The remaining obstacle was non-linear central extensions
- Key new ingredient (Theorem C): a countable-index subalgebra A of R with a continuous one-parameter family of ring homomorphisms from the inclusion back to a countable real-closed field, which makes the cohomological obstruction from central extensions vanish
- AI disclosure: 'The key countable-index deformation was first constructed by OpenAI's GPT 6 Astra during extended mathematical discussions. The argument was subsequently developed, checked and substantially rewritten by the authors'
- Acknowledgments: 'The proof emerged from three weeks of discussions with OpenAI's GPT, initially aimed at exploring SL3(R) with ring deformations and K-theoretic arguments. GPT Astra eventually produced the countable-index deformation underlying Theorem C. We then suggested extending the argument to SLn(R) and other simple Lie groups'
- Prior art the authors found while checking: 'substantial parts of the argument already appear, notably the algebraic-analytic construction of Parusiński–Rond [PR20]'; the simultaneous Puiseux embedding with prescribed residues (Lemma 2.4) and its combination with countable-index gluing 'appear to be new and creative'
- Context: in September Thom argued that OpenAI's announced non-sofic group result relied on his 2019 work with Gábor Kun and that OpenAI's attribution was misleading; this paper credits Astra openly

#### What happened
On 13 Oct 2026 Nicolas Monod and Andreas Thom posted [arXiv 2610.18464](https://arxiv.org/abs/2610.18464), proving that every Lie group can be realised as a group of permutations of a countable set. Schreier and Ulam had shown this for R in 1935, which led Ulam to ask whether it holds for all Lie groups. Linear Lie groups were known, and Conversano and Monod reduced the question to connected simple Lie groups in 2024. The remaining case was simple groups that are only central extensions of linear groups, where the cohomological obstruction lives on "wild" subgroups of countable index.

The new tool is Theorem C: a subalgebra of R of countable index together with a continuous path of ring homomorphisms from a countable real-closed field to the inclusion, built from Puiseux series over fields of analytic functions and glued along countably many pieces. The continuity lets the authors trivialise the central extension on the corresponding subgroups.

The authors give a detailed account of the AI's role. The proof "emerged from three weeks of discussions with OpenAI's GPT", which started on SL3(R). "GPT Astra eventually produced the countable-index deformation underlying Theorem C." The humans proposed the extension to SLn(R) and other simple groups, checked and rewrote everything, and found that a large part of the construction already appears in Parusiński–Rond (2020). They still call the Puiseux embedding with prescribed residues and its gluing "new and creative".

Andreas Thom is the Dresden group theorist who, in September, criticised OpenAI's framing of Astra's non-sofic group result as ignoring his work with Gábor Kun (see [Astra's ten advances](/e/2026-08-01-openai-astra-ten-advances/)). His recent Mathstodon posts did not mention this paper (checked 14 Oct). No Hacker News thread was found (Algolia, 14 Oct).

#### Why it matters
It is a classical, well-known question answered with a clearly credited AI contribution to the decisive step, and with an unusually frank account of what was new and what turned out to be in the literature. It comes from a mathematician who had publicly challenged AI-maths attribution, which makes the credit given here more significant.

Sources: [arXiv 2610.18464: Lie groups are countably representable (Monod & Thom)](https://arxiv.org/abs/2610.18464); [Andreas Thom (guest post on Tao's blog, Sept 11): On the existence of non-sofic groups](https://terrytao.wordpress.com/2026/09/11/on-the-existence-of-non-sofic-groups/)

### 2026-10-13: October 2026: the arXiv stream of conjectures settled with disclosed AI help continues (catalogue of the 2–13 Oct listings)

***; OpenAI, Anthropic, Google DeepMind, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/

Catalogue of arXiv preprints from the 2–13 October 2026 listings that settle or advance a named conjecture or open question and disclose a substantive AI contribution, with the AI role quoted from each paper. It continues the July–September catalogue, from which these blocks were split on 13 Oct 2026. GPT-6 Astra is the model named most often; Claude, Codex, Aristotle, Rethlas and Google DeepMind's Aletheia also appear. A new pattern is papers that build on OpenAI's unrefereed 6 Oct manuscripts. Almost all items are unrefereed preprints; the larger results have their own entries.

- Scope: nine listing blocks (Oct 2–5, Oct 5–6, early October, 6–8, 8–9, 5–8, 6–9, 9 Oct and 7–13 Oct), read from the PDFs; the 7–13 Oct block came from 242 swept papers; the Oct 2–5 scan alone covered 132 PDFs, of which 70+ had AI statements
- Results with their own entries: 3SUM/APSP hypotheses refuted by an algorithm Claude found; linear Hadwiger conjecture (GPT-6 Astra); KLS conjecture; Artin–Davenport for cubic forms in 10 variables; Woodin's GCH question; polynomial Littlewood–Offord ('discovered autonomously by GPT-6 Pro'); exact overlaps; LeBrun–Salamon; multiple-unicast; Jackson–Kechris–Louveau treeability; Dualization under ETH; Hilbert number; kinetic Krylov–Safonov and Pucci; Smale's sixth problem for five bodies; rank of 3×3 matrix multiplication over F_2; Erdős #522 and #993; Toms–Winter; Bolzano
- Proofs credited to a model: amicable numbers bound ('a proof, found by ChatGPT Astra', Pollack, arXiv 2610.05661); Forcade–Pollington conjecture of 1990 ('based on our extensive interactions with GPT-5.6 Pro', Campbell, 2610.12087); random-oracle unitary synthesis ('GPT 5.6 Sol … was able to solve the problem', Huang, Ramkumar & Wright, 2610.06700); noncommutative AM–GM ('found by GPT6 Astra gradually', Zhong, 2610.04874)
- Several models and agents on one result: Golod–Shafarevich towers of arithmetic surfaces 'only after a collaboration with GPT5.6 Sol, Fable and Aletheia a Gemini-powered internal agent developed at Google DeepMind' (Keller & Pagano, 2610.06461); a simpler simple nil ring 'found by Aristotle (Harmonic) with the assistance of ChatGPT' (Gubarev, 2610.05384); the automorphic commutator relation drafted by the agent system Rethlas in about 1 hour (Liu, 2610.05781)
- With Lean: an explicit counterexample to Tsirelson's problem (a binary linear system game with 1,417,152 equations in 1,889,684 variables; GPT 6 Astra and Fable 5.1; Gao, Yu & Zhi, 2610.10248); László's Toeplitz Böttcher–Wenzel conjecture refuted (Claude and Codex; Zhu & Nie, 2610.08980); quasi-Riemann constants re-tuned to Re s > 0.874957… (Codex GPT 6.1 Sol, audit by Claude Opus 5.5; Liu, 2610.12234)
- Counterexamples with AI help: Erdős–Sós bipartite-link conjecture (edge density above 0.250000356; Yang, 2610.11642, GPT-6.1 for strategies and checking); Strang's spline conjecture (search 'relied heavily on … Claude Fable 5.1'; Potu, 2610.08470); continuous Aldous–Lyons conjecture (Gelander, 2610.08581, GPT-6 Astra); two compactness conjectures of Brudnyi (Huang & Xu, 2610.11066)
- Papers building on OpenAI's unrefereed 6 Oct manuscripts, which inherit that status: de Bruijn–Newman Λ ≤ 0.13963 assuming OpenAI's 7/8 zero-free claim (2610.12803); an L^3 reverse square function estimate (2610.12681); ranks of twin-prime elliptic curves assuming OpenAI's low-corank BSD statement (2610.09370); irrationality exponent of log r (2610.10192); radius of comparison (Elliott & Niu, 2610.11143, 'carried out using ChatGPT'); Burdzy's hot-spots note 'generated by ChatGPT' (2610.12622)
- Lê's conjecture (1982): Portilla Cuadrado (2610.10418) says ChatGPT 'found a fix for the case mn = 4 and proved the conjecture, also, in the two remaining cases, mn = 2, 3', to appear in another paper; not yet posted when this catalogue was split
- Failure modes on record: GPT 5.6 Pro 'claimed a false proof of an FPRAS extension …, which GPT 6 Astra Pro managed to fix' (SK-model sampling, 2610.09882); a potential function 'initially suggested by ChatGPT-5.6 Sol' came with a lemma 'whose validity was difficult to verify', so the authors wrote their own proof (2610.04645); the 'Viva La Vida' study of a multi-agent prover (186 hours, $5,694) found that 26% of its lemma library came from rejected arguments (2610.04829)
- Human-only results of the same weeks are listed for contrast, e.g. the 4-to-1 Games Conjecture with perfect completeness (2610.12378) and Problem 101 of Yau's list ('all the proofs were developed without the use of AI tools', 2610.12238)

#### What happened
This catalogue continues [the July–September catalogue](/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/). It lists arXiv preprints from the 2–9 October 2026 listings that settle or advance a named conjecture or open question and state an AI contribution. For each paper the PDF's AI statement was read; quotes are from the papers, and nothing here was independently verified. The blocks below were written during the update runs of 6–13 October and are kept in the order and wording in which they were added (they were moved here from the summer entry on 13 Oct 2026 without changes, apart from one cross-reference). A smaller batch from 5–9 October is in [a separate entry](/e/2026-10-09-arxiv-ai-disclosed-results-early-october-2026/).

### Oct 2–5 listing (manual arXiv scan, Oct 6 morning; 132 PDFs, 70+ with AI statements)
Own entries: **3SUM/APSP hypotheses refuted** by an algorithm Claude found (Alman & Vassilevska Williams, 2610.06783; 2026-10-05-claude-refutes-3sum-apsp-hypotheses), the **linear Hadwiger conjecture** proved by GPT-6 Astra (Norin & Steiner, 2610.05291; 2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra) and the **KLS conjecture** (Song & Zhang 2610.01447; Bizeul, Klartag & Lehec 2610.05474; 2026-10-02-kls-conjecture-proved-ai-assisted).
- **Golod–Shafarevich towers of arithmetic surfaces** over Z, answering questions of Bost–Charles, Ihara, Frey–Kani–Völklein and Venkatesh (2022, linear conductor growth). Keller & Pagano, [arXiv 2610.06461](https://arxiv.org/abs/2610.06461): worked on it since 2022 and "were able to bring it to fruit only after a collaboration with GPT5.6 Sol, Fable and Aletheia a Gemini-powered internal agent developed at Google DeepMind".
- **Amicable numbers**: A(x) ≤ x·exp{−(1/2+o(1)) log x log₃x / log₂x}. Pollack, [arXiv 2610.05661](https://arxiv.org/abs/2610.05661): "We report on a proof, found by ChatGPT Astra".
- **First FPRAS for counting common bases of two matroids** (and Hadamard products of curvature-nonnegative measures). Chen & Liu, [arXiv 2610.06724](https://arxiv.org/abs/2610.06724): after a prompt to "keep trying", ChatGPT "found a solution"; the authors then recognised its stochastic-gradient-descent framework and generalised it with ChatGPT's help.
- **Simple nil ring** (Kaplansky's 1956 problem, solved by Smoktunowicz 1999): a substantially simpler construction "found by Aristotle (Harmonic) with the assistance of ChatGPT". Gubarev, [arXiv 2610.05384](https://arxiv.org/abs/2610.05384).
- **Benson–Etingof–Ostrik conjecture** for finite symmetric tensor categories. Coulembier & Flake, [arXiv 2610.04096](https://arxiv.org/abs/2610.04096): main ideas of Sections 3–4 "originally obtained using GPT-6 Astra", from a formula found by numerical experiments with Claude 4.6–5 Opus.
- **LeBrun–Salamon conjecture** for quaternionic dimension ≤ 14 (real dimension ≤ 56). Daura Serrano, Gohr, Lawn & Schedler, [arXiv 2610.06336](https://arxiv.org/abs/2610.06336): LLMs "used extensively", mainly GPT-6-Astra and GPT-5.6-Sol, with occasional Claude Opus 5/5.5 and Claude Fable.
- **Automorphic commutator relation** (Liu–Wang, relative geometric Langlands) without extra hypotheses. Shurui Liu, [arXiv 2610.05781](https://arxiv.org/abs/2610.05781): draft produced by the agentic system Rethlas in about 1 hour with OSVerify; Danus on GPT-6-Astra failed in 5 hours.
- **Gaussian elimination growth factor** under partial pivoting for Gaussian matrices. Urschel, [arXiv 2610.06785](https://arxiv.org/abs/2610.06785): "the result of human–AI interaction involving ChatGPT Sol 5.6 & Astra 6".
- **Strong freezing in the binary perceptron** (Krauth–Mézard prediction). Gong & Li, [arXiv 2610.06275](https://arxiv.org/abs/2610.06275): GPT-6 suggested the restricted partition function that completes the monotonicity argument.
- **Tight quantum query lower bounds for XEB sampling** (proves an earlier conjecture for Haar-random states). Bhateja, Esmaili & Mantri, [arXiv 2610.06721](https://arxiv.org/abs/2610.06721): "ChatGPT Astra 6.1 produced the main results for Fourier sampling" (note: GPT-6.1 Astra was never released publicly).
- **Noncommutative AM–GM inequality** holds for n ≥ 2⌈m/2⌉². Zhong, [arXiv 2610.04874](https://arxiv.org/abs/2610.04874): "The proof of Theorem 1.2 was found by GPT6 Astra gradually."
- **Two-dimensional majority rule is P-complete**. [arXiv 2610.04311](https://arxiv.org/abs/2610.04311): results "assisted by AI agents" (Claude Code with Sonnet/Opus 5.5; Codex with GPT-6 Astra and GPT-6.1 Sol).
- **Bernstein–Sato zero loci need not be linear** (counterexample found with ChatGPT Pro; Shi & Zuo, [2610.04408](https://arxiv.org/abs/2610.04408)); **Suciu's conjecture** on hyperplane arrangements (key idea proposed by GPT-5.6 Sol; [2610.06683](https://arxiv.org/abs/2610.06683)); **Boolean Hidden Matching** generalisations (two key ideas from GPT-6 Astra; [2610.06789](https://arxiv.org/abs/2610.06789)); **stabilizer extent / Clifford rank** bounds (GPT-6 Astra and GPT-5.6 Sol "used extensively, including to generate mathematical arguments"; [2610.06819](https://arxiv.org/abs/2610.06819)); **Max-k-Cut games** counterexample (developed with GPT-6 Astra; [2610.04948](https://arxiv.org/abs/2610.04948)); **(q−1)-nests** ("comparable work by a human would warrant co-authorship" for ChatGPT 5.6; [2610.05514](https://arxiv.org/abs/2610.05514)); **Poly–Raby theorem** in infinite dimensions (example obtained directly with GPT-6 Astra; [2610.06595](https://arxiv.org/abs/2610.06595)); **non-plain virtually free group** counterexample suggested by a commercial LLM ([2610.04491](https://arxiv.org/abs/2610.04491)); **Borwein–Bradley-type identities for 1/π⁴** (Claude Opus 5.5 and GPT-6; [2610.04770](https://arxiv.org/abs/2610.04770)); **OEIS conjectures** of Kimberling and others proved with Claude ([2610.04842](https://arxiv.org/abs/2610.04842), [2610.05776](https://arxiv.org/abs/2610.05776)); **steady-state response curves** parameters found by Claude Opus 5.5 ([2610.04029](https://arxiv.org/abs/2610.04029)); **isosceles-free grid colouring** as an AI-assisted proof case study ([2610.04194](https://arxiv.org/abs/2610.04194)).
- Benchmarks/tools: VeriContest's 1,325 theorems all proved via a Lean-backed Rust verifier, 1,259 in one run on Claude Opus 5.5 at a median $1.17 per proof ([2610.03994](https://arxiv.org/abs/2610.03994)); AIProver auto-formalization agent lifts Claude Code/Codex semantic correctness to 79.8%/62.4% ([2610.05367](https://arxiv.org/abs/2610.05367)).
- Smaller AI roles (exploration, checking, editing): Prym–Green up to genus 30, GW/DT4 on local P², DML(1) property (proof of Prop. 1.6 "learned from ChatGPT 6 Astra"), Binder–DeMarco, Godbersen, Newtonian capacities (auxiliary function from GPT-6.0), Chern-form positivity, ×2×3 entropy (Claude line-by-line checking), and about 25 others.

### Oct 5–6 listing (sweep Oct 7; PDFs read)
Own entries: the **Artin–Davenport conjecture** (cubic forms in 10 variables; key lemma via ChatGPT 6.0; Bonolis, Browning, Glas & Wang, 2610.08226; `2026-10-06-artin-davenport-cubic-forms-ten-variables`), **Woodin's GCH question** for strongly compact cardinals (You, 2610.07537, GPT-6 Astra; `2026-10-06-gch-above-strongly-compact-woodin-question`), the **polynomial Littlewood–Offord** bound "discovered autonomously by GPT-6 Pro" (Grebennikov, 2610.08708; `2026-10-06-polynomial-littlewood-offord-gpt-6-pro`) and **Reed–Solomon list decoding up to capacity / proximity gaps** (Sept 4–5, GPT-5.6 Sol and ChatGPT 5.6 Pro; surfaced by the TIFR exposition 2610.08610; `2026-09-04-reed-solomon-list-decoding-capacity-ai`). A second Poincaré formalization (Frenzymath, 2610.08329) was added to `2026-09-27-poincare-conjecture-lean-formalization`.
- **Erdős–Gallai cycle decomposition conjecture** (1960s): every n-vertex graph splits into O(n) cycles and edges. Jaehoon Kim (KAIST), [arXiv 2610.07840](https://arxiv.org/abs/2610.07840): "The author made extensive use of ChatGPT (OpenAI) and Claude (Anthropic) in developing the arguments and preparing the text and figures." The same theorem is family 181 of OpenAI's 6 Oct release (manuscript dated 24 Sep; see `2026-10-06-openai-math-release-722-manuscripts`).
- **Mumford–Tate conjecture in every degree for hyper-Kähler varieties** (plus weight–monodromy). Ito & Zou, [arXiv 2610.07890](https://arxiv.org/abs/2610.07890). Most results were human and pre-August, but the reduction in Proposition 2.2 "was pointed out by ChatGPT 5.6 Sol in August 2026" and made the full conjecture available; Proposition 5.7 and Example 5.11 "were found with" ChatGPT 6 Astra's help.
- **Keevash–Mubayi simplex-cluster conjecture** in its full range (a common strengthening of the Erdős–Chvátal simplex conjecture and Mubayi's cluster conjecture). Wu & Feng, [arXiv 2610.08567](https://arxiv.org/abs/2610.08567): ChatGPT used "to discuss mathematical details, assist in developing and refining proofs". The Erdős–Chvátal simplex conjecture itself had been claimed in full by Jing Huang ([arXiv 2609.14470](https://arxiv.org/abs/2609.14470), 13 Sep; no AI statement).
- **Continuous Aldous–Lyons conjecture** fails (non-co-sofic invariant random subgroups in rank-one Lie groups, p-adic groups and Aut(T)); answers a question in Gelander's ICM paper and builds on the new non-sofic groups (including OpenAI's, `2026-08-01-openai-astra-ten-advances`). Gelander, [arXiv 2610.08581](https://arxiv.org/abs/2610.08581): "The author acknowledges the assistance of OpenAI's GPT-6 Astra in developing the arguments and preparing the manuscript."
- **McLeod's 1959 conjecture** on complete homogeneous symmetric ratios. Sra, [arXiv 2610.07302](https://arxiv.org/abs/2610.07302): "developed through extensive use of an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot"; the assistant's literature survey surfaced the queuing-theory interpretation, while the tensorization trick is the author's.
- **Dirac Faber–Krahn conjecture** (strict, non-negative mass, infinite-mass boundary). Ammari & Qiu, [arXiv 2610.08665](https://arxiv.org/abs/2610.08665): numerics from ChatGPT 6 Astra suggested the key non-vanishing property, after which "it provides the detailed proof" of Theorems 5.4 and 6.1; the authors simplified and rewrote it.
- **Random-oracle unitary synthesis is impossible** (and scalable ROM-PRUs ruled out; an O(N)-design construction). Huang, Ramkumar & Wright, [arXiv 2610.06700](https://arxiv.org/abs/2610.06700): after the humans set up the definition, "GPT 5.6 Sol … was able to solve the problem" within several hours; GPT-6 Astra helped simplify it. Contrast: OpenAI's family 283 claims the worst-case Aaronson–Kuperberg unitary synthesis problem has a positive answer.
- **Strang's conjecture** (dimension of C¹ cubic splines): a D₃-symmetric counterexample with no quasi-singular vertices. Potu, [arXiv 2610.08470](https://arxiv.org/abs/2610.08470): "The search for this triangulation relied heavily on … Claude Fable 5.1."
- **Adámek–Rosický problem (2001)**: cocompletion under sifted colimits need not be finitely accessible. Kawase, [arXiv 2610.08548](https://arxiv.org/abs/2610.08548): GPT-6.1 Sol used "to explore and refine candidate constructions following a strategy devised by the author".
- **Carlet's cyclic-additive conjecture for Kasami monomials** (an NSUCRYPTO 2019 problem). Nagy, McNeil & Vajda, [arXiv 2610.07905](https://arxiv.org/abs/2610.07905): Claude Fable 5, Claude Opus 5 and ChatGPT 6 Pro in "sustained mathematical discussions"; Lean 4 formalisation with Aristotle.
- **Asymptotic Makar-Limanov rank conjecture**. Yan, Shen & Yang, [arXiv 2610.06800](https://arxiv.org/abs/2610.06800): main results "obtained through an interaction between the authors and an artificial intelligence agent system, MechMath Agent Team (MMAT)", with GPT-6 Astra.
- **Hendrey–Norin–Wood question: mad is partitionable**, giving the clustered chromatic number of bounded-mad graphs. Grzesik et al., [arXiv 2610.04645](https://arxiv.org/abs/2610.04645): the potential function "was initially suggested by ChatGPT-5.6 Sol", but its proof relied on a lemma "whose validity was difficult to verify", so the authors wrote their own proof.
- Other substantive AI roles: Kakeya compression example answering a question of arXiv 2607.27629 (Du, [2610.05404](https://arxiv.org/abs/2610.05404); ChatGPT 5.6 Sol and 6 Astra); Pólya's inequality for every Neumann eigenvalue of annuli with Codex-built certificates (Dai & Sun, [2610.05146](https://arxiv.org/abs/2610.05146)); absolutely continuous radial convex-order counterexample "found using generative AI tools" (Guéant, [2610.07404](https://arxiv.org/abs/2610.07404)); Kuperberg's quadrilateral "chord conjecture" note from "continuous collaboration" with OpenAI and Anthropic models (Vígh, [2610.08168](https://arxiv.org/abs/2610.08168)); Hardy–Littlewood maximal operator not weak (1,1) on lamplighter, BS(1,n) and Sol (Fujiwara, [2610.08121](https://arxiv.org/abs/2610.08121); GPT-5.6 Sol for Lemma 3.1); sharply repulsive points for stochastic subgradient descent (Li & Hu, [2610.07777](https://arxiv.org/abs/2610.07777)); 1,142 weight-five bivariate bicycle codes from LLM program evolution (IBM, [2610.06623](https://arxiv.org/abs/2610.06623)).
- Smaller AI roles (checking, code, editing): Binder–DeMarco conjecture disproved on an open set ([2610.05602](https://arxiv.org/abs/2610.05602); ChatGPT numerics), Prym–Green in even genera 20–30 (Payne & Willwacher, [2610.05646](https://arxiv.org/abs/2610.05646); ChatGPT Pro and Codex), GW/DT4 on local P² ([2610.06024](https://arxiv.org/abs/2610.06024); ChatGPT-6 Astra), Godbersen's conjecture ([2610.06806](https://arxiv.org/abs/2610.06806); ChatGPT for checks), SABR right-wing conjectures of Henry-Labordère ([2610.04209](https://arxiv.org/abs/2610.04209); Codex and Claude, Lean checks), Szykuła's Figure 3 automata ([2610.07044](https://arxiv.org/abs/2610.07044); Codex), Iwahori Gaiotto conjecture ([2610.08243](https://arxiv.org/abs/2610.08243); ChatGPT found three errors), Butler's conjecture and N!/k bounds ([2610.07461](https://arxiv.org/abs/2610.07461)), Clunie–Sheil-Small coefficient-difference conjecture disproved ([2610.07770](https://arxiv.org/abs/2610.07770); ChatGPT for editing and discussion), FEEC dynamo spectra ([2610.01501](https://arxiv.org/abs/2610.01501); ChatGPT and Claude Opus 5.5). Explicitly AI-free ideas: quantum gap amplification (FOCS 2026, [2610.04952](https://arxiv.org/abs/2610.04952); ChatGPT 5.4/Claude Opus 4.6 for elementary calculations only) and Cheq non-finite axiomatizability ([2609.39577](https://arxiv.org/abs/2609.39577)).

### Early October additions (arXiv, 7 Oct)
Own entries: exact overlaps conjecture (2026-10-07-exact-overlaps-conjecture-gpt-6-astra), LeBrun–Salamon (2026-10-07-lebrun-salamon-conjecture-proof), multiple-unicast conjecture refuted (2026-10-07-multiple-unicast-conjecture-false-gpt-6).
- **Explicit counterexample to Tsirelson's problem** (approximation form; Cqa ≠ Cqc). Gao, Yu & Zhi (CAS), [arXiv 2610.10248](https://arxiv.org/abs/2610.10248). A binary linear system game with 1,417,152 equations in 1,889,684 variables has a perfect commuting-operator strategy, while every finite-dimensional quantum strategy succeeds with probability strictly below 1. The full system and the separation are formalized in Lean 4 with Mathlib ([repo](https://github.com/leanhnhe091221-boop/explicit-linear-system-game-formalization)). Tsirelson's problem was already answered negatively by MIP* = RE (2020), but non-constructively. The paper builds on Wang & Zhi's explicit Connes-embedding polynomial (in the [summer catalogue](/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/)) and on Kun–Thom and Thom (2026). Disclosure: "Generative AI tools (GPT 6 Astra and Fable 5.1) were used to assist with mathematical exploration and proof development, the implementation and computational verification of the explicit construction, the writing and debugging of Lean 4 code, and the drafting and revision of this manuscript."
- **Lê's conjecture** (1982; injective germs ℂ² → ℂ³), reduction to orders 2, 3, 4. Portilla Cuadrado, [arXiv 2610.10418](https://arxiv.org/abs/2610.10418). The paper itself is human-led (ChatGPT 5.6 for references and writing), but its disclosure says that when the draft was handed to ChatGPT, "it found a flaw in the argument. After several iterations and a final long session, ChatGPT found a fix for the case mn = 4 and proved the conjecture, also, in the two remaining cases, mn = 2, 3. These cases will appear in another paper", in which the author "has minor intellectual contribution". If that follow-up holds, Lê's conjecture would be settled with the decisive cases due to ChatGPT.
- **Polishchuk–Van den Bergh conjecture on curves**: explicit S₃-action counterexamples. Pan, [arXiv 2610.10147](https://arxiv.org/abs/2610.10147). ChatGPT used "as a tool for exploring proof strategies, refining arguments, and drafting and revising the manuscript".
- **Centroaffine Bernstein conjecture** disproved (complete centroaffine extremal hypersurfaces via Calabi compositions). Xing, Sun & Xu, [arXiv 2610.09332](https://arxiv.org/abs/2610.09332). "The authors used AI models to assist with proving Theorem 1.1" (model not named).
- **Hassanzadeh–Nasrollah Nejad–Simis conjecture on Euler conductors** proved, and Ma–Zuo's integral-closure strengthening disproved. Zhang & Zuo, [arXiv 2610.09833](https://arxiv.org/abs/2610.09833). Thanks "AI models, particularly ChatGPT 6 Astra, for valuable assistance in exploring some of the proof strategies and calculations".
- **Coarse graph theory**. Georgakopoulos (Warwick), [arXiv 2610.09992](https://arxiv.org/abs/2610.09992). "A mid-range AI agent made available by the University of Warwick … CheapGPT, was used to prove Lemma 8.2" (minor role).
- **Tamagawa products in BSD, statistically** (Angdinata, Lau, Ono, Swaminathan, Vemulapalli, [arXiv 2610.10243](https://arxiv.org/abs/2610.10243)). Not an open-problem resolution: all 21 formal targets were "autoformalized in Lean by AxiomProver" (Axiom Math; [repo](https://github.com/AxiomMath/BSDTamagawa)).
- Minor-role disclosures the same day: Type-ω Vaught's conjecture for ω-stable theories (2610.09314, ChatGPT-5.6 Sol proofreading), geometric Arveson–Douglas conjecture via Veronese embeddings (2610.09836, Claude Sonnet 4.6 for spelling), Pappas's conjecture on regular integral models (2610.09475, ChatGPT found a sign error), the Cyclicity Conjecture (2610.10093, GPT-5.6 via OpenCode for simplifications and literature review; now own entry 2026-10-07-cyclicity-conjecture-positive-operators-dobrick).

### 6–8 Oct additions (sweep 2026-10-09)
Own entries: finite index over treeable, the Jackson–Kechris–Louveau problem (2026-10-06-finite-index-over-treeable-jkl-problem), Dualization lower bound under ETH found by GPT-6 Astra (2026-10-08-dualization-eth-lower-bound-gpt-6-astra), Hilbert number Ω(d² ln² d) with GPT-5.6 Sol and Lean (2026-10-08-hilbert-number-lower-bound-gpt-5-6-sol-lean), and the human-written hyperkähler SYZ proof (2026-10-08-hyperkahler-syz-conjecture-engel-mauri). Found late: the Courtade–Kumar conjecture, proved three times on 21–22 Sep with AI (2026-09-21-courtade-kumar-conjecture-proved).
- **Erdős–Sós bipartite-link conjecture** disproved: 3-graphs on 497k vertices whose links are all bipartite, with edge density 1/4 + 25/420 903 336 > 0.250000356 (the conjecture said 1/4 + o(1); flag-algebra upper bound 0.258295). Tianchi Yang (Georgia Tech), [arXiv 2610.11642](https://arxiv.org/abs/2610.11642): "The author used GPT-6.1 to assist with exploring proof strategies, checking proofs, and improving the exposition."
- **Mongardi's conjecture** (finite symplectic group actions on K3^[n]-type hyperkähler manifolds: rk(S) + ℓ(A_S) < 24 for the Leech coinvariant lattice). Fu, Wang & Zheng, [arXiv 2610.11109](https://arxiv.org/abs/2610.11109): "Large language models were used to explore proof strategies, assist in the development and refinement of mathematical arguments, and draft and revise the text."
- **Two compactness conjectures of Brudnyi** (Conjectures 2.5 and 2.11) refuted by a translation-invariant, order-continuous quasi-Banach function lattice on ℝ. Huang & Xu, [arXiv 2610.11066](https://arxiv.org/abs/2610.11066): "The authors acknowledge the assistance of ChatGPT 6 Astra. Suggestions obtained through interactions with this AI system were subsequently examined, reformulated, incorporated into the manuscript, and independently verified."
- **Bermond–Thomassen conjecture, asymptotically**: minimum out-degree 2k + O(k^{3/4} log k) forces k disjoint directed cycles. Bang-Jensen, Wang & Wang, [arXiv 2610.11517](https://arxiv.org/abs/2610.11517): "AI tools aided the construction of the sequence Z_j(v) in Lemma 5.5."
- **P. A. Smith's conjecture on local presentations of groups** (a group is recovered from every generating subset iff it is trivial or cyclic of order 2 or 3). Yu Pan, [arXiv 2610.10586](https://arxiv.org/abs/2610.10586): ChatGPT used "for exploration of proof strategies and assistance with developing and checking arguments".
- Smaller roles: triangular prism equations for near-group categories, linking unitarity to a Stark-type real-multiplication conjecture (He, Liu, Lu, Palcoux & Ren, [2610.11652](https://arxiv.org/abs/2610.11652); "The AI tool was used as a research assistant"); Volčič's rank Nullstellensatz conjecture (2610.01804, AI-tools assistance acknowledged) and Lefschetz multisections (2610.01776, AI for consistency checks and GAP code), both posted 1 Oct; vector parking functions (2610.05665, LLM for literature and figures); Hedetniemi's conjecture for uncountable complementary graphs (2610.08027, ChatGPT for grammar); found late from the leads queue: quantitative explosion of the divisible sandpile (Bou-Rabee & Panagiotis, [2609.02829](https://arxiv.org/abs/2609.02829), 2 Sep; "We acknowledge ChatGPT 5.6 for help with the proof of Proposition 4.7"). Galt–Tyutyanov conjecture on Skiba's question (2610.12295): "AI is not used in proving the conjecture", ChatGPT only for language.
- Related human work (no AI statement): Vu Khac Ky's balanced Hellinger conjecture for antipodal Boolean functions ([2610.10618](https://arxiv.org/abs/2610.10618), citing OpenAI family 119); an independent computer-assisted proof of the Chen–Raspaud conjecture for k = 4 (Nazarczuk, [2610.07968](https://arxiv.org/abs/2610.07968); research done in 2025, concurrent with Choi's computer-free proof).
- Research on the process: "Viva La Vida" (Xu, Zheng & Han, Berkeley, [2610.04829](https://arxiv.org/abs/2610.04829)) instrumented a multi-agent prover on an open problem (186 hours, $5,694). A unanimity verifier silently blocked acceptance after API failures, one verifier approved three claimed solutions that GPT rejected, and 26% of the lemma library came from rejected arguments with their refutational context removed.

### 8–9 Oct additions (manual arXiv scan, 12 Oct evening; PDFs read)
Five results from this listing have their own entries: [GPT-6 Astra's kinetic Krylov–Safonov theorem and Pucci's conjecture](/e/2026-10-09-kinetic-krylov-safonov-pucci-conjecture-gpt-6-astra/), [Smale's sixth problem for five bodies](/e/2026-10-09-smale-sixth-problem-five-bodies-lean-tomar/), [the rank of 3×3 matrix multiplication over F_2](/e/2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar/), [Erdős Problem #522](/e/2026-10-08-erdos-522-littlewood-polynomial-zeros-lean-kawada/) and [PPT entanglement distillation](/e/2026-10-08-ppt-entanglement-distillation-rains-bound-chatgpt-astra/). Also in the listing:
- **Forcade–Pollington conjecture (1990)**: if an injective map from {1, …, k} to a group of order k respects products, the group is abelian (listed as unsolved by Richard Guy). John M. Campbell (Dalhousie), [arXiv 2610.12087](https://arxiv.org/abs/2610.12087): the proof is "based on our extensive interactions with GPT-5.6 Pro".
- **Shamir secret sharing, the "one-half barrier"**: one-bit local leakage resilience for rates k/n ≥ 1/2 − δ, and improved worst-case Optimal Polynomial Intersection. Yihang Sun and Mary Wootters (Stanford), [arXiv 2610.12357](https://arxiv.org/abs/2610.12357): ChatGPT 6-Astra "produced the key truncation idea in Step 5" and, prompted with the authors' idea, "obtained a version of the balanced ensemble and Theorem 1.12"; ChatGPT 5.6-Sol "was heavily consulted" earlier.
- **Hot spots on non-smooth domains**: Krzysztof Burdzy, [arXiv 2610.12622](https://arxiv.org/abs/2610.12622), extends the smooth-domain hot-spots theorem from OpenAI's 6 Oct collection to arbitrary bounded simply connected planar domains with a simple first eigenvalue (non-strict form). "This paper was generated by ChatGPT in response to multiple prompts from the human author. KB has not checked every detail, because the theorem is, to a human, an obvious corollary." Rodrigo Bañuelos generated "an almost identical theorem and proof" at the same time; neither will submit to a journal. Burdzy adds that "The OpenAI model that generated [3] missed the theorem presented in this note" and that OpenAI's paper failed to cite de Dios Pont's counterexample for convex domains in high dimensions.
- **Quasi-Riemann hypothesis, constants**: Baiying Liu, [arXiv 2610.12234](https://arxiv.org/abs/2610.12234), re-tunes two parameters of OpenAI's 7/8 argument to get zero-free half-planes Re s > 0.874975 and Re s > 0.874957…, formalized in Lean. "OpenAI Codex (GPT 6.1 Sol Medium-Ultra) assisted with drafting, source checking, and the search for improved bounds. Anthropic Claude (Opus 5.5 Medium) assisted with auditing."
- **Eleven-variable Vasc inequality** (a cyclic sum is nonnegative for all positive 11-tuples): exact rational certificates and a Lean 4 proof of 2,031 source files that trusts native evaluation, by the "MechMath Agent Team" of the Key Laboratory of Mathematics Mechanization (Guo, Qiu, Cao, Feng, Gao), [arXiv 2610.13463](https://arxiv.org/abs/2610.13463). The paper does not describe the agents' role.
- Smaller roles: fast and stable approximate matrix multiplication, with a better border-rank bound for ⟨2,4,5⟩ (Dumitriu, Holtz, Schneider et al., [2610.13905](https://arxiv.org/abs/2610.13905); "An LLM also provided feedback on an earlier draft … highlighting an error in the proof of Theorem 4.6, which was corrected by a human").
- Related human work (AI for wording only, or stated not used): the 4-to-1 Games Conjecture with perfect completeness, Khot 2002 (Fei, Minzer & Wang, [2610.12378](https://arxiv.org/abs/2610.12378); "the mathematical ideas and the overall approach presented in this paper are due to the authors"); Problem 101 of Yau's 1982 list, negatively curved minimal surfaces in the round 4-sphere (Caniato, [2610.12238](https://arxiv.org/abs/2610.12238); "all the proofs were developed without the use of AI tools"); the four-loop pressure of cold QED, open since 1977 (Kärkkäinen et al., [2610.13873](https://arxiv.org/abs/2610.13873); "No AI tools were used to perform or validate the calculations"); a gravitational path-integral proof of the OSV conjecture (Bobev, Boruch, Iliesiu & Maskalaniec, [2610.12550](https://arxiv.org/abs/2610.12550); "The manuscript was written by the authors").

### 5–8 Oct additions (lead from the 13 Oct sweep; PDFs read 13 Oct)
Five results from this batch have their own entries: [Erdős Problem #993, unimodality of forest independence polynomials](/e/2026-09-27-erdos-993-forest-independence-polynomials-unimodal/), [the Toms–Winter conjecture](/e/2026-10-07-toms-winter-conjecture-completed-chatgpt-astra-lemma/), [Bolzano's unattended run on 3,800 open problems](/e/2026-10-07-bolzano-automated-open-problem-solving-charles-university/), [multiplier associativity is hard for resolution](/e/2026-10-06-multiplier-associativity-hard-for-resolution-gpt-5-6-sol/) and [completeness of decreasing diagrams](/e/2026-10-05-decreasing-diagrams-complete-for-confluence-rta-problem-56/). Knausgård's 83.9% distinct-zeros paper is in the [zeta-zeros entry](/e/2026-08-10-claude-riemann-zeta-zeros-two-thirds/).
- **Jackson–Ordaz conjectures on Hamilton cycles in digraphs**: a near-linear Chvátal–Erdős condition, f_2(a) = O(a (log a)^4 / (log log a)^2), towards the conjectured f_2(a) ≤ a + 1, and a disproof of the Jackson–Ordaz pancyclicity conjecture. Chengli Li and Bo Ning, [arXiv 2610.10292](https://arxiv.org/abs/2610.10292). "Early discussions with ChatGPT 5.6 explored several approaches … but yielded only partial results. … The main progress came on 9 September 2026, when extensive author-guided discussions with ChatGPT 6 led to the near-linear bound stated in Theorem 1.1."
- **Colonel Blotto games**: computing an inverse-polynomial approximate Nash equilibrium of two-player general-sum Blotto is PPAD-hard even with two battlefields, "resolving an open question of [KPF+25]"; constant-rank cases are tractable. Pollatos and Kontogiannis, [arXiv 2610.05956](https://arxiv.org/abs/2610.05956): "Initial versions of the proofs were developed with the assistance of ChatGPT 5.6 Sol and Astra."
- **Finite-time blowup for the unforced 2D incompressible porous media equation** (smooth odd initial density on the torus). Mimi Dai, [arXiv 2610.06712](https://arxiv.org/abs/2610.06712), 51 pages. The pressure-Hessian control follows "an idea from the Euler construction of OpenAI"; acknowledgement: "Publicly available ChatGPT was used to understand the finite-time blowup construction of Open AI [14] for the Euler equations, and to assist with some of the computations in the manuscript." A human-led result that transfers a technique from an AI-written manuscript to another equation.
- **Papers building on OpenAI's unrefereed manuscripts** (each result is conditional on, or derived from, an OpenAI preprint that has not been reviewed):
  - de Bruijn–Newman constant Λ ≤ 0.13963, "assuming that ζ(s) has no nontrivial zero with Re s > 7/8, as asserted in a recent unrefereed preprint of OpenAI". Hakimoglu-Brown, [arXiv 2610.12803](https://arxiv.org/abs/2610.12803): "ChatGPT assisted with the replacement strategy, analytic derivations, exact scalar and Bernstein-coefficient checks, interval verification of the stepped tables, and drafting." "The seven-eighths hypothesis remains an assumption."
  - L^3 reverse square function estimate for the light cone in four dimensions, obtained "from the critical local-smoothing and round-cone propagation statements in the September 2026 OpenAI manuscript". Temur, [arXiv 2610.12681](https://arxiv.org/abs/2610.12681); no AI-use statement found.
  - Ranks of the twin-prime elliptic curves y² = x(x−2)(x−p): equality of analytic and algebraic rank and the exact BSD formula for p ≡ 3, 5, 7 (mod 8), "assuming the low-corank Birch–Swinnerton-Dyer statement announced in the October 2026 OpenAI mathematics release". Ruihan Chen, [arXiv 2610.09370](https://arxiv.org/abs/2610.09370); no AI-use statement found.
  - Anderson model: Jaksic and Last, [arXiv 2610.11611](https://arxiv.org/abs/2610.11611), a human note on the Bethe lattice, add an appendix written "one day after OpenAI publicly released the preprint … claiming to prove purely absolutely continuous spectrum at weak disorder for the Anderson model on Z^d, d ≥ 3", observing that their structural argument shortens the last step of OpenAI's proof.

### 6–9 Oct additions (sweep 2026-10-13, first arXiv fetch after the rate-limit outage; PDFs read 13 Oct)
No new own entries from this batch. All items are preprints and none has been reviewed.
- **Isoperimetric problem in the three-dimensional cube**: the complete relative isoperimetric profile of the unit cube with a classification of all minimizers ("corner eighth-balls, edge quarter-cylinders, and coordinate slabs, with transitions at 4π/81 and 1/π") and quantitative stability with the optimal exponent two; the authors say they "resolve the isoperimetric conjecture in Milman's formulation" for all cuboids (0, β) × (0, 1)² and "the three-dimensional case of Brezis's Open Problem 10.1". Bin Deng, Jiahuan Li and Yilu Liu, [arXiv 2610.08644](https://arxiv.org/abs/2610.08644), 63 pages. "The authors used ChatGPT (OpenAI) as an auxiliary tool for language editing, exploration of some of the proof strategies and calculations." A well-known open problem if the proof holds; human-led, no expert check found yet.
- **László's conjecture on the Toeplitz Böttcher–Wenzel form is false**: for an explicit family of Toeplitz pairs the quartic form is not a sum of squares at all sufficiently large orders; it is a sum of squares for every 2 ≤ N ≤ 50 (Lean 4 through order 20), and the large-order negative theorem "has also been formally verified in Lean 4". Wenqi Zhu and Ping Nie, [arXiv 2610.08980](https://arxiv.org/abs/2610.08980): the authors "used large language model systems (Claude, Anthropic; Codex models, OpenAI) to generate candidate constructions and code, to run computations, to draft mathematical arguments". Lean-verified in part; repository github.com/erenup/toeplitz-bw-not-sos.
- **Sampling the Sherrington–Kirkpatrick model**: polynomial-time simulated annealing with inverse-polynomial total-variation error for every β < 1, through a weak Poincaré inequality. Zhe Hou, Jingcheng Liu and Yixiao Yu, [arXiv 2610.09882](https://arxiv.org/abs/2610.09882). The disclosure is unusually detailed: key ideas "were selected among a few alternatives found by GPT 5.6 Pro. GPT 5.6 Pro also claimed a false proof of an FPRAS extension …, which GPT 6 Astra Pro managed to fix. GPT 6 Astra Pro also found a separate deterministic FPTAS for the SK model, which we had to defer to a separate manuscript in view of OpenAI's recent announcements." "We verified all the proofs produced by AI."
- **Dynamic matching with abandonment** (Arnosti, Niazadeh, Nuti, Simon, [arXiv 2610.12991](https://arxiv.org/abs/2610.12991)): "One result in this paper was obtained with the help of an AI system." The polynomial-time computability of a restricted fixed point (Proposition 5.4) "was first found by OpenAI's Codex Astra, after several rounds of prompting and interaction with the authors, and was subsequently verified and simplified by the authors".
- **Deb–Sokal conjectures on higher-order Stirling triangles**: row log-concavity of the higher-order Stirling cycle and subset triangles, plus a combinatorial interpretation of the subset case. William Y.C. Chen and Elena L. Wang, [arXiv 2610.09362](https://arxiv.org/abs/2610.09362); the proofs rely on "strengthened log-concavity inequalities discovered with the assistance of ChatGPT 5.6".
- **Hurwitz vector fields with periodic orbits** (Problem 19 in Gasull's list of open problems on low-dimensional dynamics): Castañeda and Rebollo-Perdomo, [arXiv 2610.09176](https://arxiv.org/abs/2610.09176). "The construction presented in this paper emerged from an iterative and exploratory dialogue between the authors and OpenAI's ChatGPT."
- **Building on OpenAI's unrefereed manuscripts** (results inherit that status):
  - Irrationality exponent of log r is 2 for every positive rational r ≠ 1: Liu, Jiang and Zhang, [arXiv 2610.10192](https://arxiv.org/abs/2610.10192), adapting "the separated-weight method" of OpenAI's π manuscript; no AI-use statement. Details in [the π entry](/e/2026-10-06-irrationality-exponent-pi-openai/).
  - Radius of comparison equals half the mean dimension for minimal free actions of amenable groups with the uniform Rokhlin property and Cuntz comparison of open sets: George A. Elliott and Zhuang Niu, [arXiv 2610.11143](https://arxiv.org/abs/2610.11143). "The arguments in both cases are based on the recent preprint of OpenAI dealing with Z-actions"; "The work was carried out using ChatGPT."
  - Erdős similarity problem for null sequences with adjacent ratios bounded below: Kan Jiang and Lifeng Xi, [arXiv 2610.11449](https://arxiv.org/abs/2610.11449), which "adapts a finite-tree construction for geometric sequences" and cites two unpublished OpenAI manuscripts (dyadic and geometric cases); no AI-use statement found.
- Smaller roles: a sharp minimum for cube-apex bodies with Lean 4 verification (Shen and Dubovski, [2610.11014](https://arxiv.org/abs/2610.11014); "OpenAI Codex assisted mathematical exploration, Lean proof development, literature checking and manuscript preparation"); r-neutralized Pfister–Sullivan entropy (Chen, Xie, Zhou, [2610.11639](https://arxiv.org/abs/2610.11639); GPT-6 helped "in the construction of Example 3.12"); sharp Hilbert transform norms on Schatten classes (Long, [2610.10158](https://arxiv.org/abs/2610.10158); "OpenAI models as assistive tools"); constant-round approximation of dominating set (Amiri and Siebertz, [2610.09618](https://arxiv.org/abs/2610.09618); "AI agents to fill in technical details … and check the correctness of the proofs"); bi-contact plugs (Hozoori, [2610.11243](https://arxiv.org/abs/2610.11243); an LLM supplied "some computational details" of Lemma 5.8).

### 9 Oct addition (leads run, 13 Oct evening; PDF read)
- **Quantitative Eshelby stability** (solid mechanics; a stability version of the Eshelby conjecture that uniform interior strain characterizes ellipsoidal inclusions): a nonlinear two-sided relation between interior strain heterogeneity and distance from the ellipsoid family, plus a finite-shape theory checked by finite elements. Bo Ni, [arXiv 2610.13400](https://arxiv.org/abs/2610.13400), 33 pages. The paper presents itself as a case of "human-led, AI-assisted theoretical mechanics": "Generative AI supported literature organization, candidate formulation, analytical exploration, symbolic and numerical checking, code development, partial formalization, and manuscript preparation", while the researchers set objectives and validation; "Selected mathematical steps were reconstructed by an AI reviewer given the definitions and target before seeing the candidate proof", and the Lean 4 work "is not described as complete formal verification". It adds: "No claim of autonomous discovery, measured research acceleration, or AI correctness probability is made." The model is not named. The paper also introduces a curated list, Open Problems in Mechanics.

### 7–13 Oct additions (sweep 2026-10-14, first arXiv fetch after two skipped runs; 242 papers listed, ~60 PDFs read)
Own entries: the **Vaught conjecture for L_ω1,ω** disproved with a Lean-certified counterexample formalized over 8 months with Claude Code and Codex (Ackerman, Freer & Knight, 2610.15217; `2026-10-10-vaught-conjecture-infinitary-counterexample-lean`), the **Bloch constant conjecture** of 1937 disproved, Lean by Codex (Zhang, 2610.16808; `2026-10-12-bloch-constant-conjecture-disproved`), the **Erdős Matching Conjecture** for large matchings, with the k ≥ 6 cases of the key tail bound developed by generative AI (Chi, Sun & Wang, 2610.16203), **Chow polynomials of matroids not log-concave**, counterexamples found in LLM searches with Claude Opus 5.5 and ChatGPT Astra (2610.17135), **Skryabin's conjecture** on noetherian Hopf algebras, strategy from GPT-6 (Sciandra, 2610.15473), Drouot's **bulk-conductance proof "due to Claude Opus 5.5"** (2610.16208), the **TheoPhysEval** benchmark (2610.16733), and two expert write-ups of OpenAI release proofs: **Landau–Siegel zeros** with explicit constants (Banks, 2610.15483) and **Thompson's group F** (Kim & Lodha, 2610.18360). Wang & Dai's ChatGPT-assisted Lyapunov spectra (2610.14922) and Chen & Tong's AI-assisted Smale five-body paper (2610.16107) were added to the existing entries.
- **Cutwidth and pathwidth in O(log n)**, matching the LP integrality gap and improving Bansal–Katzelnick–Schwartz (FOCS 2024). Aditya Anand, [arXiv 2610.17237](https://arxiv.org/abs/2610.17237): the author proposed the greedy Seymour template; "GPT-5.6 Sol developed the proof establishing the desired guarantee … The improvement to O(log n) for the layout problems … was obtained by Claude Opus 5.5, when given the paper with these results."
- **Virtual extension complexity can be exponentially smaller than extension complexity** ("the core open problem on the topic"). Black, Hertrich, Kober & Loho, [arXiv 2610.15909](https://arxiv.org/abs/2610.15909): "The proofs of all three of our main results arose from multiple discussions with different versions of ChatGPT (ranging from 5.6 Sol to 6 Pro)"; the authors rewrote them because they "were quite difficult to parse".
- **Forsythe's conjecture / ACI(2) for real symmetric matrices** (restarted Arnoldi with s = 2). Faber, Liesen, Liesen & Tichý, [arXiv 2610.16492](https://arxiv.org/abs/2610.16492): "an extensively rewritten and simplified version of a proof found using Codex with GPT-5.6 Sol" in Ultra reasoning (26 July), with subagent cross-review; the section on its origin lists agent failure modes (getting lost in numerics, drifting notation, "LLM slop"). Compare the Colbrook–Stepaniants–Townsend claim in the summer catalogue.
- **Rényi area-law conjecture false for α < 1 in dimension ≥ 2**. Tomotaka Kuwahara, [arXiv 2610.15199](https://arxiv.org/abs/2610.15199): ChatGPT used throughout since GPT-5 (Aug 2025); "the lower-bound construction establishing the optimal scaling in this manuscript originated from a proposal by GPT-6 Astra".
- **Artin's problem for skew fields**: the first division rings infinite over their centre that are index-symmetric (Weyl skew fields and more). Lauritzen & Thomsen, [arXiv 2610.15469](https://arxiv.org/abs/2610.15469): "The central idea of this paper arose from an unexpected suggestion by GPT-6 Astra … just after its release"; GPT-6 Astra, Fable 5.1 and Opus 5.5 then helped develop generalizations.
- **Lawson surfaces ξ_{k,l} are non-algebraic**, answering Hsiang's 1967 question and Lawson's 1970 conjecture for ξ_{2,2}. Lynn & Sebastian Heller, [arXiv 2610.16853](https://arxiv.org/abs/2610.16853): main ideas are the authors'; "OpenAI Codex and ChatGPT Astra were used to explore and test these ideas and proof strategies, prepare drafts, and edit the manuscript."
- **Rudin's problem on uniformly bounded orthonormal bases of homogeneous polynomials** on complex spheres, all K ≥ 2 (Bourgain had K = 2, 3). Gonçalves, Ferreira & Radziwiłł, [arXiv 2610.14160](https://arxiv.org/abs/2610.14160): "Only the decomposition algorithm of Section 4 was developed in conjunction with generative AI."
- **Combinatorial auctions with samples**: (2+ε) prophet-with-samples mechanism with O(log log m) samples, resolving two open problems of Dütting et al. Rubinstein & Singla, [arXiv 2610.17517](https://arxiv.org/abs/2610.17517): GPT-5.5 Pro "to explore proof strategies".
- **Yau's number-theoretic conjecture** on integral points in right simplices, every n ≥ 3. Liu, Yau & Zuo, [arXiv 2610.17704](https://arxiv.org/abs/2610.17704): ChatGPT (GPT-5.6 Sol, GPT-6 Astra Pro) incl. the AIM framework, for exposition and "technical details of the proofs".
- **Lin–van den Driessche 3×3 block conjecture (2014)**, so the p-Bures–Wasserstein distance is a metric for 1 ≤ p ≤ 2. Suvrit Sra, [arXiv 2610.15889](https://arxiv.org/abs/2610.15889): "developed through extensive use of an AI assistant (OpenAI's GPT Pro). None of the claims were obtained zero-shot"; first complete proof 11 Aug 2026.
- **Spectral counting for x₁² + x₂² − x₃² = D with small x₃**. Stephan Baier, [arXiv 2610.15706](https://arxiv.org/abs/2610.15706): "written in collaboration with Claude (Opus 5.5, High)"; Claude "proposed the spectral approach" and "proved Theorems 5–8".
- **Krzyż's conjecture for the sixth coefficient** (|a₆| ≤ 2/e), by rigorous branch-and-bound. Frank Wikström, [arXiv 2610.15206](https://arxiv.org/abs/2610.15206): Claude Opus 5.5 for literature, numerics and "the design and implementation of the branch-and-bound code"; code reviewed by GPT-6 Astra.
- **Auluck–Chowla–Gupta conjecture (1942)**: p(n,k) is unimodal in k for every n, with ball-arithmetic constants and exhaustive checks to 2·10⁵. Jaideep Sai Padhi, [arXiv 2610.14084](https://arxiv.org/abs/2610.14084): "AI tools (Anthropic's Claude) were used … for the computations, the accompanying code, and the preparation of the manuscript."
- **Goodman-type bounds for all odd cycles** at edge density p ≥ 2/3. Im, Kim & Yang, [arXiv 2610.18277](https://arxiv.org/abs/2610.18277): proof search with GPT-5.5/5.5 Pro and Claude Fable 5.0, then GPT-6.0 Astra and Opus 5.5; "The Lean formalisation of the proof was carried out by Claude Fable 5.1 and Claude Opus 4.8, with only minimal input from the authors."
- **Unit distances, first half of the Dúcz–Varga conjecture**: f(n) = (m₁ + o(1))n. Li & Su, [arXiv 2610.15424](https://arxiv.org/abs/2610.15424): the key observation, write-up and Lean formalization "were produced with Claude (Anthropic)", audited with GPT.
- **Finite-order exponential mixing does not imply the CLT** for smooth diffeomorphisms. Bassols Cornudella & Castro, [arXiv 2610.17162](https://arxiv.org/abs/2610.17162): ChatGPT 6.0 (Astra) as research assistant; Lean formalization of the main results, kernel-checked.
- **BB84 security, end-to-end in Lean** (composable, finite-key, coherent attacks). Ben Shahar, Morag, Englund & Sulimany, [arXiv 2610.17286](https://arxiv.org/abs/2610.17286): "The proof itself was enabled by LLM-based autoformalization" with Claude and GPT agents; parts written by a locally run open-weight GLM-5.3-Flash "under the direction of Claude Opus 5".
- **Kahane's conjecture in the Wiener algebra** (‖Fⁿ‖_A = o(log |n|) forces F = e^{i(mx+b)}). Przemysław Ohrysko, [arXiv 2610.14981](https://arxiv.org/abs/2610.14981): "supported by OpenAI through the ChatGPT for Academic Researchers program"; AI tools "for mathematical verification and editorial support".
- **Cartan determinant conjecture for representation-finite algebras over perfect fields** (Liang Chen, [2610.14638](https://arxiv.org/abs/2610.14638): "ChatGPT assisted with this work"); **multiplicity one and van Dijk's conjecture** for connected complex symmetric spaces (Yuxuan Ge, [2610.14540](https://arxiv.org/abs/2610.14540): ChatGPT-6 Astra and Claude Opus 5.5 "for proof exploration and drafting"); **positively curved six-manifolds with SO(3)/SU(2) symmetry** ("Some of the results in this paper are discovered by ChatGPT-6-astra", Gao & Liu, [2610.14582](https://arxiv.org/abs/2610.14582)); **Tuza's conjecture when e(G) ≥ n²/4 − cn, c < 2** (ChatGPT 6 Pro for strategies and checking; Liu & Zeng, [2610.15539](https://arxiv.org/abs/2610.15539)); **de Branges–Rovnyak coefficient-formula counterexample** (ChatGPT "to develop and explore strategies for the proof of Theorem 3.1"; Ransford, [2610.15019](https://arxiv.org/abs/2610.15019)); **integral ABP estimates fail for a nonlocal operator** (ChatGPT; Jiang & Xu, [2610.16162](https://arxiv.org/abs/2610.16162)); **2005 digraph-Laplacian polygon conjecture false for n = 7** (Claude and ChatGPT for eigenvector calculations; Chebotarev, [2610.14100](https://arxiv.org/abs/2610.14100)); **Game of Cycles parity conjecture fails on trees** (Claude wrote, ran and cross-checked the programs, with independent re-implementations by separate instances; Xu, [2610.15145](https://arxiv.org/abs/2610.15145)); **failure of Fourier–Schur transference** (the key inequality "was produced by ChatGPT-5.6"; González-Pérez, Parcet, Pérez García & Wang, [2610.15881](https://arxiv.org/abs/2610.15881)); **Lipschitz viscosity solutions of the minimal surface system** (ChatGPT Edu, GPT-6.1 Sol, suggested the White–Santilli framework; Dimler, [2610.15952](https://arxiv.org/abs/2610.15952)); **cyclic steepest descent is R-superlinear a.e.** (GPT-5.6 Sol, Claude Fable 5 and GPT-6 Astra workflow; Li & Wang, [2610.15335](https://arxiv.org/abs/2610.15335)); **parallel edge ranking of trees is P-complete**, open since SODA'93 (the hardness proof is not credited to AI; a companion MPC algorithm for 2-approximate edge ranking, Theorem 1.2, "was developed through extensive interaction with ChatGPT 6 Pro"; Giliberti, Hajiaghayi & Yun, [2610.12438](https://arxiv.org/abs/2610.12438)); **NP-hardness of heterogeneous asynchronous multi-player bandits** (Claude Opus via Claude Code identified the 3-partition reduction; [2610.18514](https://arxiv.org/abs/2610.18514)).
- Second Jones problem (a series with a Cantorval of subsums and term ratio → 1/2): solved by humans ([2610.14251](https://arxiv.org/abs/2610.14251)), who note that an "artificial intelligence laboratory" (Metriq PRISM Laboratory, Zenodo 10.5281/zenodo.21434379) had earlier posted a different candidate solution "obtained with substantial use of artificial intelligence" and not peer reviewed.
- Smaller roles (exploration, checking, editing, Lean help): Lotka–Volterra thresholds (ChatGPT Pro derived the integral representations, 2610.16748), Dolbeault's 2021 conjecture (ChatGPT/Codex brainstorming, 2610.14359), reduced Embry problem (GPT-5.6 "solely to accelerate calculations", 2610.17303), SLE natural length (GPT 6, 2610.15066), monotone completeness and AW*-algebras ("Errors in a previous version were found by GPT 6 Astra", 2610.17435), Schanuel and rational-weight GNNs (ChatGPT-5.6 Sol Max, 2610.16993), dihedral Hessenberg spaces (Lean with GPT Sol 6.1 and Opus 5.5, 2610.16001), pedigree polytopes (Lean with Claude, 2610.15914), QEDIA quantum program verification in Lean (LLM prover agent, 2610.17708), Frank–Loss counterexamples (2610.15787), Tu–Deng equality cases (2610.12630), and about 40 others with editing-only statements. A Lean formalization of the Ramaekers disproof (2610.16323) was done "without any assistance from large language models".

#### Why it matters
In the first nine days of October 2026 the stream did not slow down: each arXiv listing brought several named conjectures with a disclosed AI role, and two new patterns appeared. Authors now describe the model's work in detail, including its false proofs, and other mathematicians have started to build on OpenAI's 6 October manuscripts before anyone has refereed them, so errors there would propagate. For a model with an older cutoff, the practical lesson is the same as for the summer: check the current status of a conjecture before calling it open, and check whether a claimed resolution has been verified.

Sources: [arXiv listing: mathematics, new submissions](https://arxiv.org/list/math/new); [arXiv 2610.06461: Golod–Shafarevich towers of arithmetic surfaces (GPT-5.6 Sol, Fable, Aletheia)](https://arxiv.org/abs/2610.06461); [arXiv 2610.05661: amicable numbers bound (proof found by ChatGPT Astra)](https://arxiv.org/abs/2610.05661); [arXiv 2610.10248: explicit counterexample to Tsirelson's problem (Lean 4)](https://arxiv.org/abs/2610.10248); [GitHub: explicit linear system game formalization (Lean 4)](https://github.com/leanhnhe091221-boop/explicit-linear-system-game-formalization); [arXiv 2610.10418: Lê's conjecture, reduction to orders 2, 3, 4](https://arxiv.org/abs/2610.10418); [arXiv 2610.12087: Forcade–Pollington conjecture (GPT-5.6 Pro)](https://arxiv.org/abs/2610.12087); [arXiv 2610.11642: Erdős–Sós bipartite-link conjecture disproved](https://arxiv.org/abs/2610.11642); [arXiv 2610.08980: László's Toeplitz Böttcher–Wenzel conjecture is false (Lean 4)](https://arxiv.org/abs/2610.08980); [arXiv 2610.09882: sampling the Sherrington–Kirkpatrick model (detailed AI disclosure)](https://arxiv.org/abs/2610.09882); [arXiv 2610.12622: hot spots on non-smooth domains (paper generated by ChatGPT)](https://arxiv.org/abs/2610.12622); [arXiv 2610.04829: Viva La Vida, a study of a multi-agent prover](https://arxiv.org/abs/2610.04829); [arXiv 2610.17237: cutwidth/pathwidth O(log n) (GPT-5.6 Sol proof, Claude Opus 5.5 improvement)](https://arxiv.org/abs/2610.17237); [arXiv 2610.15909: Two Linear Programs are Better than One (proofs from ChatGPT)](https://arxiv.org/abs/2610.15909); [arXiv 2610.16492: Forsythe conjecture s = 2 (proof found with Codex)](https://arxiv.org/abs/2610.16492); [arXiv 2610.13400: Quantitative Eshelby stability, 'through AI-assisted mechanics'](https://arxiv.org/abs/2610.13400)

### 2026-10-13: k-median breaks the factor-2 barrier: EPFL-led team gives a 1.95-LMP algorithm; GPT-6 Astra pushes it to 1.844 and proves a 1.8 limit, both with Lean certificates

***; EPFL, Google Research, Carnegie Mellon University, New York University, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-13-k-median-better-than-2-approximation-epfl/

arXiv 2610.18144 (13 Oct 2026), "A Better-than-2 Approximation for k-Median Part I: The Power of General Bids", by Buhai, Cohen-Addad, Fleischmann, Gupta, Jubés Monforte, Pittu, Stouras, Svensson and Vogl, replaces the linear bid in the Jain–Mahdian–Markakis–Saberi–Vazirani greedy algorithm with general bidding functions. That gives a 1.95-LMP algorithm for facility location, where the factor 2 had essentially stood for two decades, and, with a companion paper, a better-than-2 approximation for k-median. The authors say their proofs were human, but GPT-6 Astra found a bidding function reaching 1.844 and proved that the method cannot go below 1.8, both with Lean certificates.

- Paper: arXiv 2610.18144 [cs.DS], 13 Oct 2026; authors at EPFL, Google Research, Carnegie Mellon and NYU (incl. Ola Svensson, Vincent Cohen-Addad, Anupam Gupta)
- Main result: a polynomial-time 1.95-LMP (Lagrangian-multiplier-preserving) approximation for facility location using the bid β(α, d) = [α − 1.224 d]+; the previous best LMP factor was 2 − η with η ≈ 2.25 × 10⁻⁷ (Cohen-Addad, Grandoni, Lee, Schwiegelshohn)
- k-median: a 1.95-approximation opening k + O(log n) centers; with a (1.89 + ε)-approximation for stable instances in the companion Part II, this gives a better-than-2 polynomial-time approximation. Prior state: (2 + ε) (Cohen-Addad et al. 2025; Byrka et al. 2026); k-median is hard to approximate below 1 + 2/e
- GPT-6 Astra (Remark 1.5): 'GPT-6 Astra managed to improve the LMP approximation ratio to 1.844', with a 13-term bidding function, and 'managed to prove that for any admissible bidding function … the supremum of the factor revealing program … has value at least 1.8', so the approach cannot go below 1.8
- AI statement: 'The mathematical proofs presented in this work (apart from Remark 1.5) were developed solely by the authors without the use of AI.' GPT-5.6 Sol and GPT-6 Astra were used for proof-checking and writing help; 'The lean certificates mentioned in Remark 1.5 and the certificate for Theorem 4.4 were obtained by GPT-6 Astra'
- Lean certificates: github.com/epfl-thl2/project-k-median (reachable on 14 Oct 2026)
- Concurrent work, as described in the paper: OpenAI announced on 6 Oct two better-than-2 k-median algorithms, an optimal (1 + 2/e + ε)-approximation extending Byrka et al.'s iterative rounding and a (2 − σ)-approximation; a group around Byrka and Shi Li independently reached 2 − σ. These OpenAI claims were not checked here; OpenAI's repository lists them as family 125 with Lean comparator statements (own entry 2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e)

#### What happened
The greedy algorithm of Jain, Mahdian, Markakis, Saberi and Vazirani (2002–03) has been the engine behind the best k-median guarantees through its 2-LMP guarantee for facility location. A 2023 local-search analysis improved it by only about 2.25 × 10⁻⁷. In [arXiv 2610.18144](https://arxiv.org/abs/2610.18144) (13 Oct 2026), a team from EPFL, Google Research, CMU and NYU changes the algorithm itself: clients bid with general functions of their budget and distance instead of the classical linear bid. The bid [α − 1.224 d]+ gives a 1.9499-LMP guarantee, which they make "log-adaptive" to get a polynomial-time 1.95-approximation opening k + O(log n) centers. With a (1.89 + ε) algorithm for stable instances from the companion Part II, this gives a better-than-2 approximation for k-median.

The paper separates human and AI work. The main proofs "were developed solely by the authors without the use of AI". In Remark 1.5, GPT-6 Astra found a 13-term bidding function with LMP ratio 1.844, and proved that no admissible bidding function can push this factor-revealing program below 1.8. Lean certificates for both, and for the paper's Theorem 4.4, "were obtained by GPT-6 Astra" and are on GitHub.

The authors also note concurrent work. OpenAI announced on 6 Oct two better-than-2 k-median algorithms, including an optimal (1 + 2/e + ε)-approximation, and a group around Jarosław Byrka and Shi Li independently reached 2 − σ. Those claims were not checked for this entry.

#### Why it matters
Factor 2 was a well-known barrier in clustering approximation. Here it falls to human work in the same weeks as OpenAI's claimed AI-found algorithms. The AI's own contribution is typical of the stronger 2026 collaborations: it tuned the construction further and proved where the method stops, with machine-checkable certificates.

Sources: [arXiv 2610.18144: A Better-than-2 Approximation for k-Median Part I: The Power of General Bids](https://arxiv.org/abs/2610.18144); [GitHub: epfl-thl2/project-k-median (Lean certificates)](https://github.com/epfl-thl2/project-k-median)

### 2026-10-14: Li, Li & Liu: external-memory integer sorting beats the comparison bound, answering Aggarwal–Vitter (1988) by way of network coding, with ChatGPT/Codex and Lean

****; University of Toronto, Tsinghua University, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding/

On 14 Oct 2026 Baochun Li (Toronto), Zongpeng Li and Sirui Liu (Tsinghua) posted arXiv 2610.20079. It gives a new counterexample to the 2004 Li–Li conjecture and uses it to sort N integers of Θ(log N) bits in external memory with O((N/B)(1 + log N/((log(M/B))^{1−ε}(log M)^ε))) block transfers. In some parameter ranges this is asymptotically below the comparison-sorting bound of Aggarwal and Vitter. Whether integer keys can beat that bound has been open since they asked it in 1988. The sorting algorithm's correctness and complexity bounds are checked in Lean 4. The method adapts OpenAI's 2026 integer-multiplication circuits, and ChatGPT and Codex were used throughout.

- Paper: 'Undirected Network Coding and Faster External-Memory Integer Sorting', arXiv 2610.20079 (cs.DS, 14 Oct 2026)
- Bound: for N integers of Θ(log N) bits, memory M words, block size B with 2B ≤ M ≤ N: O((N/B)·(1 + log N / ((log(M/B))^{1−ε}(log M)^ε))) transfers, ε > 0 an absolute constant; the comparison bound is Θ((N/B)(1 + log(N/M)/log(M/B))) [Aggarwal–Vitter 1988]
- Context: Farhadi, Hajiaghayi, Larsen and Shi (2019) showed that if the Li–Li conjecture held, no algorithm could beat comparison sorting in ranges including B ≥ log N. The conjecture was refuted on 7 Oct by Braverman–He and by Zhang–Li–Li; this paper notes that 'invalidating the assumption behind a conditional lower bound does not itself produce a faster algorithm' and supplies one
- Builds on OpenAI's integer-multiplication work ('[Ope26]'), whose recursive linear-circuit construction is adapted to block transfers
- Lean: 'We also use Lean 4 to verify the sorting algorithm's correctness and complexity bounds, and the strict advantage of coding over routing' (github.com/rtyu-1/coded-sorting-lean)
- AI methodology (verbatim): 'We used OpenAI's ChatGPT, including Codex, to assist with exploring network-coding constructions and sorting algorithms, developing and examining mathematical arguments, implementing and debugging Lean proofs … The tools materially affected the constructions and algorithmic analyses in Sections 4–7'

#### What happened
In 2004 Zongpeng Li and Baochun Li conjectured that in undirected multiple-unicast networks network coding does no better than fractional routing. The conjecture mattered beyond networking: Farhadi, Hajiaghayi, Larsen and Shi showed in 2019 that it would rule out integer sorting faster than comparison sorting in external memory, which would close a question Aggarwal and Vitter asked in 1988. On 7 Oct 2026 the conjecture was refuted twice, both times with GPT-6 help ([entry](/e/2026-10-07-multiple-unicast-conjecture-false-gpt-6/)).

A week later Baochun Li, Zongpeng Li and Sirui Liu turned the refutation into an algorithm ([arXiv 2610.20079](https://arxiv.org/abs/2610.20079)). They build a coding counterexample suited to "address-field interchange", rearrange data with recursive XOR circuits adapted from OpenAI's integer-multiplication construction, and combine this with Columnsort and most-significant-digit radix partitioning. The result is a deterministic sort whose I/O count is asymptotically below the comparison bound in a range of parameters. The algorithm's correctness and complexity bounds, and the advantage of coding over routing, are checked in Lean 4.

AI methodology section (verbatim): "We used OpenAI's ChatGPT, including Codex, to assist with exploring network-coding constructions and sorting algorithms, developing and examining mathematical arguments, implementing and debugging Lean proofs, and preparing the manuscript and literature discussion. The tools materially affected the constructions and algorithmic analyses in Sections 4–7, the supporting arguments and formalization in Appendices A–D, and their exposition in Sections 1–3."

No Hacker News thread was found on 15 Oct (Algolia). X was not searched directly.

#### Why it matters
It is a clear chain in which one AI-assisted refutation led to an AI-assisted algorithm: a disproved conjecture removed a conditional lower bound, and within a week the conjecture's own authors used it to answer a 38-year-old question. External-memory integer sorting is a central problem of I/O-efficient algorithms. The gain is asymptotic and holds only in some parameter ranges, so its practical impact is not claimed.

Sources: [arXiv 2610.20079: Undirected Network Coding and Faster External-Memory Integer Sorting](https://arxiv.org/abs/2610.20079); [GitHub: rtyu-1/coded-sorting-lean](https://github.com/rtyu-1/coded-sorting-lean)

### 2026-10-14: Lü, Wang & Wu claim µ(p+2) = o(π(x)), a model case of the parity problem open since at least 1989, building on an OpenAI preprint with ChatGPT's help

****; Shandong University, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu/

On 14 Oct 2026 Guangshi Lü, Zhiwei Wang and Jie Wu posted arXiv 2610.20242 (52 pages). It claims a proof that Σ_{p≤x} µ(p+2) = o(π(x)): the Möbius function cancels on the shifted primes p + 2. The statement is a folklore case of the Hardy–Littlewood–Chowla conjecture, in print since Hildebrand (1989), and a standard model case for the parity problem in sieve theory. A corollary is that µ(p+2) = +1 and −1 each occur for half of the primes with p + 2 squarefree. The authors thank ChatGPT for help understanding the methods of an OpenAI Math Release preprint on prime predecessors, and for the appendix. The result is unrefereed.

- Paper: 'On the Hardy–Littlewood–Chowla conjecture', arXiv 2610.20242 (math.NT, 14 Oct 2026, 52 pages)
- Theorem 1: Σ_{p≤x} µ(p+2) = o(π(x)), equivalently Σ_{n≤x} Λ(n)µ(n+2) = o(x). The shift is fixed at 2 'for simplicity'
- Theorem 2: #{p ≤ x : µ(p+2) = κ} = (A₂/2 + o(1))π(x) for κ = ±1, with A₂ = ∏_{p>2}(1 − 1/(p(p−1))); this addresses the question in Sarnak's AIM problem 5.2 about both sign patterns along shifted primes
- Before: only averaged versions (Lichtman 2022; Lichtman–Teräväinen) or results conditional on Siegel zeros (Tao–Teräväinen 2022)
- AI statement (verbatim): 'The authors would like to thank ChatGPT for its assistance in understanding the methods in [11], and in establishing the Appendix A. The authors also thank ChatGPT for helping to check the manuscript.'
- Reference [11] is OpenAI, 'The Poisson–Dirichlet law for prime predecessors' (OpenAI Math Release, 24 Sep 2026 version), family 011 of the 6 Oct release, which claims the Ford–Konyagin–Luca conjecture. Appendix A reproves its Lemma 3.3 (a root-counting estimate) with an explicit error term 'so that this input can be verified within the present paper'
- Related OpenAI claim in the same family: infinitely many primes p with µ(p−1) = 1 ('Prime Predecessors with an Even Number of Prime Factors', 17 Sep 2026)

#### What happened
Chowla's conjecture predicts that the Möbius function µ has no correlation with its own shifts. The Hardy–Littlewood conjecture predicts how often prime patterns occur. The mixed Hardy–Littlewood–Chowla conjecture combines the two. Its simplest case, Σ_{p≤x} µ(p+h) = o(π(x)), says that p + h is as likely to have an even as an odd number of prime factors when p runs over the primes. The paper says this "is a folklore conjecture and a well-known model case for the parity problem in sieve theory", in print at least since Hildebrand (1989). Earlier results covered only averages over the shift h (Lichtman; Lichtman–Teräväinen) or assumed that Siegel zeros exist (Tao–Teräväinen).

Lü, Wang and Wu claim the case h = 2 unconditionally ([arXiv 2610.20242](https://arxiv.org/abs/2610.20242)). They prove it with Type I/Type II estimates and a "parity conjugacy" bijection (Section 4). Their root-counting input comes from OpenAI's preprint "The Poisson–Dirichlet law for prime predecessors". To make the paper self-contained, they reprove that lemma in Appendix A.

AI statement: "The authors would like to thank ChatGPT for its assistance in understanding the methods in [11], and in establishing the Appendix A. The authors also thank ChatGPT for helping to check the manuscript."

No Hacker News thread was found on 15 Oct (Algolia). X was not searched directly. No expert reaction was found yet.

#### Why it matters
If the proof holds, it gets past the parity barrier in one of the problem's simplest standard test cases. It also shows a route that is becoming common: human experts take the method of a machine-written OpenAI preprint, have a chatbot explain it, and turn it into a result OpenAI did not claim.

Sources: [arXiv 2610.20242: On the Hardy–Littlewood–Chowla conjecture (Lü, Wang, Wu)](https://arxiv.org/abs/2610.20242); [OpenAI math release CONTENTS.md (family 011: prime-factor statistics of p − 1)](https://github.com/openai/math/blob/main/CONTENTS.md); [AIM problem list: Sarnak's conjecture, problem 5.2](https://aimpl.org/sarnakconjecture/)

### 2026-10-14: Bray, Dolmen, Khuri & Schmidt prove the spacetime Penrose inequality for 'maximally parameterized' data with Claude's help, and say OpenAI's Oct 6 Penrose paper appears to finish their program

****; Duke University, Stony Brook University, Anthropic, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-14-penrose-conjecture-bray-khuri-claude-openai-provenance/

On 14 Oct 2026 Hubert Bray, Tuan Dolmen, Marcus Khuri and David Schmidt posted arXiv 2610.20559. It proves the spacetime Penrose inequality in all dimensions n ≥ 3 for initial data sets of a 'maximally parameterized' form, using a new family of generalized Schoen–Yau identities. Their AI statement says Claude helped derive and check the identities and "made some useful mathematical suggestions" that would have earned a human co-authorship. It also says OpenAI's 6 Oct manuscript claiming the full Penrose conjecture appears to complete their own program. They do not know whether their privately circulated notes reached OpenAI's training data, and they ask labs to publish training-data provenance.

- Paper: 'Proof of the Penrose Conjecture for Maximally Parameterized Initial Data Sets' (arXiv 2610.20559, math.DG, 14 Oct 2026; 38+ pages)
- Result: the spacetime Penrose inequality, ADM mass ≥ ½(A/ω_{n−1})^{(n−2)/(n−1)}, for asymptotically flat data under the dominant energy condition in every dimension n ≥ 3, when the second fundamental form k has the 'maximally parameterized' form (4.7), plus a nondegeneracy condition at the horizon. That form has n(n+1)/2 local functions, the same formal freedom as an arbitrary symmetric 2-tensor; Kerr (Boyer–Lindquist) data are an example
- Method: a new 1-parameter family of generalized Schoen–Yau identities gives a comparison metric ḡ = g + β⊗β with scalar curvature ≥ 0, which reduces the problem to the Riemannian Penrose inequality (Bi–Zhu, all dimensions). The full conjecture would follow if a certain degenerate PDE system (3.18) could be solved in general
- Abstract: 'Essentially, we believe this is what the OpenAI paper [27] posted last week on October 6, 2026 claims to have done.' Reference [27] is OpenAI, 'The spacetime Penrose inequality and enclosing area' (family 260 of the openai/math release, dated 27 Sep 2026)
- AI statement (verbatim): 'By mid-February, we began using Claude to help with the algebra (which is quite messy) leading to the new Schoen–Yau identity for general t. We used Claude to verify every identity numerically as well as algebraically … Claude also made some useful mathematical suggestions … Had a human researcher made similar mathematical suggestions, we would have invited them to be a co-author. … The help was appreciated, but not decisive.'
- On a full proof with Claude: 'We also played around with Claude looking for an existence theory to various regularized versions of our equations … The resulting discussions got very complicated … Even if they were correct, Claude would have been the lead author'
- On OpenAI: 'The OpenAI paper begins with what we recognize as regularized versions of our equations with very little motivation, a bit like a magic trick.' They call the mass release of papers 'both good for the field in some ways (new ideas) and bad for the field in other ways (mistakes, confusing exposition, … massive checking needing to be done)'
- Provenance (verbatim): 'We do not know whether our privately circulated notes influenced the OpenAI results, and under current practices we cannot know. Labs could resolve such questions by publishing training-data provenance; absent that, authors are left with timelines, so we state ours.' The gauge-invariant t = 1 identity was 'derived entirely by humans in early February 2026'
- OpenAI's claim (family 260, CONTENTS.md): 'Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions', with charged, Kerr–Newman and anti-de Sitter companions and a Lean link; not yet confirmed by experts

#### What happened
The Penrose inequality says that an asymptotically flat initial data set satisfying the dominant energy condition has ADM mass at least ½(A/ω_{n−1})^{(n−2)/(n−1)}, where A is the minimal area enclosing its black holes. Penrose proposed it in 1973. The time-symmetric (Riemannian) case was proved by Huisken–Ilmanen and by Bray in three dimensions, and extended to all dimensions by Bray–Lee and Bi–Zhu. The general spacetime case was still open. Bray and Khuri proposed in 2009–2011 to reduce it to a system of PDEs (the "generalized Jang" approach).

On 14 Oct 2026 Bray, Dolmen, Khuri and Schmidt posted [arXiv 2610.20559](https://arxiv.org/abs/2610.20559). It proves the spacetime inequality for "maximally parameterized" data in every dimension n ≥ 3. They also show that the full conjecture follows if a degenerate PDE system (their (3.18)) has solutions. In the abstract they say that, in their view, this is essentially what OpenAI's 6 Oct manuscript claims to have done.

That manuscript, "The spacetime Penrose inequality and enclosing area" (dated 27 Sep), is part of family 260 of [OpenAI's 722-manuscript release](/e/2026-10-06-openai-math-release-722-manuscripts/). The family also includes charged, Kerr–Newman and anti-de Sitter versions and is linked to a Lean formalization. Until this paper, the dataset had not covered the Penrose part of the release.

### The AI statement (Appendix C)
The authors write that they derived the gauge-invariant identity (t = 1) "entirely by humans in early February 2026". They then used Claude for the algebra of the general-t identity and to check every identity numerically and algebraically. Claude "also made some useful mathematical suggestions" while they derived the special form of k, and they write: "Had a human researcher made similar mathematical suggestions, we would have invited them to be a co-author. A younger researcher would likely have accepted, and a senior human researcher might have declined the offer. The help was appreciated, but not decisive." They also used Claude to draft short notes, and they ran their conversations "under settings that exclude them from model training".

They tried using Claude to find an existence theory for regularized versions of their equations, which in the best case would have proved the full conjecture. They stopped because the discussions became too complicated to trust, and "Even if they were correct, Claude would have been the lead author".

### On OpenAI's release
The authors say they are impressed by the productivity of OpenAI's Penrose papers, but "the mode of release concerns us". They say OpenAI's paper starts from regularized versions of their equations "with very little motivation, a bit like a magic trick", and that their own paper explains the direction OpenAI's paper takes. They note that drafts shared privately can enter AI training data if a recipient pastes them into a consumer product whose default settings allow training. They write that "no lab can demonstrate that it has not" happened, and they call this "a serious and growing problem for scholarly credit". They give their own timeline and claim credit for the program as it is defined in their paper.

No Hacker News thread was found on 15 Oct (Algolia search for the arXiv id and "Penrose"), and a web search found no press coverage yet. X was not searched directly.

#### Why it matters
This is the first time experts whose own program was apparently completed by OpenAI's mass release have said so in a paper. In the same paper they credit a different lab's model (Claude) with co-author-level suggestions. The appendix shows what is now at stake in AI-assisted mathematics: credit, how readable machine-written proofs are, and training-data provenance. If OpenAI's general claim holds, the 1973 Penrose conjecture is settled.

Sources: [arXiv 2610.20559: Proof of the Penrose Conjecture for Maximally Parameterized Initial Data Sets (Bray, Dolmen, Khuri, Schmidt)](https://arxiv.org/abs/2610.20559); [OpenAI math release CONTENTS.md (family 260: spacetime Penrose inequalities)](https://github.com/openai/math/blob/main/CONTENTS.md); [OpenAI: The spacetime Penrose inequality and enclosing area (PDF, dated 27 Sep 2026)](https://github.com/openai/math/blob/main/preprints/The-spacetime-Penrose-inequality-and-enclosing-area-September-27-2026/paper.pdf)

### 2026-10-14: Mid-October 2026: arXiv results with disclosed AI help (catalogue of the 13–15 Oct listings, plus late items from 8–12 Oct)

***; OpenAI, Anthropic, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/

Continuation of the October catalogue (which reached about 65 KB and is closed at the 13 Oct block). It lists preprints from the 13–14 Oct arXiv listings, plus late items from 10–12 Oct missed earlier, that settle or advance a named problem and disclose an AI contribution. Each AI statement is quoted from the PDF. Three results have their own entries: the rational exponents conjecture write-up, Ulam's question on Lie groups, and the better-than-2 k-median algorithm. A second block, added on 15 Oct from the 14 Oct listing, has its own entries for the Penrose inequality paper (Bray et al., Claude), the Hardy–Littlewood–Chowla case µ(p+2) = o(π(x)), and external-memory integer sorting. All items are unrefereed preprints.

- Call for stories (Oct 14): a guest post on Terence Tao's blog by Deanna Needell and Hayden Schaeffer asks mathematicians who took part in, or whose earlier work enabled, recent AI-assisted results to write to mathematiciancontributions@gmail.com, to document 'how people and AI systems work together most effectively'; contributions are confidential unless consent is given
- Own entries: rational exponents conjecture write-up by Conlon, Janzer et al. (2610.18713); Ulam's question, every Lie group is countably representable (Monod & Thom, 2610.18464, key deformation from GPT-6 Astra); better-than-2 k-median (2610.18144, GPT-6 Astra improvement to 1.844 with Lean)
- Largest stated AI role in this block: S-Hamiltonian cycles (Kumar, Mari & Sandeep, 2610.18300), which resolves an open problem of Amarilli, Lombardo and Monet (WG 2026): 'GPT 6 Pro was used to extend this result to oriented graphs and to obtain all other new results in this paper'
- Counterexamples credited to ChatGPT: prescribed scalar curvature in the null case on flat tori of dimension ≥ 5 (Li, Wu & Zhou, 2610.18432: 'The counterexample presented in this paper was suggested by ChatGPT'); three conjectures of F. Luo (2011) on minimal triangulations (Qilong Guo, 2610.13979: ChatGPT provided 'the idea for the counterexample')
- A model suggests the key tool: Szemerédi's theorem with random differences, threshold improved to p ≥ n^(−c/log log k) (Gabriel Dahia, 2610.18859): GPT 6 Astra Pro said the ideas the author asked about would not work 'but suggested that the construction from [Green–Ruzsa] could be used'
- Constructions from GPT-5.6 Sol: Zhi-Wei Sun's conjecture that 1..n can be ordered so that the reciprocals of adjacent sums add to 1 (Ai, Hu & Yan, 2610.18431: 'Most of the construction and the recursive step … were proposed and refined through our interaction with GPT-5.6 Sol')
- With Lean: central limit theorem for the high-temperature SYK model (Kamijima, Nakajima, Noda & Sasaya, 2610.18440; proof from ChatGPT 5.5/5.6 Pro discussions, Lean code AI-generated and human-reviewed); exact selection count V₇(16) = 28 comparisons (Jonathan Cremer, 2610.18740; AI coding assistants 'implemented the software and produced first drafts of the proofs')
- Smaller roles: framed LMOV conjecture for links (Liu, Peng & Zhu, 2610.18618; GPT-6 Astra helped derive local crossing identities); four Takigiku–Tsuchioka conjectures (Liuquan Wang, 2610.18666); Berndt–Straub conjectures on generalized Ramanujan polynomials, one proved and one disproved (Siddhi Pathak, 2610.18065); strong monodromy conjecture for the symmetric determinant (Lyu & Zhang, 2610.18395; ChatGPT 5.5 Pro suggested a transverse normal form); essentially tracially rank one C*-algebras (Huaxin Lin, 2610.14711; an example 'was an AI experiment')
- 15 Oct block, own entries: Penrose inequality for maximally parameterized data (Bray, Dolmen, Khuri & Schmidt, 2610.20559; Claude made co-author-level suggestions); µ(p+2) = o(π(x)) (Lü, Wang & Wu, 2610.20242; built on an OpenAI preprint with ChatGPT's help); external-memory integer sorting below the comparison bound (Li, Li & Liu, 2610.20079)
- 15 Oct block, largest AI roles: 25 of the 30 smallest open two-variable Diophantine equations solved, and 209 of 211 resistant equations shown to have no integer solutions, with Lean (Grechuk & Go, 2610.18916: 'The crucial identities from Table 2 were found by ChatGPT 6 Astra Pro'); Two-Level Posetgame is PSPACE-complete, open 'for over a decade' (Hashimoto, 2610.19558: 'The proof of Theorem 2 was obtained during a session with GPT-6 Astra'); two of Krishan–Vishwanathan's five torus-polynomial problems ('discovered and formalized in Lean using GPT-6 Astra', 2610.19174); postage-stamp h-bases with negative elements (Aristotle by Harmonic, autonomous and Lean-verified, 2610.19470)
- Human-only results of the same days, for contrast: maximum nonlinearity of 8-variable balanced Boolean functions is 116, a long-standing open problem (Lou, Xu & Wang, 2610.17897; no AI statement found); endpoint power-weight bounds for spherical maximal functions ('No AI tools were used for the mathematics or the writing', 2610.18316); p-adic inhomogeneous Littlewood note (Chow & Jones, 2610.15809, 'AI has not been used'); Bolsinov–Taimanov analytic entropy conjecture (Chen & Jiang, 2610.14065; no AI statement found)

#### What happened
This block was read from the PDFs of the arXiv papers the 14 Oct evening sweep listed as disclosing AI use (112 papers) or claiming to settle conjectures without a disclosure (86). Most 7–12 Oct items had already been processed in the morning run and are in [the October catalogue](/e/2026-10-13-ai-assisted-results-october-2026/) or in the leads file. Below are the new items, grouped by how large the stated AI role is.

### Results where the AI did most of the new work
- **S-Hamiltonian cycles and paths** (Kumar, Mari & Sandeep, [2610.18300](https://arxiv.org/abs/2610.18300), 13 Oct). The paper completes the complexity classification for undirected graphs, including NP-completeness for every finite set of odd walk lengths, which was an open problem of Amarilli, Lombardo and Monet (WG 2026), and treats digraphs. Declaration: "A preliminary version of Theorem 12 was obtained manually using a construction with loops. GPT 6 Pro was used to extend this result to oriented graphs and to obtain all other new results in this paper."
- **Prescribed scalar curvature, null case** (Yuxiang Li, Yunqing Wu & Jie Zhou, [2610.18432](https://arxiv.org/abs/2610.18432), 13 Oct). Smooth sign-changing functions with negative mean on flat tori Tⁿ, n ≥ 5, that are not the scalar curvature of any metric in the conformal class. "The counterexample presented in this paper was suggested by ChatGPT. The authors have verified its correctness."
- **Three conjectures of F. Luo (2011)** (Qilong Guo, [2610.13979](https://arxiv.org/abs/2610.13979), 11 Oct). One counterexample to all three: a minimal two-tetrahedron triangulation of the lens space L(3,1) with a null-homotopic loop edge. The author thanks ChatGPT "for providing the idea for the counterexample".
- **Zhi-Wei Sun's permutation conjecture** (Jinggao Ai, Dingsheng Hu & Junmin Yan, [2610.18431](https://arxiv.org/abs/2610.18431), 13 Oct). For n > 6 (line) and n > 7 (circle), 1, …, n can be arranged so that the reciprocals of adjacent sums add up to 1. "Most of the construction and the recursive step in this paper were proposed and refined through our interaction with GPT-5.6 Sol"; the authors checked every case by hand.
- **SYK central limit theorem** (Kamijima, Nakajima, Noda & Sasaya, [2610.18440](https://arxiv.org/abs/2610.18440), 13 Oct). A CLT for log Z_N(β) of the q-body SYK model at high temperature. The paper is written as a case study: direct attempts with ChatGPT 5.5 Pro failed; after the humans proposed working with the partition function instead, the discussion produced proofs; ChatGPT 5.6 Pro later helped replace a complicated analytic step with a combinatorial one. "In parallel, we formalized the refined proofs in Lean: the AI generated the code, and we reviewed it." Code at github.com/TokyoProbabilityGroup/SYK_CLT.
- **Exact selection complexity** (Jonathan Cremer, [2610.18740](https://arxiv.org/abs/2610.18740), 13 Oct). V₇(16) = 28 and V₅(17) = 27 comparisons, with better upper bounds for other ranks; V₇(16) was one of the two 16-element entries left open after Dörrer et al. settled n ≤ 15 (previous range 28–30, from Gasarch–Kelly–Pugh's table). Every bound has a certificate checked by an independent checker and in Lean 4. "The assistants implemented the software and produced first drafts of the proofs."

### A model suggested the key tool
- **Szemerédi's theorem with random differences** (Gabriel Dahia, [2610.18859](https://arxiv.org/abs/2610.18859), 13 Oct). The threshold is improved from p ≥ n^(−2/k+o(1)) (Briët–Castro-Silva) to p ≥ n^(−c/log log k) for large k. The author asked GPT 6 Astra Pro whether an Alon–Pham-style idea could give a small cover; "It replied that it could not use those ideas, but suggested that the construction from [Green–Ruzsa, arithmetic Kakeya] could be used", and the proof "crucially relies" on that construction.
- **Strong monodromy conjecture for the symmetric determinant** (Renjie Lyu & Xiping Zhang, [2610.18395](https://arxiv.org/abs/2610.18395)). ChatGPT 5.5 Pro helped connect the resolution to complete quadrics, computed Euler characteristics and "suggests the following transverse normal form" (Lemma 2.3).
- **Framed LMOV conjecture for links** (Kefeng Liu, Pan Peng & Shengmao Zhu, [2610.18618](https://arxiv.org/abs/2610.18618)). Built on Liu–Peng's 2012 strategy; "GPT-6 Astra assisted in deriving the local crossing identities (4.1) and (4.2)".

### Exploration, computation and checking
- **Takigiku–Tsuchioka conjectures**: eight Rogers–Ramanujan-type identities (Liuquan Wang, [2610.18666](https://arxiv.org/abs/2610.18666)); GPT-5.6 Sol and GPT-6 Astra "used to assist in exploring proof strategies, performing calculations".
- **Berndt–Straub conjectures** (Siddhi Pathak, [2610.18065](https://arxiv.org/abs/2610.18065)): the conjecture for R_k is proved; the one for S_2n is disproved (S₆(x; χ₁₆₃, χ₁₆₃) has four zeros off the unit circle), so the "Riemann hypothesis for period polynomials" can fail for Eisenstein newforms. ChatGPT (GPT-5.6 Sol, GPT-6 Astra) helped with exploration and examples.
- **C*-algebras of essentially tracial rank one** (Huaxin Lin, [2610.14711](https://arxiv.org/abs/2610.14711)), which the abstract says "completes a long-standing project": "The example in Section 14 was an AI experiment"; an argument "was initially suggested by ChatGPT-5.6-sol".

### Human-only results of the same days (for contrast)
- **Maximum nonlinearity of 8-variable balanced Boolean functions is 116** (Lou, Xu & Wang, [2610.17897](https://arxiv.org/abs/2610.17897)), a negative answer to the existence of nonlinearity 118, "a famous and long-standing open problem"; no AI statement found. Same group: covering radius of RM(1,9) is 242 (2610.17871).
- Endpoint power-weight inequalities for spherical maximal functions (Fraccaroli, Lee, Lei, Roos & Seeger, [2610.18316](https://arxiv.org/abs/2610.18316)): "No AI tools were used for the mathematics or the writing of this paper."
- p-adic inhomogeneous Littlewood (Chow & Jones, [2610.15809](https://arxiv.org/abs/2610.15809)): "AI has not been used in this work".
- Bolsinov–Taimanov analytic entropy conjecture (Chen & Jiang, [2610.14065](https://arxiv.org/abs/2610.14065)): no AI statement found.

No Hacker News threads on any of these were found on 14 Oct (Algolia search); X was not searched directly.

#### 15 Oct block (14 Oct listing and late 8–13 Oct items)
Read from the PDFs of the 15 Oct sweep (116 papers with an AI mention, 87 claiming a conjecture without one). Three results have their own entries: [the Penrose inequality paper](/e/2026-10-14-penrose-conjecture-bray-khuri-claude-openai-provenance/), [µ(p+2) = o(π(x))](/e/2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu/) and [external-memory integer sorting](/e/2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding/). Follow-ups to OpenAI's release (Shlyakhtenko on free group factors, Visser on quasi-RH, ℝ³ chromatic numbers, Solomon–Yau, hot spots) are listed in [the release entry](/e/2026-10-06-openai-math-release-722-manuscripts/).

### The model produced the key result
- **Smallest open Diophantine equations** (Bogdan Grechuk & Eugene Go, [2610.18916](https://arxiv.org/abs/2610.18916), 12 Oct). The paper solves 25 of the 30 two-variable equations of size ≤ 34 whose integer solubility was open, and the other 5 by deeper methods. It shows that 209 of 211 equations that resisted known methods have no integer solutions, with Lean proofs. The authors first tried "all available AI models, including ChatGPT 6 Astra Pro, initially with no success", until the model solved one equation. They then extracted its hidden identity into a general method, and with that method the model solved the rest. "The crucial identities from Table 2 were found by ChatGPT 6 Astra Pro, as were the long certificates in Section 5." Claude Opus 5.5 checked the identities and certificates independently.
- **Two-Level Posetgame is PSPACE-complete** (Kengo Hashimoto, [2610.19558](https://arxiv.org/abs/2610.19558)). The game is played on posets of height ≤ 2, and its complexity had "remained an open problem for over a decade". "The proof of Theorem 2 was obtained during a session with GPT-6 Astra."
- **Torus polynomials for Majority** (Saint Wesonga, [2610.19174](https://arxiv.org/abs/2610.19174), 13 Oct). The paper resolves two of the five open problems posed by Krishan and Vishwanathan (2026). "The results presented here were discovered and formalized using GPT-6 Astra", starting from the author's Lean circuit-complexity framework.
- **h-bases with negative elements** (van Doorn, Monticone & Nathanson, [2610.19470](https://arxiv.org/abs/2610.19470)). The variant of the postage stamp problem for k = 2 is answered. The comparison results "were autonomously discovered and formally verified in Lean by Aristotle", Harmonic's agent; Nathanson then wrote new human proofs, and "Aristotle's original solution made this work possible."
- **Online convex optimization with few experts** (Assos, Brahmbhatt & Elad Hazan, Princeton/Google DeepMind, [2610.20416](https://arxiv.org/abs/2610.20416)). The paper separates the information-theoretic and polynomial-time regret for Freund's question in the continuous setting, assuming the hypergraphic planted clique conjecture. GPT-6 Astra "proposed the central idea of a reduction from hypergraphic planted clique"; the authors strengthened the bound.
- **Classical cost of simulating quantum communication** (Castellano, Sekatski & Brunner, Geneva, [2610.19060](https://arxiv.org/abs/2610.19060)). No finite-average-dimension classical simulation exists for d ≥ 4, which addresses "a long-standing open question". "ChatGPT Sol 5.6 and ChatGPT Astra 6.0 have contributed via extensive dialogues with the authors. This helped prove the statements".
- **Minimum-autocorrelation constant** A ≥ 0.41037…, improving Russell's certified 0.39921 (Chen & Gu, independent researchers, [2610.18888](https://arxiv.org/abs/2610.18888)). It has a Lean-checked certificate; ChatGPT and Codex assisted "with construction and proof exploration, verification code, Lean proof development".
- **Policy mirror descent counterexample** (Li & Wei, Fudan, [2610.19853](https://arxiv.org/abs/2610.19853)). It is "a counterexample with assistance from OpenAI's GPT-5.6 model" showing that PMD with a non-decomposable mirror map need not converge in policy.

### A model suggested an essential step
- **Pucci's conjecture** (sharp Lᵖ estimates for Lu = f; Hung & Wu, [2610.19633](https://arxiv.org/abs/2610.19633)). The strategy was the authors'. To remove a restriction on q, "Through extensive interaction with AI-assisted tools (ChatGPT6 Astra), two ideas were suggested which is essential": an auxiliary parameter t, and a determinant/2-Hessian approach.
- **Karp's 2012 conjecture** that zeros of Toeplitz determinants of rising factorial series are real and negative (Henry Zweiman, [2610.18895](https://arxiv.org/abs/2610.18895)). GPT-6 Astra helped "obtain an earlier proof … that differs from the one given here".
- **Neumann eigenvalues toward Pólya's conjecture** (He, Jiang, Fanghua Lin & Wang, [2610.19455](https://arxiv.org/abs/2610.19455)). "The proof of Proposition 1.9 and those in Section 6 were initially prepared with the assistance of ChatGPT."
- **Ordinary = measurable chromatic number of ℝ³** (Koshelev, Kupavskii & Voronov, [2610.18907](https://arxiv.org/abs/2610.18907)); both lie between 7 and 15. The authors "used ChatGPT 6 Astra to obtain the main ideas from" OpenAI's plane-not-5-colourable preprint. The result closes the gap between n = 2 (OpenAI) and n ≥ 4 (Davies).
- **High-probability regret for Feel-Good Thompson sampling** (Li, Zhao & Quanquan Gu, [2610.19513](https://arxiv.org/abs/2610.19513)): "The proof ideas are partially developed by GPT-6 Astra under the authors' guidance."
- **Optimal control (LVPP method)** (Haaf & Herzog, [2610.16678](https://arxiv.org/abs/2610.16678)). It was done "in close collaboration with … Claude Code … using the Opus 5.5 and Fable 5.1 models", which "allowed us to conduct this study within approximately three weeks".
- Smaller roles: bounded-density vertex deletion, open cases of Raupach et al. (MFCS 2026) (GPT-5.6 Sol for constructions and proof strategies, [2610.20395](https://arxiv.org/abs/2610.20395)); fence conjecture of Dumas–Wolf (DeepSeek, Doubao and ChatGPT for exploration and checks, [2610.19664](https://arxiv.org/abs/2610.19664)); low-rank PPT states violating the Peres conjecture (GPT-5.6 Sol discussions, [2610.19905](https://arxiv.org/abs/2610.19905)); Bachmann–van Ittersum conjecture on ternary period polynomials (Ilin & Ken Ono with AxiomProver, Lean, [2610.19309](https://arxiv.org/abs/2610.19309)); short proofs for Erdős Problem 506 on determined circles, which note five near-simultaneous AI-assisted investigations (Cambie, Paliy & Wrona, [2610.18918](https://arxiv.org/abs/2610.18918)).

### Checking and writing only
A folklore conjecture on multiplicative exponential sums (Robles, [2610.19464](https://arxiv.org/abs/2610.19464); ChatGPT Astra 6 checking, then "Claude Fable 5.1 was used for an adversarial audit"); the Erdős–Odlyzko–Sárközy conjecture for prime-power moduli (Banks & Shparlinski, [2610.19597](https://arxiv.org/abs/2610.19597); Claude for exposition, literature and proof checking); symmetry of information for nKt (Goldberg, Hu, Lu, Lyu & Oliveira, [2610.20188](https://arxiv.org/abs/2610.20188); proofs checked with GPT 5.6/6.0); Eremenko's conjecture on higher-order Briot–Bouquet equations (Diego Marques, [2610.19243](https://arxiv.org/abs/2610.19243); ChatGPT "solely for English-language editing"; as of Oct 15 the item is not yet marked on Eremenko's [problem page](https://www.math.purdue.edu/~eremenko/uns1.html), which says he withdrew his monetary prizes in summer 2026 "in view of the AI revolution" and now marks at least eight problems as solved with AI or AI-assisted, e.g. Hille's problem on meromorphic solutions "solved completely by several people using AI in August-September 2026"); a combinatorial proof of the Chan–Robbins–Yuen volume formula, long asked for (Shyntas, [2610.19119](https://arxiv.org/abs/2610.19119); Codex for manuscript preparation); zero-sum conjectures of Z.-W. Sun (Li, Ling, Sun & Zhu, [2610.19965](https://arxiv.org/abs/2610.19965); presentation only).

### No AI statement found (for contrast)
A starlike 3D counterexample to the hot spots conjecture (Atar & Burdzy, [2610.19908](https://arxiv.org/abs/2610.19908)); Meeks' conjecture on disjoint minimal graphs in two cases (Benedetti & Nelli, [2610.20508](https://arxiv.org/abs/2610.20508)); Tunçel's 1995 conjecture on spectral self-concordant barriers (Peña & Sendov, [2610.17442](https://arxiv.org/abs/2610.17442)); a bounded-defect family refuting the conjecture that the Mondrian defect grows with n (O'Kuhn & Slaughter, [2610.19193](https://arxiv.org/abs/2610.19193)).

No Hacker News threads on these were found on 15 Oct (Algolia); X was not searched directly.

### Evening additions (15 Oct sweep, 8–14 Oct listings)
- **Lazy and invertible cellular automata** (Alcalá-Arroyo & Castillo-Ramirez, [2610.18480](https://arxiv.org/abs/2610.18480), 13 Oct; with Haobo Ma & Wenlin Zhang, [2610.20329](https://arxiv.org/abs/2610.20329), 14 Oct). The pair answers Problem 2 of the first two authors' 2026 *Theoret. Comput. Sci.* paper: for locally finite G, lazy and invertible cellular automata generate all of CA(G; A) exactly when 3 ≤ |A| < ∞, or |A| = 2 and G has no Klein four-group quotient; the Klein-four obstruction case is then described completely (submonoid of 59,712 elements, Lean-verified). Disclosures: "The authors used ChatGPT 6 Astra and Claude Opus 5.5 to assist with mathematical exploration and discovery" (2610.18480); "several models of the GPT (OpenAI) and Claude (Anthropic) families for mathematical discovery, drafting, the Lean formalization and verification scripts, and independent checking of the results" (2610.20329).
- Smaller roles: Nullstellensatz and real Nullstellensatz for modules and sheaves ([2610.15887](https://arxiv.org/abs/2610.15887); "Claude (Anthropic) and ChatGPT (OpenAI) for conceptual discussions regarding proofs and literature"); sharp stability for mixed bubbles on hyperbolic space (Wei, Yan & Yang, [2610.15920](https://arxiv.org/abs/2610.15920); ChatGPT "for exploratory discussions"); crystabelline deformation rings, wildly ramified case ([2610.16225](https://arxiv.org/abs/2610.16225); "ChatGPT-6 to help check proofs and references"); topological degree for singular N-mean field equations (Ahmedou, Tang & Wang, [2610.20281](https://arxiv.org/abs/2610.20281); ChatGPT "to check intermediate arguments and references").
- Lean-formalised results with no AI statement, all single-author and posted 9–13 Oct (often a sign of agent-built formalisations, but not stated): expand-contract plasticity of compact quasi-metric spaces ([2610.14301](https://arxiv.org/abs/2610.14301)), independent sets in bipartite graph covers ([2610.14892](https://arxiv.org/abs/2610.14892)), a 4×4 obstruction for ordered binary matrices ([2610.15370](https://arxiv.org/abs/2610.15370)), log-concavity failures in trees with perfect matchings ([2610.18340](https://arxiv.org/abs/2610.18340)).

#### Why it matters
One day's listing now routinely holds several AI-assisted resolutions of named problems. Their disclosures are increasingly specific, naming which lemma, construction or counterexample came from the model. The rational exponents write-up and Monod–Thom paper show leading experts publishing and crediting AI-originated arguments under their own names.


#### Follow-ups to OpenAI's 722-manuscript release (14 Oct listing)
Bray, Dolmen, Khuri & Schmidt prove the spacetime Penrose inequality for "maximally parameterized" data with Claude's help and say OpenAI's family 260 paper "begins with what we recognize as regularized versions of our equations with very little motivation, a bit like a magic trick"; they ask labs for training-data provenance ([2610.20559](https://arxiv.org/abs/2610.20559); own entry 2026-10-14-penrose-conjecture-bray-khuri-claude-openai-provenance). Lü, Wang & Wu claim µ(p+2) = o(π(x)) by following family 011's method, which ChatGPT helped them understand ([2610.20242](https://arxiv.org/abs/2610.20242); own entry). Dima Shlyakhtenko reformulates the free-group-factor proof (287) and extends it to semicircular C*-algebras ([2610.19590](https://arxiv.org/abs/2610.19590)). Koshelev, Kupavskii & Voronov use the plane-not-5-colourable theorem (158) to show that the ordinary and measurable chromatic numbers of ℝ³ are equal ([2610.18907](https://arxiv.org/abs/2610.18907)). Jingwen Chen uses the Solomon–Yau theorem (349) to build Allen–Cahn flows in all dimensions ([2610.20453](https://arxiv.org/abs/2610.20453)). Atar & Burdzy give a starlike 3D counterexample to hot spots, with family 369 as background ([2610.19908](https://arxiv.org/abs/2610.19908)). Matt Visser works out the prime-gap consequences of the 7/8 quasi-RH "if true", and is sceptical of the release ([2610.20058](https://arxiv.org/abs/2610.20058)). Castillo & Gordillo-Núñez criticise four 24 Sep AI-assisted orthogonal-polynomial preprints and call for "a standard that rewards structural explanation and verification rather than the proliferation of examples" ([2610.16704](https://arxiv.org/abs/2610.16704)). Two more follow-ups found in the evening: Artyom Radomskii answers the $50 O(log N) part of Erdős Problem #32 with a lemma adapted from OpenAI's "Improved Long Gaps Between Primes" ([2610.14127](https://arxiv.org/abs/2610.14127); own entry 2026-10-09-erdos-32-additive-complements-primes-radomskii); Netan Dogra & Arnab Saha cite the release in print: "Recently, an AI-generated proof of the Zilber–Pink conjecture for subvarieties of abelian varieties defined over Q was announced", adding that by Barroero–Dill it would imply the general abelian case, and prove an unramified bounded-degree variant by arithmetic jet spaces ([2610.17325](https://arxiv.org/abs/2610.17325), 12 Oct). The first arXiv paper on the integer-multiplication κ race (Dağlı, Dağlı & Dağlı, [2610.14263](https://arxiv.org/abs/2610.14263)) is noted in the Colkitt entry.

Sources: [Terence Tao blog (guest post by Needell and Schaeffer): Call for stories: human–AI collaboration in recent mathematics](https://terrytao.wordpress.com/2026/10/14/call-for-stories-human-ai-collaboration-in-recent-mathematics/); [arXiv 2610.18300: The Complexity of Hamiltonian Cycles and Paths with Prescribed Walk Lengths (GPT 6 Pro)](https://arxiv.org/abs/2610.18300); [arXiv 2610.18432: Counterexamples to prescribed scalar curvature in the null case (suggested by ChatGPT)](https://arxiv.org/abs/2610.18432); [arXiv 2610.18859: Szemerédi's theorem with random differences and arithmetic Kakeya (GPT 6 Astra Pro)](https://arxiv.org/abs/2610.18859); [arXiv 2610.18431: On permutations whose adjacent-sum reciprocals sum to one (GPT-5.6 Sol)](https://arxiv.org/abs/2610.18431); [arXiv 2610.18440: A Central Limit Theorem for High-Temperature SYK Model (ChatGPT Pro + Lean)](https://arxiv.org/abs/2610.18440); [GitHub: TokyoProbabilityGroup/SYK_CLT (Lean)](https://github.com/TokyoProbabilityGroup/SYK_CLT); [arXiv 2610.18740: Selecting the 7th Smallest of 16 Elements Takes Exactly 28 Comparisons (Lean)](https://arxiv.org/abs/2610.18740); [GitHub: JCremerr/selection-7-of-16](https://github.com/JCremerr/selection-7-of-16); [arXiv 2610.18618: A proof of the framed LMOV conjecture for links](https://arxiv.org/abs/2610.18618); [arXiv 2610.18065: On Berndt–Straub conjectures about zeros of generalized Ramanujan polynomials](https://arxiv.org/abs/2610.18065); [arXiv 2610.13979: A counterexample to three conjectures of Luo (idea from ChatGPT)](https://arxiv.org/abs/2610.13979); [arXiv 2610.18916: Certifying non-existence of integer points on curves (ChatGPT 6 Astra Pro + Lean)](https://arxiv.org/abs/2610.18916); [arXiv 2610.19558: The Two-Level Poset Game is PSPACE-Complete (GPT-6 Astra)](https://arxiv.org/abs/2610.19558); [arXiv 2610.19174: Majority and Torus Polynomials (GPT-6 Astra + Lean)](https://arxiv.org/abs/2610.19174); [arXiv 2610.19470: Positivity and negativity for h-bases for n (Aristotle)](https://arxiv.org/abs/2610.19470); [arXiv 2610.19633: Sharp Lq estimate of adjoint problem and Pucci conjecture (ChatGPT6 Astra)](https://arxiv.org/abs/2610.19633); [arXiv 2610.18907: Ordinary and measurable colorings of Euclidean three-space](https://arxiv.org/abs/2610.18907); [arXiv 2610.20416: Tracking a Small Set of Experts in Online Convex Optimization (GPT-6 Astra)](https://arxiv.org/abs/2610.20416); [arXiv 2610.17897: The Maximum Nonlinearity of 8-variable Balanced Boolean Functions is 116 (human)](https://arxiv.org/abs/2610.17897)

### 2026-10-14: Two Nature papers from the Baker lab: proteases designed from scratch with RFdiffusion cut chosen protein sequences

***; University of Washington, Institute for Protein Design; Science & math; confidence high; https://postcutoff.com/e/2026-10-14-baker-lab-de-novo-designed-proteases-nature/

On Oct 14, 2026 Nature published two papers from David Baker's lab describing enzymes that break peptide bonds, designed de novo with RFdiffusion-family tools: cysteine proteases (13 of 69 designs active, 19%, rate enhancement up to 3x10^7, folds unlike any natural protease) and zinc metalloproteases (36% of 135 designs active; the best, after four substitutions, above 10^10-fold rate enhancement) that cleave chosen sequences, including in disease-associated human proteins. Peptide bonds are stable for hundreds of years in water, so this is a much harder design target than the low-barrier reactions earlier de novo enzymes handled.

- Paper 1: 'De novo design of cysteine proteases' (Nature, online Oct 14, 2026; first author Hojae Choi, senior authors Samuel J. Pellock and David Baker). Success rate 13/69 = 19%; kcat/kuncat up to 3x10^7; designed folds unlike natural proteases (TM score < 0.50); six crystal structures within 1.2 A Ca RMSD of the design models
- Paper 2: 'Computational design of metalloproteases' (Nature, online Oct 14, 2026; first author Anqi Chen, senior authors Donald Hilvert and David Baker). Of 135 designs, 36% cleave their intended peptide substrate at the targeted bond; the best accelerates hydrolysis more than 10^8-fold, and four substitutions take it above 10^10-fold
- Targets: designed metalloproteases selectively cleave the human disease-associated proteins TDP-43, amyloid-beta and serum amyloid A, with rate enhancements up to 10^9-fold; the paper also reports caged cytokines and receptor antagonists activated by the designed proteases (abstract)
- Pipeline (Ars Technica): RFdiffusion2 generates a backbone constrained to hold the catalytic residues and a target sequence; ProteinMPNN fills in the amino acids; AlphaFold3 predicts structures. Starting points were a crayfish digestive zinc protease and a yeast cysteine protease
- Ars Technica's reading of the zinc work: designs that did not place a catalytic water often failed (14 of 89 weakly active), while adding the water as a design constraint raised this to 35 of 38
- Limitation stated by the authors (via Ars Technica): the tools can be told to satisfy structural constraints, not to achieve a specific catalytic activity, so designs are generally far less active than evolved enzymes; protein delivery and immune response are open obstacles to any therapeutic use

#### What happened
Both papers start from a natural protease's catalytic core and replace everything else with a computationally designed
protein. RFdiffusion2 builds a backbone that holds the catalytic residues in place and binds the intended substrate
sequence, ProteinMPNN chooses the amino acids, and AlphaFold3 checks the predicted structures. Crystal structures match the
design models closely, and the designed folds do not resemble natural proteases.

#### Why it matters
Earlier de novo enzyme design worked mainly on easy reactions. Cutting the peptide backbone at a chosen sequence is a long-
standing target because it would allow proteins to be made and then selectively destroyed or activated. The disease-protein
demonstrations are proof of principle only; delivery and immunogenicity remain unsolved.

Sources: [Nature: De novo design of cysteine proteases (Oct 14, 2026)](https://www.nature.com/articles/s41586-026-11143-w); [Nature: Computational design of metalloproteases (Oct 14, 2026)](https://www.nature.com/articles/s41586-026-11167-2); [Ars Technica: Researchers use AI to design proteins that cut up other proteins (John Timmer, Oct 14)](https://arstechnica.com/science/2026/10/researchers-use-ai-to-design-proteins-that-cut-up-other-proteins/); [Institute for Protein Design: RFdiffusion2, efficient enzymes from prompts](https://www.ipd.uw.edu/2025/12/rfdiffusion2-efficient-enzymes-from-prompts)

### 2026-10-14: Microsoft Research's Probabilistic Bias Correction (Nature) improves 2-to-6-week weather forecasts and won ECMWF's AI Weather Quest

**; Microsoft Research, ECMWF; Science & math; confidence high; https://postcutoff.com/e/2026-10-14-microsoft-probabilistic-bias-correction-subseasonal-forecasts/

On Oct 14, 2026 Microsoft Research announced Probabilistic Bias Correction (PBC), published in Nature: a machine-learning method that learns a forecast model's systematic errors and corrects its probabilities for the 2-to-6-week (sub-seasonal) range. Applied to ECMWF's sub-seasonal model it improved 98% of precipitation, 92% of temperature and 91% of pressure targets, and its entry won every variable and lead time in the first season of ECMWF's AI Weather Quest. Kenya's meteorological service uses it for rainy-season forecasts. Code is on GitHub.

- Method: PBC 'takes your favorite model as input and learns in what ways it tends to make errors' (Lester Mackey, Microsoft Research New England)
- Gains on ECMWF's sub-seasonal model: better on 98% of precipitation, 92% of temperature and 91% of pressure targets; it 'doubled the accuracy' of ECMWF's AI Forecasting System (AIFS), per Microsoft
- AI Weather Quest (ECMWF real-time competition launched August 2025): team MicroEnsemble's MicroDuet (PBC on ECMWF's dynamical model and the hybrid PoET model) placed first for every variable and lead time in the inaugural three-month season, ahead of six operational centres and 34 teams
- Examples given: flagged the Blizzard of 2026 (on Jan 29), Winter Storm Fern (Jan 1, 2026) and a mid-December 2025 cold outbreak (Nov 20, 2025) 19–26 days ahead
- Deployment: with Rhiza Research, the Kenya Meteorological Service Authority, UC Santa Barbara's Climate Hazards Center and KIT; KMSA's Phillip Okello says it 'improved our forecasts more than any other method we tested'
- Availability: code at github.com/mouatadid/pbc; a Microsoft Weather S2S model (six-week probabilistic guidance) is in private preview

#### What happened
Microsoft Research published a post-processing method for sub-seasonal forecasts, the weakest range between dependable two-week weather
forecasts and seasonal climate outlooks. It corrects the forecast probabilities of an existing model rather than replacing it.

#### Why it matters
Two-to-six-week forecasts matter for farming, water and disaster planning. The method is model-agnostic, open source and already used
operationally in Kenya. The Nature paper itself was not opened here; figures are from Microsoft's story.

Sources: [Microsoft Research: New AI framework bridges a critical gap in mid-range weather forecasts (Oct 14, 2026)](https://www.microsoft.com/en-us/research/story/new-ai-framework-bridges-a-critical-gap-in-mid-range-weather-forecasts/); [The Verge: Microsoft has a new machine learning tool to improve weather forecasts (not read)](https://www.theverge.com/science/1011127/ai-weather-forecast-microsoft); [GitHub: mouatadid/pbc](https://github.com/mouatadid/pbc)

### 2026-10-15: Exact asymptotic rate of Delsarte's linear programming bound for binary and spherical codes found, with key steps from Claude Opus 5.5 and GPT-5.6 Sol

****; Anthropic, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-15-delsarte-lp-bound-exact-rate-salmon-opus/

arXiv 2610.21263 (15 Oct 2026) by Andrew Salmon determines the exact asymptotic rate of Delsarte's linear programming bound for binary codes and for spherical codes, for all distances, as closed formulas in complete elliptic integrals. Upper and lower bounds match; the upper bounds connect his earlier N-qubit construction to OpenAI's August 'Ten advances' chapter on spherical codes. Gil Kalai had asked for this limit in 2024. The AI statement says Salmon and GPT-5.6 Sol formulated the upper bounds, Claude Opus 5.5 identified the key MacWilliams compatibility and recast the problem as a thin obstacle problem, and the draft was prepared with Codex. Unrefereed preprint.

- Paper: 'The rate of the Delsarte bound for binary codes and spheres', Andrew Salmon, arXiv 2610.21263 [math.CO], dated 13 Oct, listed 15 Oct 2026
- Main theorem: for 0 < δ < 1/2 and 0 < s < 1 the lim inf and lim sup of the normalised Delsarte LP optima coincide and equal explicit rate functions J(δ) (binary) and F_S(s) (spherical), parametrised by complete elliptic integrals K, E
- Background: the MRRW (1977) and Kabatiansky–Levenshtein (1978) bounds come from feasible solutions of the Delsarte LP and were the best asymptotic upper bounds for decades; OpenAI's 'Ten advances' Chapter 2 (Aug 2026) improved both; whether the LP itself could do better, and by how much, was open (Kalai 2024, Questions 3–4; Barg–Jaffe 2001 numerics; Samorodnitsky, Navon–Samorodnitsky lower bounds)
- Builds on: Salmon's earlier 'The half-rate linear programming bound for binary codes is 1/2 − 1/π' (arXiv 2609.03736), and OpenAI's Chapter 1 proof of the Afkhami-Jeddi–Cohn–Hartman–de Laat–Tajdini conjecture for the Cohn–Elkies bound
- Method: lower bounds via potential theory, an obstacle problem solved in closed form; upper bounds from interlacing spectra under the Markov–Krein correspondence; the two match
- AI statement (quoted): 'The author and GPT-5.6 Sol formulated the upper bounds for spheres and binary codes during the writing of [Sal26] … The author initially worked with Sol and Astra models on incremental strengthenings of the Navon–Samorodnitsky argument … Opus 5.5 identified the compatibility of the MacWilliams transform with the predicted relation … From there, Opus, with light direction from the author, recast the half-rate result as a thin obstacle problem … and in doing so identified the right generalization of [Sal26, Lemma 3.3]. The draft was prepared with Codex.'

#### What happened
Delsarte's 1973 linear program gives the best known general upper bounds on the size of binary codes and spherical codes. For
fifty years the asymptotic bounds actually used, MRRW (1977) and Kabatiansky–Levenshtein (1978), came from particular feasible
solutions of that program, and nobody knew what the program's true limit was. OpenAI's August 2026 "Ten advances" improved both
bounds; Gil Kalai had asked in 2024 for the limit itself.

On 15 Oct 2026 Andrew Salmon posted [arXiv 2610.21263](https://arxiv.org/abs/2610.21263), which computes it: for binary codes at every
relative distance below 1/2 and spherical codes at every angle, the normalised Delsarte optimum converges, and the limit is an
explicit function of complete elliptic integrals. Lower bounds come from an obstacle problem in potential theory, upper bounds
from interlacing spectra of N-qubit constructions related to OpenAI's Chapter 2, and the two match. It extends his September
paper, which found the binary rate at the special distance 1/2 − 1/π.

The AI section is unusually specific. Salmon and GPT-5.6 Sol formulated the upper bounds; work with "Sol and Astra models" on
Navon–Samorodnitsky-style arguments improved parts but did not finish; the human realisation of what had to be proved came "on a
human timescale"; then "Opus 5.5 identified the compatibility of the MacWilliams transform with the predicted relation", and "with
light direction from the author" recast the problem as a thin obstacle problem and found the right generalisation of his earlier
lemma. "The draft was prepared with Codex."

No Hacker News thread or press coverage was found on 16 Oct (Algolia); X was not searched.

#### Why it matters
It closes a classical question about the strongest general method for bounding codes, which also matters for sphere packing. It is
also a case where models from two labs contributed distinct ideas, recorded step by step by the author.

Sources: [arXiv 2610.21263: The rate of the Delsarte bound for binary codes and spheres](https://arxiv.org/abs/2610.21263); [arXiv 2609.03736: The half-rate linear programming bound for binary codes is 1/2 − 1/π (Salmon)](https://arxiv.org/abs/2609.03736); [Gil Kalai: On the limit of the linear programming bound for codes and packing (2024)](https://gilkalai.wordpress.com/2024/01/21/on-the-limit-of-the-linear-programming-bound-for-codes-and-packing/); [OpenAI: Ten advances in mathematics and theoretical computer science (PDF)](https://cdn.openai.com/pdf/ten-proofs-oai.pdf)

### 2026-10-15: Hopf's Euler characteristic conjecture proved under T³ symmetry; the first proof came from ChatGPT-6-Astra 'with minimal human input'

****; OpenAI, University of Münster; Science & math; confidence medium; https://postcutoff.com/e/2026-10-15-hopf-conjecture-t3-symmetry-chatgpt-astra/

arXiv 2610.22173 (15 Oct 2026) by Jan Nienhaus and Dennis Wulle proves that every closed even-dimensional manifold with positive sectional curvature and an isometric 3-torus action has positive Euler characteristic, and extends it to orbifolds. This is the Hopf conjecture under T³ symmetry; Nienhaus had proved the T⁴ case in 2022. The AI disclosure says the starting point was "a proof of Theorem A obtained with OpenAI's ChatGPT-6-Astra and minimal human input". Unrefereed preprint.

- Paper: 'The Hopf conjecture with T³-symmetry', Jan Nienhaus & Dennis Wulle, arXiv 2610.22173 [math.DG], 15 Oct 2026
- Theorem A: a closed positively curved manifold of even dimension with an effective isometric T³-action has χ(M) > 0. Theorem B: each fixed-point component has the rational cohomology ring of S^f, CP^(f/2), HP^(f/4) or S² × HP^((f−2)/4); also proved for orbifolds
- Context: Hopf (1930s) conjectured χ > 0 for all even-dimensional positively curved manifolds; the Grove symmetry program proved it under ever smaller torus actions; Nienhaus's 2022 paper (arXiv 2211.13151) needed T⁴. The authors call the new result 'a capstone for … the 4-periodicity program'
- AI disclosure (quoted): 'The starting point for this paper was a proof of Theorem A obtained with OpenAI's ChatGPT-6-Astra and minimal human input, using the c1-approach outlined in Appendix A. This proves vanishing of the odd cohomology of the fixed point components, but no additional structure. The alternative p1-route was discovered interactively … The observations that the new route allows for structure results on the cohomology of fixed point components and the generalization to orbifolds are fully human … This document contains no machine-generated text.'
- Footnote on the general conjecture: 'There has been an announcement of a counterexample to the Hopf conjecture in dimension 6, but to the knowledge of the authors this work is entirely AI and not verified by geometers' (the footnote gives no citation; the matching announcement is Guo, Fang and Lu's S³×S³ metric credited to the Odin agent, arXiv 2609.11980, see 2026-09-06-odin-ai-agent-positive-curvature-s3xs3-hopf-sign-conjecture)

#### What happened
Heinz Hopf conjectured in the 1930s that every even-dimensional closed manifold of positive sectional curvature has positive Euler
characteristic. The general case is out of reach, so since the 1990s geometers have proved it under symmetry assumptions, needing
smaller and smaller torus actions. In 2022 Jan Nienhaus proved it for isometric T⁴-actions.

On 15 Oct 2026 Nienhaus and Dennis Wulle posted [arXiv 2610.22173](https://arxiv.org/abs/2610.22173), which needs only a T³-action
and also covers orbifolds. Their Theorem B gives more: the fixed-point components have the rational cohomology of a sphere, a
complex or quaternionic projective space, or S² × HP. The authors call this "a capstone" of the 4-periodicity approach used in all
major work on the problem in the last 25 years.

The AI disclosure says the first proof of Theorem A came from ChatGPT-6-Astra "with minimal human input" (the "c1-approach" in
Appendix A). A second route was found "interactively", and the structure results and the orbifold extension are "fully human". The
paper "contains no machine-generated text". A footnote mentions an announced AI counterexample to the Hopf conjecture in dimension 6
that, as far as the authors know, is "not verified by geometers". The paper does not say who announced it, and we have not
identified it (open lead).

No Hacker News or web discussion was found on 16 Oct (Algolia, web search); X was not searched.

#### Why it matters
It is a step in a well-known geometric program, done by its main experts, where the main theorem was first proved by a model.
It also shows specialists treating unverified AI-generated claims (the dimension-6 counterexample) with open caution while using
AI-generated proofs they can check.

Sources: [arXiv 2610.22173: The Hopf conjecture with T³-symmetry](https://arxiv.org/abs/2610.22173); [arXiv 2211.13151: An improved four-periodicity theorem and a conjecture of Hopf with symmetry (Nienhaus, T⁴ case)](https://arxiv.org/abs/2211.13151); [University of Münster: Young Research Fellow Jan Nienhaus (collaboration with Dennis Wulle)](https://www.uni-muenster.de/MathematicsMuenster/news/artikel/2026/06-23.shtml)

### 2026-10-15: All 16,453,493 imaginary quadratic fields with class number ≤ 2000 determined unconditionally, using OpenAI's Siegel-zero proof made explicit in Lean with Claude Opus 5.5

****; OpenAI, Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/

arXiv 2610.21511 (15 Oct 2026) by Iwao Kimura makes the argument of OpenAI's October manuscript on Landau–Siegel zeros explicit, (1 − β) log q ≥ 5811 × 10⁻⁷ for every real zero of every primitive real character, proves it and the resulting class-number bound in Lean, and with a sieve computation determines all imaginary quadratic fields with class number at most 2000: exactly 16,453,493, the largest |d| being 1,062,350,083. Unconditional lists previously stopped at h ≤ 100. Claude Opus 5.5 (via Claude Code) did the explicit constants, Lean files, sieve program and most of the text under the author's direction. The same day Bryce Kerr posted a simpler ChatGPT-assisted proof of the Siegel-zero theorem (2610.21099).

- Paper: 'Imaginary quadratic fields of class number at most 2000', Iwao Kimura, arXiv 2610.21511 [math.NT], 15 Oct 2026
- Theorem 1 (Lean): for χ primitive real of conductor q ≥ 3 and L(β, χ) = 0, (1 − β) log q ≥ 5811 · 10⁻⁷, and ≥ 1193/200000 once log q ≥ 35.0131
- Theorem 2: exactly 16,453,493 imaginary quadratic fields have class number ≤ 2000; largest |d| = 1,062,350,083 (h = 1978). For h ≤ 1000: 4,115,897 fields, largest |d| = 227,932,027 (h = 996). Matches the GRH-conditional data of Cremona–Sutherland and Holmin et al.
- Prior state: Watkins (2004) determined the fields with h ≤ 100; 'To our knowledge, no unconditional list beyond h ≤ 100 was known'
- Status of the parts: the class-number bound (Theorem 10) is proved in Lean and checked by the Lean kernel, a replay and an independent kernel (nanoda), reusing OpenAI's Lean formalization of the Siegel-zero manuscript, so it 'does not depend on the correctness of the text of [SZ]'; the sieve computation for 6·2²⁸ ≤ |d| ≤ 1.65·10¹⁵ is argued in the paper but its program is not formally verified; small discriminants come from Mosunov–Jacobson tables
- AI statement (Section 8): 'The following was done with Claude Opus 5.5 (Anthropic), used through Claude Code, under the author's direction: (a) the explicit form of the argument of [SZ] … (b) the Lean files … (c) the sieve program … (d) the text of Sections 2 to 7 and of this section, written from the author's outline'. 'The author posed the problem and made the decisions that shaped the work' and 'worked through the proofs in the paper'
- Companion item the same day: Bryce Kerr, 'A simpler proof that there are no Landau–Siegel zeros', arXiv 2610.21099 (4 pages): replaces OpenAI's interpolation-determinant analysis with the pigeonhole principle and a result of Masser; 'ChatGPT assisted with the mathematical development and preparation of this paper'; Kerr thanks William Banks for a draft of his exposition

#### What happened
Siegel's 1935 theorem says class numbers of imaginary quadratic fields grow, but its constant is ineffective because a real
"Siegel zero" of a Dirichlet L-function could not be ruled out. That is why complete lists of fields with a given class number
stopped at h ≤ 100 (Watkins, 2004), apart from GRH-conditional tables.

OpenAI's internal model claimed on 1 Oct 2026 that Siegel zeros do not exist, with a Lean formalization but no explicit constant.
William Banks wrote out an explicit version on 11 Oct. On 15 Oct Iwao Kimura posted [arXiv 2610.21511](https://arxiv.org/abs/2610.21511),
which proves its own explicit constant, (1 − β) log q ≥ 0.0005811, in Lean (reusing OpenAI's formalization, not Banks's text), derives
an explicit class-number lower bound, and runs Watkins's sieve up to |d| ≈ 1.65 · 10¹⁵. Result: exactly 16,453,493 imaginary
quadratic fields have class number at most 2000, the largest discriminant in absolute value being 1,062,350,083.

The AI roles are listed in a dedicated section. OpenAI's model produced the Siegel-zero manuscript and its Lean proof ("We took no
part in them"). Claude Opus 5.5 in Claude Code worked out the explicit constants, wrote the Lean files and certificates, the sieve
program and comparison scripts, and drafted Sections 2–7 "from the author's outline". The author made the design decisions,
including basing the result on a formal proof rather than on OpenAI's text, and checked the proofs and sources.

The same listing has Bryce Kerr's four-page [arXiv 2610.21099](https://arxiv.org/abs/2610.21099), "A simpler proof that there are no
Landau–Siegel zeros", which replaces OpenAI's interpolation determinant with a pigeonhole argument and Masser's theorem. "ChatGPT
assisted with the mathematical development and preparation of this paper."

No Hacker News thread or press coverage was found on 16 Oct (Algolia, web search); X was not searched.

#### Why it matters
This is the first concrete consequence drawn from OpenAI's Siegel-zero claim, and it is built so that it does not trust the claim's
prose: the needed bounds are machine-checked. If the claim holds, effective results that number theory waited 90 years for become
routine computations. Two independent rewrites (Banks, Kerr) and now a Lean-based application within two weeks also show how fast
specialists are absorbing and checking the October release.

Caveats: unrefereed; the large sieve computation is not formally verified (it agrees with GRH-conditional tables where they overlap).

Sources: [arXiv 2610.21511: Imaginary quadratic fields of class number at most 2000 (Kimura)](https://arxiv.org/abs/2610.21511); [arXiv 2610.21099: A simpler proof that there are no Landau–Siegel zeros (Kerr)](https://arxiv.org/abs/2610.21099); [arXiv 2610.15483: On the OpenAI proof that Landau-Siegel zeros do not exist (Banks)](https://arxiv.org/abs/2610.15483); [OpenAI: Uniform exclusion of Landau–Siegel zeros (1 Oct manuscript)](https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf)

### 2026-10-15: CERN and Mistral AI sign a statement of intent to build a 'Large Physics Model', a foundation model for collider physics

***; CERN, Mistral AI; Science & math; confidence high; https://postcutoff.com/e/2026-10-15-cern-mistral-large-physics-model/

On Oct 15, 2026 CERN and Mistral AI signed a statement of intent for a long-term partnership on foundation models for science. The first goal is a "Large Physics Model", described by CERN as "the first foundation model for collider physics", trained on billions of simulated particle collisions and meant to replace some of the specialised algorithms that reconstruct and identify particles. No budget or timeline was given.

- Signed: statement of intent for a long-term strategic partnership, announced Thursday Oct 15, 2026 (Reuters via bdnews24)
- Model: a 'Large Physics Model', trained on billions of simulated collisions; CERN: 'the first foundation model for collider physics'; aim to replace some particle reconstruction and identification algorithms
- Data context: CERN holds about one exabyte of LHC experimental data; collision data are expected to grow tenfold after the LHC upgrade around 2030
- Enrica Porcari, CERN CIO: 'High-energy physics provides a unique environment in which artificial intelligence can be developed and tested'
- CERN's release (read Oct 19) calls it a 'statement of intent' for a long-term strategic partnership; Mistral's side is quoted through Johannes Brandstetter, Vice-President of AI for Engineering; intended uses include reconstructing collisions, identifying particles, flagging unexpected signals and evaluating future detector designs
- Not disclosed: budget, timeline, compute, or whether weights will be open. Some outlets call the model 'Flare'; we did not see that name in a primary source
- Mistral on X (Oct 15, ~90k views): 'The ambition: a single Large Physics Model, trained on billions of simulated collisions.'

#### What happened
CERN and Mistral announced the partnership on Oct 15. Both say the models could later help other sciences. CERN's own press release did not load for us; the details come
from Reuters (via bdnews24) and Mistral's post.

#### Why it matters
It brings a European frontier-model lab into the core data analysis of the world's largest particle-physics lab, at a time when
Google DeepMind, OpenAI and Anthropic are all building AI-for-science programmes. It is still only a statement of intent: no model,
budget or schedule exists yet, so the calendar should check for a funded project or first results.

Sources: [CERN: CERN and Mistral join forces to accelerate AI for science (official release, Oct 15)](https://home.cern/cern-and-mistral-join-forces-to-accelerate-ai-for-science/); [Hacker News discussion (237 points, Oct 19)](https://news.ycombinator.com/item?id=50153630); [Mistral AI on X: CERN and Mistral join forces (Oct 15)](https://x.com/MistralAI/status/2110684732103041474); [bdnews24 (Reuters): CERN and Mistral partner to build AI foundation models for physics](https://bdnews24.com/technology/w4tk7zc7dx)

### 2026-10-15: Ghys's 1995 conjecture proved in the codimension-one case (incl. complex dimension 3) with Shanghai AI Lab's Intern-S2; Peking University formalizes it in Lean (~550k lines)

***; Shanghai AI Laboratory, Peking University; Science & math; confidence high; https://postcutoff.com/e/2026-10-15-ghys-conjecture-codimension-one-intern-s2-lean/

arXiv 2609.37695 (v1 29 Sept, v3 13 Oct 2026), "Codimension-one holomorphic Anosov diffeomorphisms" by Jiesong Zhang, proves that every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This verifies Étienne Ghys's 1995 conjecture in the codimension-one case and so in complex dimension three. The v3 declares that an internal version of Shanghai AI Lab's Intern-S2 helped develop the ideas over many rounds of dialogue; Bin Dong and Ruochuan Liu's Peking University team formalized the main theorems in Lean with its open-source SHEAF agent. Shanghai AI Lab announced it on 15 Oct.

- Paper: arXiv 2609.37695 [math.DS], single author Jiesong Zhang; v3 (13 Oct) 'Added a declaration of AI use, a statement on proof formalisation, and acknowledgements. 19 pages'
- Abstract: 'every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This verifies a conjecture of Ghys in the codimension-one case and, in particular, in complex dimension three.' (Complex dimension 2 already follows from Ghys 1995, Theorem A)
- Declaration of AI use (v3): 'The author used an internal version of Intern-S2 to assist in developing the mathematical ideas and arguments presented in this paper through multiple rounds of dialogue … The author has independently verified all mathematical arguments and takes full responsibility for the manuscript.'
- Formalisation (v3): Theorem A and Theorems 1.1–1.3, plus the Ghys 1995 results used, 'have been formalised in Lean by the Bin Dong–Ruochuan Liu team in Peking University, using the open-source formalisation agent SHEAF'; carried out by Jiedong Jiang and Bin Wu; code at github.com/frenzymath/HolomorphicAnosov (repo created 13 Oct 2026)
- Shanghai AI Lab (15 Oct): Intern-S2 (书生-S2) produced a ~400-page derivation report in its first round; over 7 rounds the problem graph grew from 15 to 389 nodes, with 1,000+ pages of derivations; the Lean check took about 5 days with up to 100 agents in parallel and ~550,000 lines of Lean (lab's figures, not checked by us)
- Who led: the lab and Global Times credit a team led by Disheng Xu (Great Bay University, visiting scholar at the lab) with Jiesong Zhang (KTH, per the lab); the arXiv paper lists only Zhang as author and thanks Xu for discussions and for hosting his stay at the lab
- Lab's account: Junyi Xie called the complex case 'very difficult' and reportedly tried GPT-6 on it without success (lab claim, not independently verified)
- Section 5 builds a holomorphic Anosov automorphism of a nilmanifold whose invariant distributions are smooth but not holomorphic, so the paper's regularity theorem fails in higher codimension; the full conjecture remains open

#### What happened
On 15 Oct 2026 the Shanghai AI Laboratory said its Intern-S2 model (书生-S2) had helped prove the three-dimensional case of a
conjecture Étienne Ghys posed in 1995 about holomorphic Anosov systems, and that a Peking University team had checked the
proof in Lean. The paper behind it is Jiesong Zhang's arXiv preprint 2609.37695, first posted on 29 Sept. Its third version
(13 Oct) adds a declaration of AI use and a formalization statement. It proves the whole codimension-one case. Complex
dimension three follows, since dimension two was already settled by Ghys.

The paper credits Intern-S2 with helping "in developing the mathematical ideas and arguments … through multiple rounds of
dialogue" and says the author verified all the arguments himself. The press accounts describe a team led by Disheng Xu,
while the arXiv author line names only Zhang, who thanks Xu for discussions. The figures on derivation pages, graph nodes,
agents and lines of Lean code come from the lab's announcement.

#### Why it matters
Most AI-assisted proofs of named conjectures in this dataset used US models (GPT-6 Astra, Gemini, Claude). This one was
done with a Chinese lab's model and presented by Chinese state media as a national milestone. The Lean formalization, done with a separate agent system (SHEAF), is public, and it
covers the paper's main theorems as well as the 1995 results it relies on. The general conjecture, beyond codimension one,
is still open.

Sources: [arXiv 2609.37695: Codimension-one holomorphic Anosov diffeomorphisms (J. Zhang)](https://arxiv.org/abs/2609.37695); [GitHub: frenzymath/HolomorphicAnosov (Lean formalization)](https://github.com/frenzymath/HolomorphicAnosov); [GitHub: frenzymath/SHEAF (formalization agent)](https://github.com/frenzymath/SHEAF); [Shanghai AI Laboratory announcement (Chinese, 15 Oct)](https://www.shlab.org.cn/news/5444307); [Global Times: Chinese scientists use homegrown LLM to solve world-class math problem](https://www.globaltimes.cn/page/202610/1372364.shtml); [Pandaily: Mathematicians use Intern-S2 to prove 3D Ghys conjecture, verified in Lean](https://pandaily.com/shanghai-ai-lab-intern-s2-ghys-conjecture-proof-human-ai-collaboration-lean-sheaf); [HyperAI: Chinese scientists use Shusheng-S2 to prove Ghys conjecture](https://hyper.ai/en/stories/4cc34ccf7304036ca06b0eac04605283)

### 2026-10-15: Harvard–Anthropic team (with Mikhail Lukin) uses Claude Mythos 5 in an evolutionary search to beat state-of-the-art quantum circuits on seven of eight tasks

***; Harvard University, Anthropic, QuEra Computing; Science & math; confidence medium; https://postcutoff.com/e/2026-10-15-llm-discovered-quantum-circuits-harvard-anthropic-mythos/

arXiv 2610.22367 (15 Oct 2026) by Andi Gu, Alek Kemeny (Anthropic), Susanne Yelin and Mikhail Lukin pairs Claude Mythos 5, run inside the open-source OpenEvolve framework, with an independent evaluator that verifies and scores reversible circuits. On eight compilation tasks (decoders for decoded quantum interferometry, HHL rotations, Coulomb and Gaussian functions, chemistry and lattice-gauge simulation, modular inversion), the discovered circuits reduce both Toffoli and qubit counts versus the state of the art in seven, with Toffoli×qubit gains of 4.1× to 39×; in the eighth it found a new reversible architecture making divsteps practical for prime-field modular inversion.

- Paper: 'Discovering state-of-the-art quantum circuits with large language models', Andi Gu (Harvard; Anthropic STEM Fellows Program), Alek H. Kemeny (Anthropic), Susanne F. Yelin (Harvard), Mikhail D. Lukin (Harvard; QuEra), arXiv 2610.22367 [quant-ph], 15 Oct 2026
- Setup (quoted): 'We search over these circuit-generating programs with OpenEvolve, using Claude Mythos 5 to propose architectures and program rewrites … Starting from a blank program, it chooses the algorithm, representation, and circuit structure and produces the discovered circuits without substantive human intervention'; MAP-Elites keeps Toffoli/qubit trade-offs
- Table I, Toffoli×qubit gain over baseline: Reed–Solomon DQI decoder 12× (5.1M → 440k Toffolis); HHL rotation angle 32×; Coulomb potential 9.9×; fixed-point Gaussian 39×; FeS alias–Select 4.1×; non-abelian lattice-gauge simulation 7.7×; Hermitian DQI decoder: first reversible compilation (9.3·10⁵ Toffolis vs 10⁸ naive)
- Modular inversion: a new reversible architecture that makes divsteps, a classical algorithm 'previously impractical for reversible integer arithmetic', viable; separately a specialised Luo construction with 3.4× fewer Toffolis at five extra qubits
- Authors' framing: 'some apparent costs of quantum arithmetic reflect limitations of existing reversible constructions rather than of the algorithms themselves'
- Anthropic blog (Oct 16, 'Designing more efficient quantum circuits with Claude'): Claude was given plain-language specifications for eight circuits plus an automated evaluator and worked 'largely on its own' in a closed loop with no human input between rounds; one design makes the costliest step of a leading quantum optimization algorithm about 12x cheaper. Context it gives: FeMoco ground-state estimates fell from ~50 trillion Toffoli gates (2017) to ~340 million through circuit redesign
- Anthropic's X post announcing it (Oct 16) had ~128k views on day one: 'Claude Mythos 5 designed quantum circuits that improve on the best published ones, sometimes by an order of magnitude or more'

#### What happened
Quantum algorithms often spend most of their cost on classical subroutines (arithmetic, decoding, function evaluation) that must be
written as reversible circuits. Optimizers usually only apply equivalence-preserving rewrites to a fixed design.

On 15 Oct 2026 a Harvard group led by Mikhail Lukin, with an Anthropic co-author and an Anthropic STEM Fellow as first author, posted
[arXiv 2610.22367](https://arxiv.org/abs/2610.22367). Claude Mythos 5, inside OpenEvolve, proposes whole circuit-generating programs
from a task specification and a blank start; a separate evaluator checks correctness on basis states and scores Toffoli count
times peak qubits. Across eight tasks from optimisation, chemistry, simulation and number theory, the search beat published
constructions in seven, often by an order of magnitude, for example a 12× lower cost for the Reed–Solomon decoder in decoded
quantum interferometry. In the eighth it found a reversible design for the divstep modular-inversion algorithm.

No Hacker News thread was found on 16 Oct (Algolia); X was not searched.

#### Why it matters
Resource estimates for useful quantum algorithms depend on such subroutines; large cuts in their cost move estimates for
applications like DQI and lattice-gauge simulation. It is also an AlphaEvolve-style result with a frontier commercial model and an
open-source framework, by a leading experimental quantum group.

Sources: [Anthropic: Designing more efficient quantum circuits with Claude (Oct 16, 2026)](https://www.anthropic.com/research/designing-efficient-quantum-circuits); [Anthropic on X: STEM Fellows quantum circuits post](https://x.com/AnthropicAI/status/2111152183240511640); [arXiv 2610.22367: Discovering state-of-the-art quantum circuits with large language models](https://arxiv.org/abs/2610.22367)

### 2026-10-15: Google DeepMind shows Antigravity as a science workbench: an agent finds and characterizes a candidate bacteriocin against Streptococcus pseudopneumoniae (wet-lab tests pending)

**; Google DeepMind; Science & math; confidence high; https://postcutoff.com/e/2026-10-15-deepmind-antigravity-antimicrobial-peptide-walkthrough/

On 15 Oct 2026 Google DeepMind researchers (Pengfei Tian, Maxwell Bileschi, Zoi Tsangalidou, Sukhdeep Singh, Lucy Colwell) published a walkthrough of Google Antigravity used as an "agentic science workbench". With Gemini 3.8 Flash the agent narrowed UniProt to a 97-residue Lactococcin 972-family bacteriocin from Streptococcus pseudopneumoniae, found no close reviewed homolog, matched its AlphaFold structure to a known bacteriocin fold, read its operon context, proposed a cell-division-targeting mechanism and drafted an experimental plan. The lead and other candidates are in wet-lab validation; no results are reported.

- Authors: Pengfei Tian, Maxwell L. Bileschi, Zoi Tsangalidou, Sukhdeep Singh, Lucy J. Colwell (Google DeepMind); model selected in screenshots: Gemini 3.8 Flash
- Candidate: UniProt A0A0U0D591, 97 amino acids, Lactococcin 972 family, from Streptococcus pseudopneumoniae (respiratory infections)
- Pipeline inside one agent session: UniProt filtering → ProtNLM function prediction → BLAST against Swiss-Prot (EMBL-EBI) → AlphaFold model, remote structural homologs and superposition → genomic context (ABC transporter permease, ATP-binding partner, second bacteriocin, XRE regulator) → literature review → plan for expression, susceptibility, target-engagement and toxicity tests
- Status: 'the lead and several other candidates' from the pipeline are in wet-lab validation; no hit rates or experimental data; no paper or code linked
- New tool shown: a Biomolecular Structure Viewer embedded in Antigravity, a shared 3D workspace where researcher and agent annotate structures

#### What happened
DeepMind published a step-by-step account of one agentic research session in Antigravity, Google's agent IDE, used here for
biology rather than code. The agent searched protein databases, called domain tools (ProtNLM, BLAST, the AlphaFold
database), compared structures in an embedded 3D viewer and wrote up a mechanism hypothesis and an experimental plan for a
bacteriocin-like peptide.

#### Why it matters
It shows Google positioning Antigravity as a general science workbench. The output is a hypothesis, though: no wet-lab
result is reported yet, so the post demonstrates a workflow, not a discovery.

Sources: [Google DeepMind blog: Accelerating the search for antimicrobial peptides with Google Antigravity](https://deepmind.google/blog/accelerating-antimicrobial-discovery/)

### 2026-10-16: Brennan's 1978 conjecture: Smirnov, Binder, Ivrii and Prause check OpenAI's AI-generated proof and publish a much shorter proof of a stronger result

*****; OpenAI, University of Toronto, Tel Aviv University, Åbo Akademi University, University of Geneva; Science & math; confidence high; https://postcutoff.com/e/2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov/

On 16 Oct 2026 Ilia Binder, Oleg Ivrii, István Prause and Stanislav Smirnov (Fields Medal 2010) posted arXiv 2610.23853, "Brennan's Conjecture and the Universal Integral Means Spectrum". It responds to two preprints from OpenAI's 6 Oct maths release (family 072, dated 24 Sept) that claim a proof of Brennan's 1978 conjecture on conformal maps and a disproof of Kraetzer's t²/4 conjecture at t = −1. "While validating these results", the four found much of the AI argument redundant. Within a day they had a "drastically shorter proof of a significantly stronger statement": the bounded universal integral-means spectrum satisfies B_b(t) ≤ t²/4 on [−2, 2], with strict inequality except at 0 and ±2. Their verdict on the AI's work: "while none of the techniques appear unprecedented, their combination in this context was missed by the human experts."

- Paper: Binder, Ivrii, Prause, Smirnov, 'Brennan's Conjecture and the Universal Integral Means Spectrum', arXiv 2610.23853 [math.CV], submitted 16 Oct 2026, announced 19 Oct
- Brennan's conjecture (1978): for every conformal map g from a simply connected proper plane domain onto the disc, ∫|g′|^p dA < ∞ for 4/3 < p < 4; equivalent to B(−2) = 1 for the universal integral-means spectrum (Carleson–Makarov reduction)
- OpenAI's claim: family 072 of github.com/openai/math, 'Brennan's conjecture and sharp inverse-square integral means' and 'A strict inverse-first-power bound for univalent functions' (both dated 24 Sept 2026). They prove Brennan and B_b(−1) < 1/4, which contradicts Kraetzer's prediction B_b(t) = t²/4. OpenAI's Lean scope note says the formalization covers the Brennan interval, B_S(−2) = 1 and B_b(−1) < 1/4
- New theorem (human-written): B_b(t) ≤ t²/4 for t ∈ [−2, 2], strict on ]−2, 2[ \ {0}; by Makarov's identity also B(t) < t²/4 on ]−2, 6 − 4√2[ \ {0}. This strengthens OpenAI's single-point disproof at t = −1 to the whole interval
- On the AI proof: 'Two OpenAI preprints released in October 2026 presented a dramatic advance.' The AI proofs 'contain a significant number of derivations that, while correct, are redundant and do not seem to contribute either to strength or clarity of the results'
- AI disclosure: 'When the OpenAI proofs appeared, at first we were puzzled by a seemingly mysterious chain of thought. However, within a day we independently realized that many of the arguments were redundant… Only some of us used AI agents to help with understanding, and the write up below was done without AI help'
- Tone: 'this was a pleasant and fruitful AI-human interaction in mathematics: there was a spectacular progress, human community was brought together, new results are about to follow, and no robots or humans were harmed.' The abstract ends with Rick Blaine's line from Casablanca: 'AI, we hope this is the beginning of a beautiful friendship'
- Next: the authors say further progress 'would be computationally intensive, and we have tried to use AI tools to this effect. The results will be reported in a separate preprint'
- Context: Wikipedia's Brennan conjecture article (checked 19 Oct) lists OpenAI's September preprint as an unverified claim. No Hacker News thread and no high-reach X post about the new paper found on the morning of 19 Oct

#### What happened
Brennan's conjecture (James Brennan, 1978) concerns how wildly a conformal map can distort area near the boundary: for a conformal map g of a simply connected domain onto the unit disc, |g′|^p should be area-integrable for every 4/3 < p < 4. Carleson and Makarov reduced it, with many other questions in geometric function theory, to the universal integral-means spectrum B(t), and Brennan's conjecture becomes B(−2) = 1. Kraetzer had conjectured more generally that B_b(t) = t²/4 for |t| ≤ 2. The authors note that evidence against that formula had recently been building up.

OpenAI's 6 Oct release of 722 AI-written manuscripts (see 2026-10-06-openai-math-release-722-manuscripts) included family 072, two papers dated 24 Sept that claim Brennan's conjecture and a strict bound B_b(−1) < 1/4, which refutes Kraetzer at t = −1. OpenAI lists a Lean formalization for both.

Ten days later Binder (Toronto), Ivrii (Tel Aviv), Prause (Åbo Akademi, Turku) and Smirnov (Geneva), all long-time workers on this spectrum, posted their response. They describe the AI argument (superharmonic potentials, a critical-point map whose Jacobian sign separates maxima from saddles, a weighted signed-area inequality, and a probabilistic averaging step for the strict bound) and trace each ingredient to earlier literature. Then they give a short proof of a stronger theorem, written without AI.

#### Why it matters
This is the first famous claim in OpenAI's release that the leading specialists have publicly checked, and the verdict is positive. It also gives a concrete expert assessment of what the AI did: correct but bloated proofs, with no new techniques, combined in a way the field had missed. Their wording ("puzzled by a seemingly mysterious chain of thought", "within a day") shows how human understanding can catch up with an AI proof. It is the same pattern as Willwacher's simplified Deligne–Drinfeld proof and Angeltveit's Smith–Toda paper posted the same day.

Sources: [arXiv 2610.23853: Brennan's Conjecture and the Universal Integral Means Spectrum (Binder, Ivrii, Prause, Smirnov)](https://arxiv.org/abs/2610.23853); [OpenAI math release: Brennan's conjecture and sharp inverse-square integral means (family 072, PDF)](https://github.com/openai/math/blob/main/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026/paper.pdf); [OpenAI math release: A strict inverse-first-power bound for univalent functions (family 072, PDF)](https://github.com/openai/math/blob/main/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026/paper.pdf); [OpenAI math release: Lean scope note for family 072](https://github.com/openai/math/blob/main/lean/docs/072.md); [Wikipedia: Brennan conjecture](https://en.wikipedia.org/wiki/Brennan_conjecture)

### 2026-10-16: New Gauss circle problem bound E(X) = O(X^(209/670+ε)) breaks Huxley's 5/16 threshold; authors used ChatGPT and Codex on proof strategies and checking

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-16-gauss-circle-problem-209-670-bound/

arXiv 2610.21730 (v1 dated 15 Oct, announced 16 Oct 2026), "A New Bound for the Gauss Circle Problem" by Xi Cen, Xitao Gao, Cheng Zhang and Junyong Zhang, proves E(X) = O_ε(X^(209/670+ε)) for the lattice-point error in a circle of radius √X. 209/670 ≈ 0.31194 improves Li and Yang's 0.3144831… and is 3/5360 below 5/16, "the 5/16 threshold suggested by Huxley as a possible limit of the Bombieri–Iwaniec method at the time". The conjecture is exponent 1/4. The AI disclosure says the authors "used ChatGPT and Codex to discuss possible proof strategies, assist in drafting and checking parts of the mathematical arguments". Unrefereed preprint.

- Paper: 'A New Bound for the Gauss Circle Problem', Xi Cen, Xitao Gao, Cheng Zhang & Junyong Zhang, arXiv 2610.21730 [math.NT]
- Result: |E(X)| ≲_ε X^(209/670+ε) for all real X ≥ 4, where E(X) = #{(a,b) ∈ ℤ² : a² + b² ≤ X} − πX; 209/670 = 5/16 − 3/5360
- Previous records named in the paper: Gauss O(X^(1/2)); Huxley X^(131/416)(log X)^(18637/8320) (≈ 0.31490); Li–Yang 0.3144831759741…, using Guth–Maldague small-cap decoupling; Bourgain–Watt linked the first-spacing problem to decoupling. Conjectured: X^(1/4+ε)
- Method: a cone distribution estimate from Guth–Maldague's amplitude-dependent decoupling that keeps fourth- and sixth-power contributions separate in the double large sieve, and an explicit inversion of Taylor-coefficient constraints that cancels the linear error in a pair-counting argument; plus Robert–Sargos and Watt spacing estimates and the Vinogradov mean value theorem
- AI disclosure (quoted): 'During the preparation of this manuscript, the authors used ChatGPT and Codex to discuss possible proof strategies, assist in drafting and checking parts of the mathematical arguments, organize references, and improve the clarity and readability of the exposition. All mathematical statements and proofs in the final manuscript were written, reviewed, and verified by the authors.' No model version is named
- No Hacker News thread or press coverage found by Oct 17

#### What happened
The preprint appeared in arXiv's number-theory listing on 16 Oct 2026. It refines the Bombieri–Iwaniec/Huxley method using
Guth–Maldague decoupling, building on Li and Yang's recent improvement, and gets the exponent 209/670. The authors present 5/16 as a
historical threshold: Huxley had suggested it as a possible limit of the method, and Popov and Li–Yang repeated the remark.

#### Why it matters
The Gauss circle problem is a textbook open problem, and exponent records move by tiny amounts every decade or so. Crossing 5/16 is
symbolically important. The AI role is stated broadly (strategy discussion, drafting and checking parts of the arguments) without
saying which ideas came from the models, so the result should be read as human-led with AI assistance until experts check it.

Sources: [arXiv 2610.21730: A New Bound for the Gauss Circle Problem](https://arxiv.org/abs/2610.21730)

### 2026-10-16: Lonely Runner Conjecture: Hu and Jing get the first constant-factor gain over the union bound, (1+c)/2n, after 'many rounds of discussion' with ChatGPT 6 Pro and Astra

****; OpenAI, University of Chinese Academy of Sciences, Ohio State University; Science & math; confidence high; https://postcutoff.com/e/2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt/

Xiyu Hu (UCAS) and Yifan Jing (Ohio State) posted arXiv 2610.23790 on 16 Oct 2026. They prove that for every set V of n ≥ 2 distinct positive speeds there is a time when all runners are more than (1+c)/(2n) from the stationary one, for an absolute c > 0. This is the first improvement over the trivial union bound 1/(2n) by a constant factor. Earlier gains, by Tao and Bedert, shrink to zero as n grows. The conjecture predicts 1/(n+1). ChatGPT 6 Pro and ChatGPT 6 Astra suggested the GCD-sum approach, pointed to the Green–Walker refinement, fixed gaps in a proof and helped strengthen Green–Walker. The authors stress that "the models were unable to improve the lonely runner bound on their own".

- Paper: Xiyu Hu and Yifan Jing, 'Near optimal gap in the Lonely Runner Conjecture', arXiv 2610.23790 [math.NT], submitted 16 Oct 2026, 24 pp.
- Theorem 1.1: there is an absolute c > 0 such that ML(V) > (1+c)/(2n) for every set V of n ≥ 2 distinct positive integers; in fact the set of good times has measure at least an absolute κ0, and c = κ0/2
- Previous bounds (paper): Tao, 1/(2n) + c·log n/(n²(log log n)²); Bedert (2025), 1/(2n) + n^(−5/3−o(1)); Bedert, communicated, n^(−3/2−o(1)). 'All these bounds still give a relative improvement tending to zero'
- Corollary: a lower bound P(ε) ≥ (1+c)/(2ε) for the rotation-and-dilation pyjama problem (Kravitz–Leng equivalence)
- Method: a refinement of Green and Walker's large-GCD argument plus a new 'physical spillover inequality' for interval multiplicities (Proposition 4.4)
- AI statement: 'The main strategy and mathematical framework of the paper were developed by the authors. ChatGPT 6 Astra and ChatGPT 6 Pro were used primarily to verify arguments, turn preliminary or informal proofs into rigorous proofs, and explore possible refinements… the models were unable to improve the lonely runner bound on their own'
- AI contributions named: 'ChatGPT 6 Pro suggested introducing suitable GCD sum estimates'; 'the AI tools pointed the authors toward the Green–Walker refinement'; 'Some small gaps in the first version of his proof were later fixed by ChatGPT 6 Pro'; ChatGPT 6 Pro 'assisted the second author in refining the Green–Walker argument'; Astra used for language polishing
- No Hacker News thread or high-reach X post found on 19 Oct morning

#### What happened
In the lonely runner problem, n+1 runners with distinct constant speeds start together on a unit circular track. The conjecture says each runner is at some moment at least 1/(n+1) away from all the others. Equivalently, for n speeds there is a time t when every v·t is at least 1/(n+1) from the nearest integer. A simple union bound gives 1/(2n), and decades of work had only added lower-order terms.

Hu and Jing push the bound to (1+c)/(2n). Their AI statement gives an unusually detailed history: Hu's partial results under modularity conditions, ChatGPT 6 Pro's suggestion of GCD sums to combine information across primes, the models pointing to Green–Walker, Jing's spillover inequality (with small gaps fixed by ChatGPT 6 Pro), and a strengthened Green–Walker theorem worked out with the model. "These discussions did not produce any uniform results" until the human spillover idea arrived.

#### Why it matters
It is a substantial result on a well-known problem, and its disclosure is a clear record of the division of labour. The models acted as a knowledgeable collaborator that supplies tools and fixes proofs, while the decisive idea came from the humans. The authors put it as "an interesting example of the importance of human mathematician input".

Sources: [arXiv 2610.23790: Near optimal gap in the Lonely Runner Conjecture (Hu, Jing)](https://arxiv.org/abs/2610.23790); [Wikipedia: Lonely runner conjecture](https://en.wikipedia.org/wiki/Lonely_runner_conjecture); [Bedert, Riesz products and the Lonely Runner Conjecture (arXiv 2511.16636, prior bound)](https://arxiv.org/abs/2511.16636)

### 2026-10-16: Maximum number of Hamiltonian cycles in a tournament determined asymptotically, (cosh 1/cos 1)·(n−1)!/2ⁿ, answering Alon and proving Wormald's conjecture, with proofs 'originally found by GPT-5.6'

****; OpenAI, Princeton University; Science & math; confidence high; https://postcutoff.com/e/2026-10-16-max-hamiltonian-cycles-tournaments-gpt-5-6-wormald/

Alexander Divoux and Shouda Wang (Princeton) posted arXiv 2610.23755 on 16 Oct 2026. They prove that C(n), the maximum number of directed Hamiltonian cycles in an n-vertex tournament, is (cosh 1/cos 1 ± o(1))·(n−1)!/2ⁿ. The bound is attained asymptotically by the cyclic regular tournament on Z/nZ. This answers a question of Noga Alon, proves a conjecture of Nicholas Wormald, and also determines the maximum number of Hamiltonian paths asymptotically. "We provide our own exposition of these proofs, which were originally found by GPT-5.6." GPT-5.6 first proved the lower bound, following the authors' hint to look for permanent estimates. A separate GPT-5.6 instance, given that proof, then generalized it to the matching upper bound.

- Paper: Alexander Divoux and Shouda Wang, 'The maximum number of Hamiltonian cycles in tournaments', arXiv 2610.23755 [math.CO], submitted 16 Oct 2026, ~48 pp.
- Main theorem: C(n) = (cosh 1 / cos 1 ± o(1))·(n−1)!/2ⁿ as n → ∞; extremal example: vertices Z/nZ, edge i→j iff j − i mod n < n/2
- History (paper): Szele (1943) gave P(n) ≥ n!/2^(n−1), often called the first use of the probabilistic method, and an upper bound P(n) = O(n!·2^(−3n/4)); Alon (1990) C(n) = O(n^(3/2)(n−1)!/2ⁿ); Friedgut–Kahn O(n^α(n−1)!/2ⁿ), α ≈ 1.249; Adler–Alon–Ross C(n) ≥ (e − o(1))(n−1)!/2ⁿ; Wormald conjectured the regular cyclic tournament is extremal
- The constant cosh 1/cos 1 was first guessed from numerics by Jay Pantone's computer program and communicated to Alon (acknowledgments)
- AI role (acknowledgments): 'A proof was found by GPT-5.6 following some strategies supplied by the authors (the relevant one being a suggestion to search hard for suitable permanent estimates). In a surprising turn of events, after being input this proof and a few hopeful ideas, a separate instance of GPT-5.6 was able to generalize its techniques into an argument for the matching upper bound'
- Human role: 'Many of the arguments were eventually replaced or simplified by the authors, and others originally came from the literature. The content of the present paper was written by the authors.' Noga Alon is thanked 'for many helpful discussions and comments'
- Same pair (with Lowen) disproved Rota's 1970 matroid unimodality conjecture with ChatGPT 5.6 Pro in July (2026-07-24-rota-flats-unimodality-disproved)
- No Hacker News thread or high-reach X post found on 19 Oct morning

#### What happened
A tournament is a complete graph with every edge given a direction. Counting directed Hamiltonian cycles in tournaments is one of the oldest problems in probabilistic combinatorics. Szele's 1943 bound is often called the first application of the probabilistic method, and the maximum has been pinned down only up to polynomial factors since Alon's 1990 paper. Numerics by Jay Pantone suggested the constant cosh 1/cos 1.

Divoux and Wang first tried to prove the conjectured lower bound for the cyclic tournament, since it "seemed promising for AI models to tackle". GPT-5.6 did this with permanent estimates. A second GPT-5.6 instance, given the proof, extended the technique to the upper bound for every tournament, which settles the asymptotics. The paper is the authors' own exposition, and they replaced or simplified many of the model's arguments.

#### Why it matters
The result closes a classic extremal question from the start of the probabilistic method, and the authors say both directions of the proof came from the model. The upper bound, the harder direction, was a generalization the authors had not expected ("in a surprising turn of events").

Sources: [arXiv 2610.23755: The maximum number of Hamiltonian cycles in tournaments (Divoux, Wang)](https://arxiv.org/abs/2610.23755)

### 2026-10-16: OpenAI's AI-generated proof of a 2D entanglement area law fully formalized in Lean by Axiom (AxiomProver tooling + Codex), with no sorry

****; Axiom Math, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-16-openai-2d-area-law-lean-formalized-axiom/

On Oct 16, 2026 Axiom Math published AxiomMath/AreaLaw2D, a Lean 4 formalization of "A two-dimensional area law from a global spectral gap" (family 265 of OpenAI's openai/math release, dated Sept 24, 2026). Theorem 1.1 (entanglement entropy of the unique ground state of a gapped finite-range Hamiltonian on any finite square-lattice domain is bounded by a constant times the boundary size), Corollary 1.2 and every intermediate proposition are proved with no `sorry`, depending only on Lean's three standard axioms, and checked against a separate challenge statement with the Lean comparator. The README says the proofs were "produced by Axiom's agent tooling and OpenAI Codex, directed by Ruiz Chen". OpenAI's own catalogue had listed no Lean proof for this family. The formal proof covers the statement layer Axiom wrote from the paper; no human expert review of the paper is reported yet.

- Repo: github.com/AxiomMath/AreaLaw2D, created 2026-10-16, last pushed 2026-10-17; Lean v4.33.0, Mathlib v4.33.0, Physlib
- Formalized: Theorem 1.1 (area law on finite induced subdomains of the square lattice from a global spectral gap), Corollary 1.2, Propositions 4.3, 9.6, 10.2, 11.2 and the lemmas of Sections 2, 3 and 11
- Audit: 'The build contains no sorry. Every result, including OpenAI.area_law, depends only on the axioms propext, Classical.choice and Quot.sound'; a Challenge/Solution/Comparator pair restates Theorem 1.1 importing only Mathlib and Physlib
- Statement layer (OpenAI/Statements.lean) 'was frozen before any proof was written'; deviations from the paper's conventions are logged in docs/oai265/Memory.md (e.g. proofs over ℂ transferred to any RCLike field)
- AI role (README, quoted): 'The proofs were produced by Axiom's agent tooling and OpenAI Codex, directed by Ruiz Chen, who froze and audited the statement layer and checked each result independently.'
- Builds on Sirui Lu's TNLean (tensor networks in Lean), 'whose independent formalization of this paper covers Sections 2 to 4 and much of the geometry of Section 11', and on Axiom's internal quantum-arealaw library begun in June 2026 for the Brandão–Horodecki 1D area law
- HISTORY.md: the oai265 statement layer was started 2026-10-12; the full development took about four days
- Paper: OpenAI, 'A two-dimensional area law from a global spectral gap', Sept 24, 2026 (openai/math family 265, 'Area laws and tensor networks for two-dimensional gapped systems'); the same family claims polynomial-bond-dimension PEPS approximations, which this repo does not formalize
- Context: the 1D area law was proved by Hastings (2007); a general 2D area law for gapped systems is a long-standing open problem (only special cases such as 2D frustration-free systems, Anshu–Arad–Gosset, were known). Isaac Kim listed '1. Proof of area law in 2D' first among the 'shocking' quantum claims in OpenAI's release

#### What happened
OpenAI's Oct 6 maths release (see the 722-manuscripts entry) included family 265, "Area laws and tensor networks for
two-dimensional gapped systems". Its paper "A two-dimensional area law from a global spectral gap" (dated Sept 24, 2026) claims an
entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced subdomains of the square
lattice, assuming only a uniform spectral gap of the full Hamiltonian and bounds on the local interactions. Unlike several other
famous claims in the release, OpenAI's catalogue gives no Lean link for this family.

On Oct 16 Axiom Math published AxiomMath/AreaLaw2D, which proves Theorem 1.1, Corollary 1.2 and all the exported intermediate
propositions in Lean 4 with no `sorry`. The statement layer was written first and frozen. A separate challenge file restates the
main theorem with every needed definition copied in, and the Lean comparator accepted the solution. According to the README,
the proofs were "produced by Axiom's agent tooling and OpenAI Codex, directed by Ruiz Chen, who froze and audited the statement
layer and checked each result independently". Parts of the supporting theory were ported from Sirui Lu's TNLean and QICLean
libraries. TNLean has its own partial, independent formalization of the same paper (Sections 2–4 and much of Section 11).

The commit history (HISTORY.md) shows Axiom's internal quantum-arealaw library started in June 2026 with the Brandão–Horodecki 1D
results. The OpenAI-specific work began on Oct 12 and the main theorem was finished by Oct 16. The working notes log every deviation
from the paper's conventions, for example a citation defect in a hypothesis docstring and the choice to prove over ℂ and transfer to any RCLike field.

No Axiom blog post, X announcement or Hacker News thread about the repo was found by Oct 18 evening. We found it through the sweep's GitHub
watch of AxiomMath.

#### Why it matters
Experts listed the 2D area law first among the "shocking" quantum claims in OpenAI's release: Hastings proved the 1D case in 2007
and the general 2D gapped case has been open since. A full machine-checked proof removes doubt about the logic, though
not about whether the formal statement matches the physics claim. That match rests on Axiom's audited statement layer, which
physicists can now read. It is also a case of AI agents verifying another lab's AI-generated mathematics within weeks of
publication.

Sources: [AxiomMath/AreaLaw2D (Lean 4 formalization, README)](https://github.com/AxiomMath/AreaLaw2D); [AreaLaw2D HISTORY.md (commit history of the development)](https://github.com/AxiomMath/AreaLaw2D/blob/main/HISTORY.md); [OpenAI: A two-dimensional area law from a global spectral gap (paper PDF, openai/math)](https://github.com/openai/math/blob/main/preprints/A-two-dimensional-area-law-from-a-global-spectral-gap-September-24-2026/paper.pdf); [openai/math CONTENTS.md (family 265 description)](https://github.com/openai/math/blob/main/CONTENTS.md); [LionSR/TNLean (Sirui Lu's tensor-network library in Lean)](https://github.com/LionSR/TNLean); [Isaac Kim on X: quantum claims in OpenAI's release](https://x.com/Isaac__kim/status/2107607429437657563)

### 2026-10-16: Smith–Toda complex V(4) exists for all primes p ≥ 11: Angeltveit extends and simplifies OpenAI's AI-written p = 1009 construction, with ChatGPT's help

****; OpenAI, Australian National University; Science & math; confidence high; https://postcutoff.com/e/2026-10-16-smith-toda-v4-angeltveit-improves-openai/

Vigleik Angeltveit (ANU) posted arXiv 2610.23215 on 16 Oct 2026. It proves that the Smith–Toda complex V(4), a finite p-local spectrum with BP-homology BP∗/(p, v1, v2, v3, v4), exists for every prime p ≥ 11. This improves "the recently announced result by OpenAI that V(4) exists at the prime 1009" (openai/math family 308). He replaces three sections of OpenAI's argument with a reference to Toda, a simpler May spectral sequence computation and a Toda-bracket trick. He found no mathematical errors in the OpenAI papers, but called them "a bit suspicious", and speculates that their ultrafilters were copied in by the AI model from an irrelevant reference. ChatGPT was "used extensively as an interactive research assistant" for his own paper.

- Paper: Vigleik Angeltveit, 'The Smith-Toda complex V(4) exists for p ≥ 11', arXiv 2610.23215 [math.AT], submitted 16 Oct 2026
- Theorem 1.1: for p ≥ 7, v4 is a permanent cycle in the Adams–Novikov spectral sequence for π∗V(3); for p ≥ 11 there is a finite p-local spectrum X with BP∗X ≅ BP∗/(p, v1, v2, v3, v4)
- OpenAI papers addressed: 'A finite Smith–Toda complex V(4) at the prime 1009' and 'Finite Smith–Toda Complexes at Varying Primes' (V(n) exists at some prime for every n), both dated 23 Sept 2026, family 308 of github.com/openai/math, which has no Lean scope note
- Critique: 'We thought both of these papers looked a bit suspicious. We have not found any mathematical errors, but both of them make some strange choices… 1009 is suspiciously high.' The second paper 'uses ultrafilters throughout… We speculate that ultrafilters came from one of the references, and the AI model left them in'
- Simplifications: OpenAI's Section 4 replaced by a citation of Toda; Section 6 by 'a much simpler May spectral sequence calculation'; Section 7 by 'a simple Toda bracket juggling trick'; the hardest step of the OpenAI paper gets 'a much shorter argument (Theorem 3.8)'
- AI disclosure: 'ChatGPT (OpenAI) was used extensively as an interactive research assistant… including for checking calculations, locating relevant literature, and exploring simplifications of the arguments in the OpenAI Smith-Toda papers. In particular, conversations with ChatGPT helped lead to the reformulation of the final Adams-Novikov obstruction in terms of Toda's classical bracket-juggling argument. The author independently checked the mathematical arguments'
- No Hacker News thread or high-reach X post found on 19 Oct morning

#### What happened
Smith–Toda complexes V(n) are basic building blocks in chromatic homotopy theory: finite spectra whose Brown–Peterson homology is BP∗ modulo (p, v1, …, vn). Per Angeltveit's background section (citing Toda), V(3) exists for p ≥ 7 and has the ring structure needed here only for p ≥ 11.

OpenAI's 6 Oct release contained two papers on the question (family 308). Angeltveit worked through the first, kept its overall plan and two differentials that must vanish, and found shorter routes for the hard parts. That gave the construction for every prime from 11 up. Conversations with ChatGPT helped him find the Toda-bracket reformulation of the last obstruction.

#### Why it matters
With the Brennan paper posted the same day (2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov), this is a second case of a specialist checking an OpenAI release paper, finding it correct, and quickly improving it. Angeltveit's remarks about the odd prime 1009 and leftover ultrafilters are a rare concrete description of what an AI-written paper looks like to an expert: valid, but shaped by its search process rather than by the natural statement.

Sources: [arXiv 2610.23215: The Smith-Toda complex V(4) exists for p ≥ 11 (Angeltveit)](https://arxiv.org/abs/2610.23215); [OpenAI math release: A finite Smith–Toda complex V(4) at the prime 1009 (PDF)](https://github.com/openai/math/blob/main/preprints/A-finite-Smith-Toda-complex-V4-at-the-prime-1009-September-23-2026/paper.pdf); [OpenAI math release: Finite Smith–Toda Complexes at Varying Primes (PDF)](https://github.com/openai/math/blob/main/preprints/Finite-Smith-Toda-Complexes-at-Varying-Primes-September-23-2026/paper.pdf)

### 2026-10-16: Alpöge and Sussman say Claude has an (unreadable) existence argument for 4D Yang–Mills on ℝ⁴, by finishing Bałaban's 1980s program

****; Anthropic, Northwestern University; Science & math; confidence medium; https://postcutoff.com/e/2026-10-16-yang-mills-claude-balaban-program-alpoge-sussman/

Northwestern mathematician Ethan Sussman and Anthropic's Levent Alpöge run a public GitHub repo (since 6 Oct 2026) in which Claude, steered by Alpöge, tries to complete Tadeusz Bałaban's renormalisation-group program for constructing quantum Yang–Mills theory in four dimensions. On 16 Oct Alpöge added to the README: "At the moment we do (appear to) have a YM4 (i.e. ℝ⁴, not just in finite volume) existence argument / slop, but it is extremely long and basically not understandable as is." The repo holds a 153-page "zeroth draft" (SU(2), continuum limits on ℝ⁴ with Osterwalder–Schrader properties, conditional on listed inputs; no mass gap) and about 26 MB of raw LaTeX. Unverified; the authors say there are "certainly gaps and mistakes".

- Repo: github.com/ethanwsussman/Yang--Mills-LLM-experiment, 'an attempt with Levent Alpoge to get Claude to finish Balaban's program to construct Yang--Mills'; created 6 Oct 2026, last upload 17 Oct; 47 stars on 18 Oct
- README (16 Oct, signed 'LA'): 'we do (appear to) have a YM4 (i.e. \R^4, not just in finite volume) existence argument / slop, but it is extremely long and basically not understandable as is'
- Same note: 'we have been made aware of emails going around from groups now working on the exact same thing, making sudden progress off of the arguments above … Please, we need to not incentivize rushing out 10,000-page slop'
- Plan: start with QED3 and YM3 (3-D, 'where the UV situation is friendlier'), show 'some of the coolest YM4 arguments (e.g. winning O(4)-invariance via an RP trick)' there, write a human exposition and 'formalize what we can'
- Zeroth draft (PDF dated 6 Oct, 153 pp.): 'SU(2) Yang–Mills theory in four dimensions via Bałaban's renormalisation group: construction, axioms and non-triviality'; it claims continuum limits on tori and on ℝ⁴ with OS regularity, reflection positivity and a Hilbert space with vacuum and non-negative Hamiltonian, plus non-triviality, 'assuming the proof inputs declared clause by clause in Section 9'
- Not claimed in the draft: 'uniqueness of the infinite-volume limit …, the cluster property, a mass gap'; 'the present paper does not address the mass gap'. Full O(4) invariance rests on a displayed hypothesis
- Origin (README.pdf, 6 Oct): Sussman first asked Codex to reproduce Dimock's QED3 work along Bałaban's lines; 'After several days, the model (5.6 Sol) reported success', then about two more weeks of compute for infinite volume and the axioms; 'I do not know whether the manuscript is correct'. He then emailed Alpöge 'whom I recognized as having an excellent track record of getting the models to produce correct results'
- README.pdf on the Claude output: 'raw LLM outputs, not manuscripts or anything meant to be read by a human … There are certainly gaps and mistakes. I cannot tell how reliable the reported progress is'; it proposes a working group 'for the production, formalization, digestion, and rewriting of an AI-written manuscript on Yang–Mills'
- Spin-off: Scott Armstrong's Lean 4 project scottnarmstrong/balaban-lean (created 13 Oct) formalizes Bałaban's related O(N) spin-model papers; ~800,000 Lean lines by 18 Oct, 'most of the Lean code and design notes are written by AI agents (Anthropic's Claude and OpenAI's Codex models)'; its acknowledgements say the Sussman–Alpöge account 'suggested the route we now follow for the large-field gas'
- Context: Kalshi's 'Yang-Mills solved next before 2045?' contract rose about 20 points to ~20% on 17 Oct (tech-insider.org), which the article attributes to AI-maths attention and a cluster of self-published Yang–Mills preprints; it does not mention this repo

#### What happened
Building four-dimensional quantum Yang–Mills theory rigorously is the existence half of the Clay Millennium problem "Yang–Mills
existence and mass gap" (the other half, the mass gap, is considered harder). In 1985–1989 Tadeusz Bałaban published a long
renormalisation-group program that is widely thought to contain the seeds of a construction in finite volume, but it was never
finished.

On 6 Oct 2026 Ethan Sussman, a Boas Assistant Professor at Northwestern, opened the public repository
[Yang--Mills-LLM-experiment](https://github.com/ethanwsussman/Yang--Mills-LLM-experiment). Its `README.pdf` explains the idea.
Bałaban's approach "would require superhuman levels of work", and LLMs could supply that work. As a first test, Sussman asked
Codex to redo Dimock's papers on QED in three dimensions. "After several days, the model (5.6 Sol) reported success." After about
two more weeks of compute it also reported the infinite-volume limit and the axioms. He writes "I do not know whether the
manuscript is correct." He then emailed Levent Alpöge (Anthropic), who has been "coaxing Claude" along Bałaban's line for YM4.

The repo holds a 153-page "zeroth draft" dated 6 Oct and three raw LaTeX files of about 26 MB in total ("theorem_0_complete_account_raw").
The draft states a conditional theorem for SU(2) with Wilson's lattice action. It claims continuum limits on tori and on ℝ⁴ with
Osterwalder–Schrader regularity and reflection positivity, a Hilbert space with a vacuum, and non-triviality. These rest on
"proof inputs declared clause by clause in Section 9", and on some supporting statements "that we do not claim as proved, two of
them not yet written". The draft says plainly that "the present paper does not address the mass gap".

On 16 Oct Alpöge added a note to the README: "At the moment we do (appear to) have a YM4 (i.e. ℝ⁴, not just in finite volume)
existence argument / slop, but it is extremely long and basically not understandable as is." The plan is to show the key
arguments first in the easier 3-D theories (QED3, YM3), to write a human exposition and to formalize what they can. He added that
other groups were "now working on the exact same thing, making sudden progress off of the arguments above", and asked people not
to "incentivize rushing out 10,000-page slop". A new upload followed on 17 Oct.

Scott Armstrong's Lean project [balaban-lean](https://github.com/scottnarmstrong/balaban-lean), started on 13 Oct, formalizes
Bałaban's related papers on the O(N) spin model. It is written mostly by Claude and Codex agents, has about 800,000 lines so far
and has not proved its two main theorems. It credits the Sussman–Alpöge account with the route it now follows for the
large-field gas.

#### Reactions
- On the Erdős Problems forum, George Coote posted it on 16 Oct as "Alpoge claims possible construction of a 4D Yang-Mills theory
  on SU(2)… He confesses the proof is not yet human readable." Another user replied that prediction markets were "not fully
  convinced, yet".
- Kalshi's "Yang-Mills solved next before 2045?" contract rose about 20 points to roughly 20% on 17 Oct
  ([tech-insider.org](https://tech-insider.org/yang-mills-solved-next-before-2045-kalshi-odds/)). The article puts the move down to
  general AI-maths attention and several self-published Yang–Mills "solutions" on Zenodo. It does not mention this repo, so the
  two are not linked here.
- We found no press coverage, HN thread or expert assessment by 18 Oct. Alpöge's and Sussman's X timelines could not be read
  through fxtwitter.

#### Why it matters
This is the first public claim by a frontier-lab mathematician that a model has produced an existence argument for 4-D Yang–Mills
on all of ℝ⁴. It comes with strong warnings from its own authors: it is not a proof anyone has checked, the mass gap is not
addressed, and parts are conditional. It also shows a new way of working. A human-abandoned program is handed to models at scale,
and the human job becomes digesting the output and formalizing it in Lean (Armstrong's project). It follows the OpenAI
Navier–Stokes controversy, and Alpöge himself raises the risk of a race to post very long unreadable manuscripts.

Sources: [GitHub: ethanwsussman/Yang--Mills-LLM-experiment (README, zeroth draft, raw Claude output)](https://github.com/ethanwsussman/Yang--Mills-LLM-experiment); [README.pdf: proposal for an LLM-aided attack on YM4 (6 Oct 2026)](https://github.com/ethanwsussman/Yang--Mills-LLM-experiment/blob/main/README.pdf); [ym4_manuscript_zeroth_draft.pdf (153 pages)](https://github.com/ethanwsussman/Yang--Mills-LLM-experiment/blob/main/ym4_manuscript_zeroth_draft.pdf); [GitHub: scottnarmstrong/balaban-lean (Lean 4, Claude and Codex agents)](https://github.com/scottnarmstrong/balaban-lean); [Erdős Problems forum, 'AI Contributions 3' (George Coote's post, 16 Oct)](https://www.erdosproblems.com/forum/thread/AI%20Contributions%203); [tech-insider.org: Yang-Mills 'Solved Next Before 2045?' odds jump 20 points on Kalshi (17 Oct)](https://tech-insider.org/yang-mills-solved-next-before-2045-kalshi-odds/)

### 2026-10-16: arXiv results with disclosed AI help, 15 Oct 2026 listing: monodromy conjecture, Fried counterexamples, Godefroy–Lancien–Zizler question and more

***; OpenAI, Anthropic, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-16-ai-assisted-results-october-15-listing/

Continuation of the mid-October catalogue (closed at about 32 KB). It lists preprints from the 15 Oct 2026 arXiv listing (and a few late 14 Oct items) that settle or advance named problems and disclose an AI contribution, each AI statement quoted from the PDF. Own entries: Smale's mean value conjecture (GPT-6 Astra's Lean disproof and the Jatar–Ng bound M > 1.1018), the Hopf conjecture under T³ symmetry, all imaginary quadratic fields with class number ≤ 2000, the exact rate of Delsarte's LP bound, and LLM-discovered quantum circuits. Highlights here: Esterov's strong monodromy conjecture for high-order poles, where ChatGPT Astra "suddenly invented" a birational inequality; Becker–Lawton's GPT-6.1 Sol counterexamples to Fried's conjecture; and Cúth–Kalenda's negative answer to a Godefroy–Lancien–Zizler question. All items are unrefereed preprints.

- Own entries from this listing: Smale's mean value constant (Jatar & Ng, 2610.21472, building on GPT-6 Astra's Lean disproof); Hopf conjecture with T³-symmetry (Nienhaus & Wulle, 2610.22173); imaginary quadratic fields of class number ≤ 2000 (Kimura, 2610.21511, Claude Opus 5.5) with Kerr's simpler Siegel-zero proof (2610.21099); rate of the Delsarte bound (Salmon, 2610.21263, Opus 5.5 and GPT-5.6 Sol); quantum circuits found by Claude Mythos 5 (Gu, Kemeny, Yelin & Lukin, 2610.22367)
- AI 'invented' the key input: strong local monodromy conjecture for poles of order ≥ n − 1 (Alexander Esterov, 2610.22111): ChatGPT Astra 6.0 was asked to combine three inputs; 'The AI model generated a proof, but, instead of using (3top), suddenly invented (3), with a proof based on deep MMP techniques'; Esterov had started 'with the belief that such an extension was unrealistic'; the raw output is attached to the arXiv record 'as a non-human-verified output'
- Counterexamples from a one-line prompt: ordinary Fried conjecture fails for real-analytic metrics arbitrarily close to hyperbolic ones in dimension 3 (Simon Becker & Benjamin Lawton, 2610.22281): 'S.B. initially prompted GPT-6.1 Sol (Ultra reasoning, in Codex) with "Prove Conjecture 9" … The resulting argument led instead to the counterexample'; Lawton used Claude and Claude Code incl. Lean, which covers 'conditional algebraic and limiting steps' only
- Long-standing question answered: the Lipschitz-free space over a separable Banach space is complemented in its bidual iff the space is isomorphic to Hilbert space, answering negatively a question of Godefroy, Lancien and Zizler for ℓ₁ (Marek Cúth & Ondřej Kalenda, 2610.21818): Sol 5.6 and Astra models were asked 'to suggest solutions to intermediate problems … Through several iterations, this process led to the main ideas of the proof'
- Constructions and counterexamples from models: strong coloring numbers of d-dimensional grids grow as r^(d−1) (Piotr Micek & Torsten Ueckerdt, 2610.21506; 'The construction in Lemma 3 was found by OpenAI's GPT-5.6 Sol'); three families of counterexamples to the Marian–Oprea strange duality conjecture on K3 surfaces (Shizhuo Zhang & Yu Zhao, 2610.21272; 'suggested by GPT-6 Pro and GPT-Astra in ChatGPT')
- Follow-ups to OpenAI's manuscript release: Zhi-Wei Sun shows Hilbert's tenth problem over ℚ is undecidable already with 86 unknowns, starting from 'Hilbert's Tenth Problem over the field Q … was solved negatively by OpenAI' (2610.20691, 14 Oct; no AI statement found); a quantitative planar Falconer distance theorem developing 'the multiscale method of the September 2026 manuscript of OpenAI' (Cheng, Fang, Li, Qu & Xiao, 2610.22222)
- Human results with no AI statement found: the Bardos–Tartar conjecture for 2D Navier–Stokes (Barnes, Liew & Titi, 2610.21530); a counterexample to Plotnick's 1983 conjecture on knot exteriors (Hofmann & Nicholson, 2610.21051); Isaac's Condition (TU) false, Lean 4-verified, single author (Ralf Stephan, 2610.21899)

#### What happened
Items below come from the PDFs of the 15 Oct 2026 arXiv listing (papers dated 14–15 Oct), read on 16 Oct. Each AI statement is
quoted or closely paraphrased from the paper. None is refereed.

### Own entries
- [Smale's mean value conjecture](/e/2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra/): Jatar & Ng ([2610.21472](https://arxiv.org/abs/2610.21472)) build on GPT-6 Astra's September Lean disproof of the constant-1 form and prove Smale's constant exceeds 1.1018.
- [Hopf conjecture with T³-symmetry](/e/2026-10-15-hopf-conjecture-t3-symmetry-chatgpt-astra/) (Nienhaus & Wulle, [2610.22173](https://arxiv.org/abs/2610.22173)): first proof of the main theorem "obtained with OpenAI's ChatGPT-6-Astra and minimal human input".
- [Imaginary quadratic fields of class number ≤ 2000](/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/) (Kimura, [2610.21511](https://arxiv.org/abs/2610.21511)), built on OpenAI's Siegel-zero proof with Claude Opus 5.5 and Lean, plus Kerr's simpler proof ([2610.21099](https://arxiv.org/abs/2610.21099)).
- [Exact rate of Delsarte's LP bound](/e/2026-10-15-delsarte-lp-bound-exact-rate-salmon-opus/) (Salmon, [2610.21263](https://arxiv.org/abs/2610.21263)).
- [Quantum circuits discovered by Claude Mythos 5](/e/2026-10-15-llm-discovered-quantum-circuits-harvard-anthropic-mythos/) (Gu, Kemeny, Yelin & Lukin, [2610.22367](https://arxiv.org/abs/2610.22367)).

### Large AI roles
- **Strong monodromy conjecture for high-order poles** (Alexander Esterov, [2610.22111](https://arxiv.org/abs/2610.22111)). Proves the strong local monodromy conjecture for poles of order at least n − 1 of the naive motivic zeta function of a polynomial in n variables, with explicit eigencycles. The abstract says the proof "combines topological techniques developed by the author with a fact of birational geometry discovered by AI". Section 1.2: given three inputs, ChatGPT Astra 6.0 "generated a proof, but, instead of using (3top), suddenly invented (3), with a proof based on deep MMP techniques". Esterov had believed such an extension "unrealistic"; the model's original output is attached to the arXiv record "as a non-human-verified output". (A smaller monodromy case, the symmetric determinant, was in the mid-October catalogue.)
- **Fried conjecture counterexamples** (Simon Becker & Benjamin Lawton, [2610.22281](https://arxiv.org/abs/2610.22281)). For real-analytic negatively curved metrics arbitrarily close to hyperbolic ones on 3-manifolds, |Ruelle zeta(0)| / torsion² tends to zero along acyclic unitary characters, refuting Fried's identity (stated as Conjecture 9 in a Zworski survey). "S.B. initially prompted GPT-6.1 Sol (Ultra reasoning, in Codex) with 'Prove Conjecture 9' … The resulting argument led instead to the counterexample." Lawton used Claude and Claude Code, including Lean for the "conditional algebraic and limiting steps"; geometric and spectral inputs are not formalized.
- **Godefroy–Lancien–Zizler question** (Marek Cúth & Ondřej Kalenda, [2610.21818](https://arxiv.org/abs/2610.21818)). The Lipschitz-free space over a separable Banach space is complemented in its bidual iff the space is Hilbertian, which answers negatively a "long-standing question" for ℓ₁. Sol 5.6 and Astra suggested solutions to intermediate problems; "Through several iterations, this process led to the main ideas of the proof."
- **Strong coloring numbers of grids** (Piotr Micek & Torsten Ueckerdt, [2610.21506](https://arxiv.org/abs/2610.21506)): scol_r(G(n,d)) = Ω_d(r^(d−1)), matching the volume upper bound. "The construction in Lemma 3 was found by OpenAI's GPT-5.6 Sol."
- **Strange duality on K3 surfaces** (Shizhuo Zhang & Yu Zhao, [2610.21272](https://arxiv.org/abs/2610.21272)): three families of counterexamples to the Marian–Oprea conjecture (the map is either zero or an isomorphism) and a refined statement. "The counterexamples in this paper were suggested by GPT-6 Pro and GPT-Astra in ChatGPT."
- **Jung–Kim conjecture** on centres of walled Brauer algebras (Shunsuke Hirota, [2610.21576](https://arxiv.org/abs/2610.21576)): the idea of Lemma 2.8 and the argument avoiding stable module categories "originated in suggestions from Astra", which also proposed the formula for the arc-algebra centre.
- **Positroids, Theta and Delta conjectures** (Hai Zhu, [2610.21894](https://arxiv.org/abs/2610.21894)): "the author used GPT-5.5 to prove Lemmas 5.7 and 5.10"; the core recursions were "made without AI assistance".
- **Quadratic APN permutations in odd dimension** (Hiroki Fukui, [2610.21169](https://arxiv.org/abs/2610.21169), submitted to IEEE Trans. Inf. Theory): Claude, Claude Code and ChatGPT "proposed and drafted proofs"; Lean code written with Claude Code, checked by the Lean kernel and an independent checker.
- **Nullstellensatz over large fields** (Pablo Cubides Kovacsics & Sebastián Uribe Rincón, [2610.21039](https://arxiv.org/abs/2610.21039)): calls itself "perhaps the authors last attempt to provide a contribution more than 80% human"; still, Claude Opus 4.8 suggested extending Theorem 4.4 to every non-large field and gave early proofs of Theorems 5.11 and 5.15; DeepSeek V4 helped the presentation.
- **Entanglement-assisted spectrum of graphs** (Nidhal Mghirbi & Seth Douglas, [2610.20775](https://arxiv.org/abs/2610.20775), 14 Oct): a continuum of Rényi points beside the Lovász number; generative AI used "throughout … including for mathematical exploration, proof development and critique".
- **Evolution algebras survey and open problems** (Kaygorodov, Martín González & Viruel, [2610.22248](https://arxiv.org/abs/2610.22248)): Harmonic's Aristotle helped with literature search, error checking, experiments and Lean verification of the new observations in Section 2.1.

### Smaller roles
Galois deformation rings in characteristic two (Yufan Luo, [2610.21298](https://arxiv.org/abs/2610.21298); ChatGPT searched for examples that became Theorems 1.7–1.9); interlacing conjecture for Sylvester–Kac singular values (Tang & Shi, [2610.21702](https://arxiv.org/abs/2610.21702); ChatGPT 6 Astra "only for language polishing and for screening counterexamples"); area-vanishing square peg conjecture (Lorenzo Valerio, [2610.21076](https://arxiv.org/abs/2610.21076); Gemini 3.1 Pro and GPT-6 Astra to find errors, "minimal use"); generalized Viterbo conjecture counterexample (Broćić, Serraille & Trifa, [2610.22175](https://arxiv.org/abs/2610.22175); thanks for GPT 6 Astra access, "AI-assisted generalizations" to appear in a later version); asymptotic Erdős–Frankl–Füredi cover-free conjecture (Xinqi Huang & Hong Liu, [2610.22021](https://arxiv.org/abs/2610.22021); AI for literature and editing, and it "suggested a simplification" of a calculation); coarse embeddability characterisations (Li, Zhang & Zhu, [2610.21632](https://arxiv.org/abs/2610.21632); ChatGPT for "exploring mathematical ideas"); Motsch–Tadmor propagation of chaos (Kldiashvili, Rossi & Trussardi, [2610.21828](https://arxiv.org/abs/2610.21828); ChatGPT-5.6 Sol for brainstorming); quantitative planar Falconer (Cheng, Fang, Li, Qu & Xiao, [2610.22222](https://arxiv.org/abs/2610.22222); an AI tool for "local mathematical reasoning"); generalized Erdős–Ginzburg–Ziv constants in rank three (Zhang & Zhang, [2610.21577](https://arxiv.org/abs/2610.21577); routine calculations); King–Tollu–Toumazet conjecture on rectangular Kostka coefficients (Deepanshu Prasad, [2610.21921](https://arxiv.org/abs/2610.21921); Claude for editing only); nilpotent multicorrelation sequences (Andreevski & Hernández, [2610.22076](https://arxiv.org/abs/2610.22076); "No AI tools were used to produce any of the results", only a GPT-6 Astra Max referee-style report).

### Follow-ups to OpenAI's manuscript release
Zhi-Wei Sun, who had brought Hilbert's tenth problem over ℤ down to 10 unknowns, now proves it undecidable over ℚ with 86 unknowns, starting from "Hilbert's Tenth Problem over the field Q of rational numbers was solved negatively by OpenAI based on a five-contact height estimate and a pole-parity construction" ([2610.20691](https://arxiv.org/abs/2610.20691); no AI statement found). Cheng, Fang, Li, Qu & Xiao give a quantitative planar Falconer theorem by developing "the multiscale method of the September 2026 manuscript of OpenAI" ([2610.22222](https://arxiv.org/abs/2610.22222)). Kimura and Kerr (own entry) build on the Siegel-zero manuscript; Salmon (own entry) on the August "Ten advances". The Hopf paper notes an announced AI counterexample to the Hopf conjecture in dimension 6 "not verified by geometers".

### No AI statement found (for contrast)
The **Bardos–Tartar conjecture** (1973): for 2D Navier–Stokes with periodic boundary conditions, the set of solutions at a positive time is dense in L² (Barnes, Liew & Titi, [2610.21530](https://arxiv.org/abs/2610.21530), dedicated to Claude Bardos); a counterexample to Plotnick's conjecture on knot exteriors (Hofmann & Nicholson, [2610.21051](https://arxiv.org/abs/2610.21051)); Isaac's Condition (TU) for simple normality of √s is false, all results "machine-verified in Lean 4", single author (Ralf Stephan, [2610.21899](https://arxiv.org/abs/2610.21899)); uniform non-homogeneous vector bundles against the Drézet and Ballico–Ellia conjecture (Macaulay2, [2610.20208](https://arxiv.org/abs/2610.20208)).

No Hacker News threads on any of these were found on 16 Oct (Algolia); X was not searched directly.

#### Why it matters
The disclosures keep getting more specific: which lemma, which construction, even the one-line prompt ("Prove Conjecture 9") that led a model to a counterexample instead. Two experts (Esterov, Nienhaus–Wulle) say a model produced the key step of a result in their own research program, and the October OpenAI release is already being used as a black box by other authors (Sun, Kimura).

Sources: [arXiv 2610.22111: The monodromy conjecture for high-order poles, and norms of residue antiderivatives (Esterov)](https://arxiv.org/abs/2610.22111); [arXiv 2610.22281: Counterexamples to the ordinary Fried conjecture near hyperbolic metrics (Becker & Lawton)](https://arxiv.org/abs/2610.22281); [arXiv 2610.21818: Lipschitz-free spaces over separable Banach spaces and their complementability in the bidual (Cúth & Kalenda)](https://arxiv.org/abs/2610.21818); [arXiv 2610.21506: Strong coloring numbers of grids (GPT-5.6 Sol construction)](https://arxiv.org/abs/2610.21506); [arXiv 2610.21272: A refinement of the strange duality conjecture on K3 surfaces (GPT-6 Pro counterexamples)](https://arxiv.org/abs/2610.21272); [arXiv 2610.21576: A proof of the Jung–Kim conjecture on the centers of walled Brauer algebras (GPT-6 Astra)](https://arxiv.org/abs/2610.21576); [arXiv 2610.21894: Algebras connecting positroids, the Theta and Delta conjectures (GPT-5.5 lemmas)](https://arxiv.org/abs/2610.21894); [arXiv 2610.21169: Optimal reduced representatives of quadratic APN permutations (Claude Code + Lean)](https://arxiv.org/abs/2610.21169); [arXiv 2610.21039: Holomorphy rings and the Nullstellensatz over large fields (Claude Opus 4.8)](https://arxiv.org/abs/2610.21039); [arXiv 2610.20691: 86 unknowns for Hilbert's tenth problem over the rationals (Zhi-Wei Sun)](https://arxiv.org/abs/2610.20691); [arXiv 2610.22222: A quantitative form of the planar Falconer distance theorem](https://arxiv.org/abs/2610.22222); [arXiv 2610.21530: The Bardos-Tartar conjecture for the Navier-Stokes equations](https://arxiv.org/abs/2610.21530)

### 2026-10-16: Sankar Das Sarma's exciton-superfluidity theory paper: derivations, code, figures and text produced by Claude Opus 5.5, with Codex (GPT-6 Astra) acting as referee

***; University of Maryland, Anthropic, OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-16-das-sarma-exciton-superfluidity-claude-opus-codex-referee/

arXiv 2610.22037 (dated Oct 16, announced 16 Oct 2026), "Exciton superfluidity in two-dimensional bilayers at zero and high magnetic fields" by Sankar Das Sarma and Yang-Zhi Chou (University of Maryland), studies how interlayer exciton coherence evolves with layer separation. Its acknowledgments, written "in accordance with the policy of the American Physical Society", say that for the zero-field theory "the analytic derivations, the numerical codes and calculations, the figures, and the text were produced with the AI system Claude (Anthropic; model identifier claude-opus-5-5)" under the author's direction, and that successive drafts "were criticized by a second AI system, Codex (OpenAI, GPT-6-Astra), acting as a referee". Unrefereed preprint.

- Paper: 'Exciton superfluidity in two-dimensional bilayers at zero and high magnetic fields', Sankar Das Sarma & Yang-Zhi Chou, Condensed Matter Theory Center and Joint Quantum Institute, University of Maryland; arXiv 2610.22037 [cond-mat.mes-hall]; 13 appendices (A–M)
- Physics claims: in the ladder approximation the apparent plasmon enhancement of pairing is 'largely cancelled by the second-order crossed-rung correction'; at large separation the pairing scale falls exponentially with squared separation while the instability persists; at νT = 1 a Hartree–Fock collective-mode instability favours 'an excitonic supersolid'; the authors conjecture the quantum Hall bilayer evolves continuously into weak pairing of composite fermions, which could mimic a finite-temperature critical separation without a quantum phase transition
- AI statement (quoted): 'This work made substantial use of artificial intelligence tools … The author posed the problems, specified the approach, and directed the work at each stage. For the zero-field theory (Secs. III–X and Appendices A–F) the analytic derivations, the numerical codes and calculations, the figures, and the text were produced with the AI system Claude (Anthropic; model identifier claude-opus-5-5) under that direction. Successive drafts of that part were criticized by a second AI system, Codex (OpenAI, GPT-6-Astra), acting as a referee, and were revised in response.'
- Checks named in the paper: the plasmon-induced coupling by two independent quadratures, the gap-equation solver against the BCS result, and limiting forms of the net second-order coupling against the full quadrature (Appendices D and F)
- The same AI was used to revise the νT = 1 theory text (Sec. XI, Appendices G–M) and to write the abstract, introduction, conclusion and Secs. II and XII; data 'available from the authors upon reasonable request'
- No Hacker News thread or press coverage found by Oct 17

#### What happened
The preprint appeared on arXiv's condensed-matter listing of 16 Oct 2026. Its long acknowledgments section splits the work by
section: the zero-field theory (Secs. III–X, Appendices A–F) was produced by Claude Opus 5.5 under the author's direction, and a
second model, Codex running GPT-6 Astra, critiqued successive drafts "acting as a referee". The quantum Hall part was human-derived,
with AI help on the text.

#### Why it matters
Most AI disclosures in physics papers so far cover code or editing. This one comes from a prominent condensed-matter theorist (University of Maryland Condensed Matter Theory Center) and
puts the analytic derivations themselves, plus code, figures and text, on the model, with an AI-vs-AI review loop before human
submission. It is a concrete template for the APS disclosure policy the paper cites.

Sources: [arXiv 2610.22037: Exciton superfluidity in two-dimensional bilayers at zero and high magnetic fields](https://arxiv.org/abs/2610.22037)

### 2026-10-16: Buzzard, Avigad, Morrison and other Lean leaders answer AI 'formalize everything' pushes: most of mathematics could be formalized within three years, but only with curated, reviewed libraries

***; Lean community, Mathlib; Science & math; confidence high; https://postcutoff.com/e/2026-10-16-lean-leaders-how-to-formalize-all-of-mathematics/

On Oct 16, 2026 Jeremy Avigad, Kevin Buzzard, Johan Commelin, Rémy Degenne, Kim Morrison, Oliver Nash and Adam Topaz published "How to Formalize All of Mathematics" on the Xena blog, responding to calls from "AI enthusiasts" to auto-formalize the whole literature. They criticise AI dumps such as Anthropic's 13-million-line Fermat formalization and OpenAI's math release, set out ten shared values (defer to human-curated Mathlib, public review, open publication, standards met before release), and say most of mathematics could be formalized "within three years, and possibly much sooner". Terence Tao amplified it the same day.

- Examples criticised: Math, Inc.'s attempted takeover of the 8-dimensional sphere-packing Lean project (the original team rewrote the AI code); Anthropic's Fermat's Last Theorem formalization (13 million lines in 11 days); OpenAI's Navier–Stokes blow-up with Lean formalization and its large GitHub math release (retractions, informal/formal mismatches, hard-to-read write-ups)
- Examples praised: OpenAI's unit-distance conjecture work and Anthropic's Riemann zeta zeros work
- Main argument: a formal artifact has value only if people use it to do more mathematics; poor formalization could waste millions of dollars and set the field back
- Projects described: Mathlib (65 reviewers, 950 contributors; 2026 Jean-Pierre Demailly Prize), DeepMind's Formal Conjectures, the Annals Challenge (statements of recent Annals theorems), Tau Ceti (AI-written and AI-reviewed, ~3M lines), Formal Frontier (41 libraries, ~200k lines), Palomar registry
- Asks: contribute to Mathlib; scale Tau Ceti and Formal Frontier tenfold within months; expert roadmaps and blueprints; keep autoformalization tools under mathematicians' control; non-profit or academic hosting
- Tao (Oct 16): questions about formal mathematics mirror those about informal mathematics, a tradeoff between speed, scale and understanding

#### What happened
The leading maintainers of Lean's mathematical library wrote a joint position paper on how AI-scale formalization should be done.
They do not oppose AI formalization; they want it, and set a three-year horizon. Their objection is to unreviewed, unaudited code
released on the promise of later cleanup, and to libraries that duplicate Mathlib instead of building on it.

#### Why it matters
After Anthropic's Fermat formalization and OpenAI's mass release, the Lean community's leaders set out the standards they expect AI
formalization to meet. They also named the projects (Tau Ceti, Formal Frontier) they think should be scaled up.

Sources: [Xena blog: How to Formalize All of Mathematics (Avigad, Buzzard, Commelin, Degenne, Morrison, Nash, Topaz; Oct 16, 2026)](https://xenaproject.wordpress.com/2026/10/16/how-to-formalize-all-of-mathematics/); [Terence Tao: How to Formalize All of Mathematics (pointer post)](https://terrytao.wordpress.com/2026/10/16/how-to-formalize-all-of-mathematics/)

### 2026-10-16: Lee–Loh–Sudakov conjecture on cuts in both directions proved; proof first found by a multi-model AI framework (ChatGPT 6 Astra, 6.1 Sol, Claude Opus 5.5) and verified in Lean

***; Altifina, Charles University, OpenAI, Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-10-16-lee-loh-sudakov-directed-cuts-altifina-lean/

arXiv 2610.21381 (v1 dated 15 Oct, announced 16 Oct 2026) by Frederic Folz (Altifina Discovery Lab) and Mykhaylo Tyomkyn (Charles University) proves that for every d ≥ 2 and all large m, every digraph with m edges and minimum outdegree ≥ d has a vertex bipartition with both directed cuts at least ⌈(d−1)m/(2(2d−1)) + d²/(2(2d−1))⌉, sharp for infinitely many m. This proves the Lee–Loh–Sudakov conjecture (2013) and answers Alex Scott's 2005 question for large graphs. The AI statement says the key proof was "first discovered by the Altifina research framework" combining ChatGPT 6 Astra, 6.1 Sol and Claude Opus 5.5, and that the proofs "have been verified in Lean". Unrefereed preprint.

- Paper: 'Optimally large cuts in both directions', Frederic Folz & Mykhaylo Tyomkyn, arXiv 2610.21381 [math.CO]
- Results: the sharp bound above for all large m (Theorem 1.1); ⌈m/6⌉ achievable for d = 2 and every m; a stability result describing near-extremal digraphs (Theorem 1.4)
- Background: Scott (Surveys in Combinatorics 2005) asked for the largest c(d) such that every digraph with minimum outdegree ≥ d has a bipartition with both directed cuts ≥ c·m; Lee, Loh & Sudakov ('Judicious partitions of directed graphs', arXiv 1305.2986, RSA 2016) conjectured c_d = (d−1)/(2(2d−1)) and showed it cannot be raised
- AI statement (quoted): 'The proof of Theorem 1.2 was first discovered by the Altifina research framework, which combines multiple AI models—in this work, OpenAI's ChatGPT 6 Astra and 6.1 Sol, as well as Anthropic's Claude Opus 5.5. The proof was revised and extended by the second author, aided by ChatGPT 6 Astra, leading to the main results of this paper, Theorems 1.1 and 1.4. The proofs have been verified in Lean, and a certificate is available upon request. The presentation is human-written and proofread with ChatGPT 6 Astra.'
- Abstract: 'The proofs were obtained by LLMs, with human input. They have been revised by the authors and verified in Lean.' The Lean code is not public (certificate 'available upon request')
- First author's affiliation is Altifina Discovery Lab (Überherrn, Germany), the company behind the framework; no Altifina announcement or Hacker News thread found by Oct 17

#### What happened
Folz and Tyomkyn's preprint appeared in arXiv's 16 Oct 2026 announcement. A random-partition argument needs a deterministic
criterion (built from a partition of components into induced stars, going back to Erdős, Gyárfás and Kohayakawa) to guarantee
both directed cuts are large at once. The authors say that criterion's proof (Theorem 1.2) was first found by the Altifina framework,
then revised and extended by Tyomkyn with ChatGPT 6 Astra into the sharp theorem and the stability result.

#### Why it matters
It is one more named conjecture by well-known combinatorialists (Po-Shen Loh, Benny Sudakov) settled with an AI-first proof. The
Lean claim cannot be checked yet because the certificate is not public. The first author's affiliation shows a small company
running a multi-model "research framework" that combines OpenAI and Anthropic models.

Sources: [arXiv 2610.21381: Optimally large cuts in both directions (Folz & Tyomkyn)](https://arxiv.org/abs/2610.21381); [arXiv 1305.2986: Judicious partitions of directed graphs (Lee, Loh & Sudakov)](https://arxiv.org/abs/1305.2986)

### 2026-10-16: Equiregular counterexample to the sub-Riemannian Sard conjecture: Claude Opus 5/5.5 wrote the construction and first proofs, Lean-formalized with Codex (Wang, Zelenko)

***; Texas A&M University, Anthropic, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-16-sub-riemannian-sard-equiregular-counterexample-claude/

arXiv 2610.24080 (16 Oct 2026) by Bangjie Wang and Igor Zelenko constructs an equiregular rank-two distribution on an open set diffeomorphic to R⁹, with growth vector (2, 3, 5, 8, 9), whose endpoint map from the origin has no regular values. This settles the equiregular case left open by Lerario, Rizzi and Tiberio's disproof of the Sard conjecture of Zhitomirskii and Montgomery. Zelenko proposed lifting their example; "the mathematical implementation of this strategy was generated … in interactive sessions with Anthropic's Claude [Claude Opus 5 and Claude Opus 5.5]". Theorems 1.1–1.3 are formalized in Lean 4 with Codex, with no unproved statements. Unrefereed.

- Paper: 'An equiregular counterexample to the Sard conjecture for sub-Riemannian endpoint maps', Bangjie Wang and Igor Zelenko, arXiv 2610.24080 [math.DG], 16 Oct 2026
- Result: equiregular rank-two distribution on an open set ≅ R⁹ with small growth vector (2, 3, 5, 8, 9), 'maximal and generic for rank-two distributions in dimension nine', whose endpoint map has no regular values. The earlier Lerario–Rizzi–Tiberio counterexample (rank two on R⁴) 'is neither equiregular nor real-analytic'
- Announced, proofs deferred: an equiregular counterexample in dimension twelve with constant Tanaka symbol, to appear in a separate paper
- AI disclosure (follows the Leiden Declaration), verbatim: 'The mathematical implementation of this strategy was generated in September and October 2026 in interactive sessions with Anthropic's Claude [Claude Opus 5 and Claude Opus 5.5], used through the Claude desktop application. This includes the explicit structure (1.2), the first versions of the proofs of Theorems 1.1–1.3, in particular the formulation and solution of the moment problem of Sections 4–6, the statements in Appendices A and B, the explicit dimension-twelve structure announced in Appendix A, and first drafts of most of the text'. ChatGPT (6 Astra) later reviewed drafts against the formalization reports
- Human role: the lifting strategy and the Tanaka-symbol and abnormal-dynamics questions came from Zelenko; 'Both authors checked all arguments line by line'; Sections 4–6 replace a single section of the AI draft
- Lean: Wang formalized Theorems 1.1–1.3 in Lean 4.30.0 / Mathlib with Codex; 'The development contains no unproved statements and uses only the standard axioms'; Appendices A and B are not formalized; the formalization 'detected two defects in an earlier draft'. Sources: Zenodo 10.5281/zenodo.23310061. The formal statements were written by the authors, who checked that they match the paper

#### What happened
The Sard conjecture in sub-Riemannian geometry asks whether the endpoints of abnormal (singular) horizontal curves form a set of
measure zero. Lerario, Rizzi and Tiberio found a counterexample, but it relied on a degenerate distribution. Zelenko suggested
lifting their example to higher dimension so that it becomes equiregular. Claude worked out the explicit structure and the hard
analytic moment problem. The authors rewrote the analytic core, and Wang formalized the main theorems in Lean with Codex.

#### Why it matters
It shows that the failure of the Sard property is not caused by degenerate growth. It is also one of the most detailed AI
disclosures so far, listing which parts came from Claude, which from the humans, and what the formalization caught.

Sources: [arXiv 2610.24080: An equiregular counterexample to the Sard conjecture for sub-Riemannian endpoint maps](https://arxiv.org/abs/2610.24080); [Zenodo: Lean 4 formalization (10.5281/zenodo.23310061)](https://doi.org/10.5281/zenodo.23310061)

### 2026-10-17: Erdős–Hajnal conjecture proved for every path P_t: Tung H. Nguyen, with ChatGPT-6 Astra proving key lemmas; a DeepMind internal model reached the same result independently, and Claude Opus 5.5 manuscripts claim P6 and all paths

*****; University of Oxford, OpenAI, Google DeepMind, Google Research, Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-10-17-erdos-hajnal-all-paths-nguyen-astra-deepmind/

arXiv 2610.24887 (17 Oct 2026), "Erdős–Hajnal for excluding paths" by Tung H. Nguyen (Oxford), proves that every path P_t has the Erdős–Hajnal property: P_t-free graphs on n vertices have a clique or stable set of size n^c. The previous frontier was P5 (Nguyen–Scott–Seymour, 2023). The paper credits ChatGPT-6 Astra with the full purity version of Lemma B.2 and with a proof of Lemma 3.9 for r = 5, which the author generalized; the main ideas are the author's. The paper also reports that a Google DeepMind internal model independently proved the P6 case and then all paths, with a Lean formalization (to be posted on GitHub, not arXiv), and that two Claude Opus 5.5 manuscripts on GitHub claim P6 and all P_t, without independent expert review.

- Paper: 'Erdős–Hajnal for excluding paths', Tung H. Nguyen (Titchmarsh Research Fellow, Oxford), arXiv 2610.24887 [math.CO], 17 Oct 2026. Theorem 1.2: 'P_t has the Erdős–Hajnal property for all t ≥ 1.' Theorem 1.3 extends it to the subdivided claws S_{1,2,t} and S'_{1,2,t} (covering the bull, E and the bird)
- Conjecture (Erdős and Hajnal, 1977): for every graph H there is b with every H-free graph G having a clique or stable set of size |G|^(1/b). It remains open in general
- New ingredient (abstract): 'a novel Gyárfás path argument based on the ideas of bookkeeping blockades and splitting paths'
- AI declaration, verbatim: 'ChatGPT-6 Astra was mathematically involved in Subsections 3.3 and 4.1 and Section B; the ideas in the rest of this paper are entirely due to the author.' The author asked Astra for a weaker version of Lemma B.2 and 'Astra then succeeded in proving the full purity version'; after discussions directed by the author, 'Astra discovered a proof of Lemma 3.9 for r = 5', which the author extended to all r; Subsection 4.1 was 'cleaned up by Astra'. Writing by the author with copy-editing from ChatGPT-6 Luna; figures by ChatGPT-6 Sol
- Google DeepMind (concurrent work section): a team 'including Junehyuk Jung, Joonkyung Lee, Thang Luong, and Aranyak Mehta' told the author that 'a DeepMind internal model had produced a proof of the Erdős–Hajnal property of P6; the proof was verified by Joonkyung Lee and accompanied by a Lean formalisation'. They then 'obtained further results including the P_t case'. The Google team will post its AI-generated manuscript and Lean proof on GitHub rather than arXiv, 'since they want to ensure that their material on the arXiv meets the standards of the mathematical community'. 'At the time of submission, neither team had seen the other's manuscript'
- Claude manuscripts named in the paper: 'an anonymous GitHub repository contains an AI-generated manuscript claiming to prove the Erdős–Hajnal property of P6, produced by Claude Opus 5.5; and another GitHub repository by Yuwen Zhou contains a manuscript claiming to prove Theorem 1.2 (also available on Zenodo), assisted by Claude Opus 5.5 … We are not aware of independent expert reviews of either manuscript'
- The P6 repository is machine-qed/erdos-hajnal-six-vertex-path (created 27 Sep 2026): a 24-page paper plus Lean 4, 'produced by Claude Opus 5.5 (Anthropic) and has not been refereed'; it proves a polynomial Rödl property with τ < 10^−15, not computed numerically
- Yuwen Zhou (UBC Okanagan), OctopusTide/erdos-hajnal-all-paths (created 8 Oct 2026, Zenodo DOI 10.5281/zenodo.23270286): the Lean main theorem AllPathsLocal.eh_all_paths takes three hypotheses (hRodl, a combinatorial lemma hcomb and EH for P6), so the formal statement is conditional on inputs supplied as assumptions, as the README shows
- Not found as of 20 Oct morning: the DeepMind manuscript or repository (google-deepmind GitHub org checked; newest maths repo is cryptids, 9 Oct), an announcement by DeepMind, or expert reactions on HN

#### What happened
The Erdős–Hajnal conjecture says that forbidding any fixed induced subgraph H forces a polynomial-size clique or stable set,
far more than the logarithmic size guaranteed in general. Paths were a central test case: P5 was settled in 2023 by Nguyen, Scott
and Seymour. Nguyen's new paper does every path. Its AI declaration is unusually exact about which lemmas GPT-6 Astra proved.

The paper's "Concurrent work" section reports a race. A Google DeepMind and Google Research team contacted Nguyen to say an internal
model had proved the P6 case, checked by Joonkyung Lee and in Lean. After Nguyen told them about his P_t result, the model was run
further and also produced the P_t case. The two teams agreed to publish separately and on different platforms. Two GitHub projects
built with Claude Opus 5.5 also claim P6 and all paths.

#### Why it matters
It is a major case of one of the central open problems in graph theory, reached almost at once by a human mathematician working
with one lab's model and, on his account, by another lab's model working largely on its own. DeepMind's choice to keep its
AI-generated manuscript off arXiv reflects the current argument over AI-written papers.

Sources: [arXiv 2610.24887: Erdős–Hajnal for excluding paths (Tung H. Nguyen)](https://arxiv.org/abs/2610.24887); [machine-qed/erdos-hajnal-six-vertex-path: P6 proof produced by Claude Opus 5.5, with Lean (unrefereed)](https://github.com/machine-qed/erdos-hajnal-six-vertex-path); [OctopusTide/erdos-hajnal-all-paths: Yuwen Zhou's all-paths manuscript and Lean code](https://github.com/OctopusTide/erdos-hajnal-all-paths); [Zenodo record of Zhou's manuscript](https://doi.org/10.5281/zenodo.23270286)

### 2026-10-17: A non-sofic word-hyperbolic group: Sandhu and Thom build one from OpenAI's non-residually-finite hyperbolic group, starting from an AI-generated construction; Douglas–Mghirbi give an independent AI-assisted proof with Lean

****; TU Dresden, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-17-non-sofic-hyperbolic-group-sandhu-thom/

arXiv 2610.24836 (17 Oct 2026), "An example of a non-sofic hyperbolic group" by Harjas Sandhu and Andreas Thom, constructs "a torsion-free, two-dimensional, word-hyperbolic group which is not sofic", building on OpenAI's construction of a non-residually finite hyperbolic group (OpenAI release, family 252). The paper "is based on an AI-generated construction obtained by Harjas Sandhu on October 7, 2026"; Thom worked through the proof and "OpenAI's GPT Astra helped draft this document". It cites "an independent AI-assisted proof" by Seth Douglas and Nidhal Mghirbi (GitHub, 7 Oct), who show that OpenAI's group itself is not sofic and formalize this in Lean on top of OpenAI's development. Both are unrefereed and depend on OpenAI's unrefereed manuscript.

- Paper: 'An example of a non-sofic hyperbolic group', Harjas Sandhu and Andreas Thom, arXiv 2610.24836 [math.GR], 17 Oct 2026. Abstract: 'We construct a torsion-free, two-dimensional, word-hyperbolic group which is not sofic. Our construction builds on OpenAI's construction of a non-residually finite hyperbolic group [14].'
- Background (paper): 'The question whether all groups are sofic was open for a long time. The first proof of the existence of a non-sofic group was given by OpenAI' (Astra's 'Nonsofic groups exist', Ten Advances, Aug 2026). Kun and Thom built non-sofic wreath products; Chapman and Peled gave a conditional non-sofic hyperbolic group from random simplicial complexes (arXiv 2509.21566)
- Construction: fundamental group of a triangle complex with a locally CAT(−1) metric, built from marked lines in D(5, q) over F_q^5, with real versus finite-field rank comparison; Ian Agol pointed to Dvir's finite-field Kakeya work for the interpolation step (Lemma 2.1)
- Acknowledgments, verbatim: 'This paper is based on an AI-generated construction obtained by Harjas Sandhu on October 7, 2026, which was in turn based on the OpenAI breakthrough [14]. The second author worked through the proof to make it readable and to reduce unnecessary overhead. OpenAI's GPT Astra helped draft this document. … An independent AI-assisted proof was given by Douglas–Mghirbi [5].' They 'view this document as a commentary on OpenAI 252'
- Douglas–Mghirbi, 'Non-soficity of OpenAI's torsion-free hyperbolic group' (GitHub Apsiape/hyperbolic-group-not-sofic, created 7 Oct 2026, Zenodo DOI 10.5281/zenodo.23202474): OpenAI's rank argument against finite quotients 'survives approximation', so the same group is not sofic. 'The group and every input to the argument are OpenAI's; the approximate argument, its constants and the formalization are ours.' The Lean code builds on OpenAI's development
- Douglas–Mghirbi v1.1.0 adds: for division rings K of the construction's characteristic q, K[G] has no Linnell division ring; G is not bi-orderable
- Context: in September Thom argued that OpenAI's August non-sofic result relied on his 2019 work with Kun (see the Astra ten-advances entry). Here he is co-author of a paper building on OpenAI's later construction
- Dependency: OpenAI family 252, 'A torsion-free hyperbolic group that is not residually finite', is an unrefereed manuscript from the 6 Oct release (Lean-formalized according to OpenAI's catalogue). Whether every hyperbolic group is residually finite is itself a famous open question (Gromov)

#### What happened
A group is sofic if it can be approximated by finite permutation groups. Every amenable and every residually finite group is
sofic, and for decades no non-sofic group was known. OpenAI's Astra claimed the first one in August. Its 6 October release added
a torsion-free hyperbolic group that is not residually finite (family 252). Within a day, two efforts used that group to get a
non-sofic hyperbolic group. Harjas Sandhu obtained an AI-generated construction on 7 October, and Andreas Thom turned it into a
readable proof. Seth Douglas and Nidhal Mghirbi showed directly that OpenAI's group is not sofic and formalized the step in Lean.

#### Why it matters
Hyperbolic groups were the main candidates in the search for a non-sofic group. A concrete hyperbolic example answers a
question that geometric group theorists have asked for a long time. Its trustworthiness depends on OpenAI's family-252
manuscript, which no referee has checked yet. That makes the residual-finiteness claim in OpenAI's release more important to
verify.

Sources: [arXiv 2610.24836: An example of a non-sofic hyperbolic group (Sandhu, Thom)](https://arxiv.org/abs/2610.24836); [Douglas & Mghirbi: Non-soficity of OpenAI's torsion-free hyperbolic group (note + Lean)](https://github.com/Apsiape/hyperbolic-group-not-sofic); [Zenodo record of the Douglas–Mghirbi note](https://doi.org/10.5281/zenodo.23202474); [openai/math (family 252: a torsion-free hyperbolic group that is not residually finite)](https://github.com/openai/math)

### 2026-10-17: arXiv results with disclosed AI help, 16 Oct 2026 announcement (second pass): Suffridge's conjecture, two Wahl conjectures, the 6-D Tarski problem and more

***; OpenAI, Anthropic, Google DeepMind, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-17-ai-assisted-results-october-16-listing/

Continuation of the 2026-10-16 catalogue. arXiv's announcement of Fri 16 Oct 2026 (papers with v1 dated 12–15 Oct; the earlier catalogue calls it the "15 Oct listing") was re-read on 17 Oct from arxiv.org/list/<category>/new, because export.arxiv.org returned HTTP 429 for three runs. Every new submission in math, math-ph, cs.CC/DS/DM/LO/IT, hep-th, gr-qc, quant-ph, cond-mat, physics.chem-ph and q-bio was screened, and the PDFs of keyword hits and of all uncited maths, CS and hep papers were scanned for AI statements. Own entries: the Gauss circle problem below 5/16 (ChatGPT and Codex help), the Lee–Loh–Sudakov directed-cut conjecture (Altifina framework, Lean-verified) and Das Sarma & Chou's exciton-superfluidity theory produced with Claude Opus 5.5 and refereed by Codex. Highlights here: Claude Opus 5.5 found the first proof that settles Suffridge's conjecture and two Brevig–Saksman conjectures; ChatGPT-6 Astra helped prove Wahl's Main Conjecture for ℚ-Gorenstein singularities and LLMs helped Beke, Park, Shin & Stipsicz prove Wahl's rational-homology-disk conjecture; the optimal Θ((log n)³) query bound for 6-D Tarski fixed points. All unrefereed.

- Own entries from this pass: Gauss circle problem exponent 209/670 < 5/16 (Cen, Gao, Zhang & Zhang, 2610.21730; ChatGPT and Codex used 'to discuss possible proof strategies, assist in drafting and checking parts of the mathematical arguments'); Lee–Loh–Sudakov conjecture (Folz & Tyomkyn, 2610.21381; proof 'first discovered by the Altifina research framework' with ChatGPT 6 Astra, 6.1 Sol and Claude Opus 5.5, 'verified in Lean'); Das Sarma & Chou, 2610.22037 (derivations, code, figures and text of the zero-field theory 'produced with the AI system Claude', Codex/GPT-6-Astra 'acting as a referee')
- Claude found the first proof: every extremal of a bounded linear functional on H^p, 0 < p < 1, is outer, resolving 'two conjectures of the first-named author and Saksman and one conjecture of Suffridge' (Ole Fredrik Brevig & Øistein Søvik, 2610.21788): 'An initial proof of Theorem 1 was found by Claude Opus 5.5 when it was tasked with resolving the conjectures from [1]'; the published proof is a simpler human one via de Leeuw–Rudin
- Wahl's Main Conjecture for non-Gorenstein ℚ-Gorenstein surface singularities, plus his ℚ-Gorenstein and Rational Conjectures (Heesang Park & Dongsoo Shin, 2610.21092): 'The authors proposed various approaches … and used ChatGPT-6 Astra to examine their feasibility'; companion paper 2610.21093 shows arbitrarily negative Wahl defects, with Astra 'finding explicit gluing maps for the construction of the counterexamples'; with Beke and Stipsicz they also prove Wahl's conjecture on rational homology disk smoothings (2610.20991: 'The authors used large language models in proving some statements, drafting the paper, and identifying errors')
- Optimal bound: deterministic query complexity of Tarski fixed points in six dimensions is Θ((log n)³) (Kristóf Zólomy, 2610.21785; 'ChatGPT-6 Astra was used to assist in developing the lower-bound argument'), after 'several papers obtained improved lower bounds using AI-assisted proofs' (Chen–Gao–Li–Rubinstein–Yannakakis 2610.07055; Liu–Wang)
- Questions answered with GPT help: sharp point-degree bound for points in ℂ^d, answering questions of Basit–Dvir–Saraf–Wolf and Averkov–Schymura (Kroeker & Lund, 2610.21415; 'GPT-6 Astra to assist with proof development'); a folklore conjecture on exponential sums with multiplicative coefficients, optimal constants 4/√6 and 1 (Haozhe Gou, 2610.22345; Sol 'suggested' the key reduction and Fourier expansion, Astra built a counterexample)
- Conjectures settled with models in the idea phase: Lévy-process abrupt/strongly-eroded dichotomy and counterexamples to Vigon's point-hitting conjecture (González Cázares, Kramer-Bang & Mijatović, 2610.20916; ChatGPT-5.6/6.1 and Gemini Flash 3.6 'used to develop the ideas'); T² vanishing criterion settling the Constantinescu–Klein–Nguyen–Singh–Venturello conjecture (Qihang Wang, 2610.20649; multi-model workflow, results 'await further verification by domain experts')
- Jason Bell (2610.22123) gives an undecidable problem in dynamics and a counterexample to a Bell–Satriano–Sierra conjecture, after giving ChatGPT Plus 'some of my rough mathematical thoughts' and asking it 'to turn them into a paper'; it left him 'with the eerie sense that … we may not like where all of this ultimately leads'
- Agent pipelines: Yukich's 1989 question on 2-D empirical measures answered with 'a reusable AI-agent workflow' and 'formally verified in Lean' (2610.21070); a Peking University paper whose 'entire research pipeline' was run by the GPT-5.6 Sol-powered system Colombo (2610.22208); key n-representability proofs 'derived by OpenAI's Astra-6 pro model' (Rubin, Google Quantum AI, & DePrince, 2610.20737)
- George Grätzer (2610.20802), after ChatGPT supplied one of two short proofs of the A^A theorem: 'ChatGPT is fully competitive with my own mathematical research abilities'
- Fully AI-developed CS results: Eccentricity Shortest Path is polynomial for every fixed treewidth, answering questions of Bhyravarapu et al. (IWOCA 2023) and Chakraborty et al. (Babu, Jacob, Kumar & Sandeep, 2610.20769: 'All results obtained in this paper were developed using OpenAI's GPT-5.6/6 Pro')
- Sweep 18 Oct additions (same announcement): Hock & Shadrin prove the higher-genus closed-string prediction of the Aganagic–Vafa conjecture for torus-knot augmentation curves (2610.21585), stating 'AI agents were used substantially in producing and polishing the proofs'; routine AI statements in 2610.20825 (Claude for discussion and symbolic checks), 2610.21210 (ChatGPT 5.5 for literature), 2610.21462 (ChatGPT 6 drafted the numerical appendix), 2610.21501, 2610.21559, 2610.21599, 2610.21975, 2610.22255

#### What happened
On 17 Oct the sweep's arXiv API step was still blocked (HTTP 429 on export.arxiv.org since the 16 Oct fetch), and arXiv's RSS
feeds were empty because arXiv does not announce on Saturdays. The HTML listing pages (`/list/<category>/new`, with abstracts)
still worked. They gave 615 new submissions from the 16 Oct announcement; 32 were already cited in the dataset. 119 matched
keywords (conjecture, counterexample, open problem, Erdős, Lean, model names); their PDFs and those of the remaining uncited
maths, CS and hep papers were read with pdftotext and searched for AI statements. Each statement below is quoted or closely
paraphrased from the PDF. None of the papers is refereed, and no Hacker News thread was found for any of them.

### Named problems settled or advanced
- **Suffridge's conjecture and two Brevig–Saksman conjectures** (Ole Fredrik Brevig & Øistein Søvik, [2610.21788](https://arxiv.org/abs/2610.21788)).
  If 0 < p < 1, every norm-attaining element of the unit sphere of H^p for a bounded linear functional is an outer function;
  this complements Duren–Romberg–Shields on the failure of Hahn–Banach in H^p. AI: "An initial proof of Theorem 1 was found by
  Claude Opus 5.5 when it was tasked with resolving the conjectures from [1]. That proof was variational and based on ideas going
  back to F. Riesz … Corollary 2 (b) led the authors to the work of de Leeuw and Rudin [2], where they found the main idea in the
  much simpler proof presented below."
- **Wahl's Main Conjecture for ℚ-Gorenstein singularities** (Heesang Park & Dongsoo Shin, [2610.21092](https://arxiv.org/abs/2610.21092)).
  Δ ≥ 0 with equality exactly in the quasi-homogeneous case, also proving Wahl's ℚ-Gorenstein and Rational Conjectures; Δ + 4p_g ≥ 0
  in general. AI: ChatGPT-6 Astra searched and analysed Wahl's and Lima-Pereira et al.'s work; "The authors proposed various
  approaches to generalizing the ideas in these papers and used ChatGPT-6 Astra to examine their feasibility." Companion
  [2610.21093](https://arxiv.org/abs/2610.21093) builds singularities with arbitrarily negative Wahl defect; there Astra also
  helped by "finding explicit gluing maps for the construction of the counterexamples".
- **Wahl's conjecture on rational homology disk smoothings** (Márton Beke, Heesang Park, Dongsoo Shin & András I. Stipsicz,
  [2610.20991](https://arxiv.org/abs/2610.20991)). A normal surface singularity whose minimal good resolution graph has at least two
  nodes has no rational homology disk smoothing; with earlier work this completes the classification of singularities admitting such
  smoothings "and proves Wahl's conjecture", and also his Rational Conjecture. AI: "The authors used large language models in
  proving some statements, drafting the paper, and identifying errors."
- **Six-dimensional Tarski problem** (Kristóf Zólomy, [2610.21785](https://arxiv.org/abs/2610.21785)). Lower bound
  Ω((log n)^(2+⌊log₃(d/2)⌋)), giving the optimal Θ((log n)³) deterministic query complexity for d = 6. "ChatGPT-6 Astra was used
  to assist in developing the lower-bound argument, working out the technical details of the manuscript, preparing the figures, and
  checking the arguments." The abstract says "several papers obtained improved lower bounds using AI-assisted proofs" (the
  references include Chen, Gao, Li, Rubinstein & Yannakakis, arXiv 2610.07055, and Liu & Wang).
- **Points on spanned lines; questions of Basit–Dvir–Saraf–Wolf and Averkov–Schymura** (Matthew E. Kroeker & Ben Lund,
  [2610.21415](https://arxiv.org/abs/2610.21415)). Some point lies on at least n + 1 − 2n/(d+1) spanned lines (sharp in every
  dimension), removing the large-rank assumption from Geelen–Nelson–Walsh; n²/40 ordinary lines in ℂ³ under a plane condition.
  AI: "The authors used GPT-6 Astra to assist with proof development and writing."
- **Exponential sums with multiplicative coefficients** (Haozhe Gou, [2610.22345](https://arxiv.org/abs/2610.22345)). Optimal
  constants 4/√6 (multiplicative) and 1 (completely multiplicative), confirming "a folklore conjecture … motivated by the work of
  Montgomery and Vaughan (1977)". AI: "ChatGPT 5.6 Sol suggested modifying this reduction to impose complete multiplicativity only at
  the primes not dividing q … and also suggested the subsequent Fourier expansion"; "ChatGPT 6 Astra also helped construct the second
  counterexample in Proposition 1.1(ii)." Ramaré's identity and the pretentious large sieve were the author's ideas.
- **Lévy-process dichotomy** (Jorge González Cázares, David Kramer-Bang & Aleksandar Mijatović, [2610.20916](https://arxiv.org/abs/2610.20916)).
  Every infinite-variation Lévy process is abrupt or strongly eroded, resolving a conjecture from Ann. IHP 2024; counterexamples to
  Vigon's point-hitting conjecture. AI: "ChatGPT-5.6 and 6.1 and Gemini Flash 3.6 were used to develop the ideas, particularly from
  Fourier analysis, needed and used in the proof … and in the construction of the examples."
- **Matroid cotangent cohomology** (Qihang Wang, [2610.20649](https://arxiv.org/abs/2610.20649)). T² vanishes exactly when the
  matroid has no U₁,₄-minor, "settling the conjecture of Constantinescu, Klein, Nguyen, Singh, and Venturello", plus obstructions of
  every order. AI: "A generative-AI workflow using OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, Grok 4.6, and OpenAI GPT-6 Astra
  developed the hyperplane-graph and cup-product formulas, higher-order obstruction constructions, and primary-quotient module
  calculations … The results await further verification by domain experts." Acknowledges Gewu Intelligence Lab.
- **Undecidability in dynamics** (Jason Bell, [2610.22123](https://arxiv.org/abs/2610.22123)). Extending Lee and Nam, the set of n
  with φⁿ(Y) meeting Z gives an undecidable problem for a fixed variety and map, and a counterexample to a Bell–Satriano–Sierra
  conjecture. In the acknowledgments: "ChatGPT offered me one free month of what they call a 'Plus' subscription … gave it some of
  my rough mathematical thoughts on this problem and asked it to turn them into a paper. Aside from the fact that I could not
  initially understand much of what it had written, it was largely a success." He then rewrote it.
- **Hegselmann–Krause dynamics** (S. Rasoul Etesami & Behrouz Touri, [2610.20783](https://arxiv.org/abs/2610.20783)). A six-agent
  counterexample to convergence of scalar heterogeneous HK dynamics (after Hegarty, Ognissanti & Wedin's counterexamples in arXiv 2610.03229).
  "Generative AI models ChatGPT and Claude have been used for the refinement of the counterexample and the exposition."
- **Crossing numbers of simplicial complexes** (Alfredo Hubard, Cheikh Saliou Ndiaye & Oriol Solé-Pi, [2610.21017](https://arxiv.org/abs/2610.21017)).
  Higher-dimensional crossing lemma, a generalised Harary–Hill conjecture, and a simple proof of Harary–Hill for antipodal drawings.
  "An LLM, after being given our calculations for B2,S, discovered the upper bound on B2. Prompted with the suggestion to use Gale
  duality, an LLM found the proof Theorem 24."
- **Eccentricity Shortest Path** (Anand Babu, Ashwin Jacob, H. Kumar & R. B. Sandeep, [2610.20769](https://arxiv.org/abs/2610.20769)).
  FPT and XP algorithms for many structural parameters, answering two published questions. "All results obtained in this paper were
  developed using OpenAI's GPT-5.6/6 Pro. These models were also used to generate, refine, and write the proofs and exposition."
- **Lacunary maximal operators** (Xiaojing Zhou, [2610.21197](https://arxiv.org/abs/2610.21197)). Negative answer to the weak-type
  (1,1) question of Seeger and Wright (2011) and Wright's Open Problems 14 and 15 (2016). Minor AI role: "The idea of applying
  Markov's inequality to obtain (2.3) in the proof of Lemma 2.1 arose from discussions with ChatGPT."

### Other substantive AI contributions
- **Yukich's 1989 question** (Weitao Huang, Yichen Wu, Renzhen Wang & Deyu Meng, [2610.21070](https://arxiv.org/abs/2610.21070)).
  Exact leading constant c* = κ₁/(2√π) for the bounded-Lipschitz (W₁) distance of 2-D uniform empirical measures, in expectation and
  almost surely. AI: "we developed a reusable AI-agent workflow for mathematical research, which we plan to release, and applied it to
  Yukich's question. The agents assisted in exploring proof strategies, testing conjectures through counterexample search, and refining
  mathematical arguments … The final proof has been formally verified in Lean."
- **Fermionic n-representability** (Nicholas C. Rubin, Google Quantum AI, & A. Eugene DePrince III, [2610.20737](https://arxiv.org/abs/2610.20737)).
  A complete family of two-body witnesses at each level of a weighted sum-of-squares hierarchy (exponential cost, as the authors say).
  AI: "The key components of the constructive solution are the work of humans, but some key proofs come from LLMs. Specifically, the
  reduction of degree-4 multiplier polynomials to degree-2 multiplier polynomials … along with the proof for the sufficiency of
  wnΣη,≤2 in Section III B were derived by OpenAI's Astra-6 pro model." Gemini 3.1 Pro suggested a counterexample to an earlier
  completeness claim.
- **Duffus's A^A question, two short proofs** (George Grätzer, [2610.20802](https://arxiv.org/abs/2610.20802)). One proof is his, the
  other is ChatGPT's (Section 4), set as a challenge: "My conclusion from this experiment is that ChatGPT is fully competitive with my
  own mathematical research abilities. If I look just a year ahead … I expect the comparison will be overwhelmingly in ChatGPT's favor."
- **An auto-research system writes an optimisation paper** (Wendao Wu, …, Zhouchen Lin, Peking University, [2610.22208](https://arxiv.org/abs/2610.22208)).
  Matching gradient–Hessian query lower bounds. "Nearly the entire research pipeline for this paper was carried out by Colombo, our
  laboratory's internal auto-research system, powered by GPT-5.6 Sol." Similar in convex optimisation: Rudomanov, Fomin, Gasnikov &
  Stonyakin's (L₀,L₁)-smooth oracle complexity ([2610.21461](https://arxiv.org/abs/2610.21461): "The models Codex Astra 6 and Claude Opus
  5.5 were used in obtaining the theoretical results"); Liang & Luo's matching stochastic bound ([2610.21692](https://arxiv.org/abs/2610.21692):
  "The construction in the lower-bound was discovered with the assistance of ChatGPT 5.5 Pro").
- **Lean-checked physics lemma** (Sarang Gopalakrishnan & Romain Vasseur, [2610.20724](https://arxiv.org/abs/2610.20724)). A bound on
  long-lived local coherences in quantum-group-symmetric spin chains; the overlap-inequality proof "was generated with the assistance of
  large language models with minimal human input, and has been formalized and machine-checked in Lean 4 (Mathlib)".
- **Zeta second moment along Gram points** (Dundulis, Garunkštis, Laurinčikas & Šimėnas, [2610.22107](https://arxiv.org/abs/2610.22107)).
  Proofs "developed with assistance from ChatGPT in exchanges on 21 August 2026"; the authors note their 7/8 and 11/12 thresholds also
  appear in OpenAI's "Quasi-Riemann Hypothesis" preprints released on 6 Oct.
- **Heegaard Floer pillowcase** (Eaman Eftekhary & Marzieh Souri, [2610.21153](https://arxiv.org/abs/2610.21153)): "The bi-grading theorem
  (Theorem 2.6) was formulated and proved through correspondence with GPT-6", and "GPT-6 provided a counterexample to this conjecture"
  (the authors' own guess about all slope pairs).
- **Gewu Intelligence Lab again** (Qihang Wang & Weiye Li, [2610.22082](https://arxiv.org/abs/2610.22082)): exact exponent 2^⌊k/2⌋ for the
  integral saturation of Bouc's hook cycles; "OpenAI GPT-6.1 Sol developed the saturation, integral-action, finite-state and fixed-space
  arguments … and drafted the expanded manuscript".

- Preprocessed 3SUM from reals to integers (Nick Fischer & Jonas Schmidt, [2610.21961](https://arxiv.org/abs/2610.21961)): ChatGPT 5.6
  Sol "suggested the general approach based on linear algebra", the rest was developed without AI.
- Optimal gap-majority lemma for bounded-round communication (Pachara Sawettamalya, [2610.22313](https://arxiv.org/abs/2610.22313)):
  GPT 5.6 Sol proposed an entropy bound and "identified the need for the 'smoothing' assumption, addressing a gap that the author had
  missed".
- Quantum no-signalling-assisted zero-error capacity (Xin Li & Yinan Li, [2610.21071](https://arxiv.org/abs/2610.21071)): ChatGPT used
  "in developing the mathematical arguments and proofs in IV-E and V-B–V-G".
- Range weighted LIS queries (Keisuke Korogi, [2610.20983](https://arxiv.org/abs/2610.20983)): ChatGPT "assisted with literature
  searches, algorithm design, proof development and checking, counterexample searches, implementation".
- Lotka–Volterra fixed-point abundances (Ph. Jacquod, [2610.22183](https://arxiv.org/abs/2610.22183)): free GPT-5.6 Luna, Claude
  Sonnet 5.5 and Gemini 3.6 Flash, chats public; Gemini linked the expansion to Catalan and Narayana numbers, and "Claude spontaneously
  suggested to instead use the cavity method".
- Gravitational-wave tails to all orders (Debanjan Karan & Babli Khatun, [2610.20770](https://arxiv.org/abs/2610.20770)): ChatGPT-6
  Astra and Claude Opus 5.5 for algebra; "Claude Opus 5.5 identified the agreement of eq.(3.12) with ref.[59]".
- Star-ordered rotor loops in Clifford algebras (Nik Jenzer, [2610.21593](https://arxiv.org/abs/2610.21593)): "the content authorship
  lies with the language model" (Claude Sonnet/Opus), the author holding editorial control.
- Quantum McKay correspondence in type D (Andrea Brini & Jingxiang Ma, [2610.22359](https://arxiv.org/abs/2610.22359)): Claude Code
  (Fable 5.1) review of their Wolfram code "identified for us the source of the discrepancy" in the literature; detailed disclosure
  following "the AIxMaths upheaval of the summer of 2026".
- Two-variable logic on 2-D structures, settling three finite-satisfiability cases open for two decades (Bednarczyk, Kojelis & Szymański, [2610.20937](https://arxiv.org/abs/2610.20937)): results
  human ("We developed the main results and first versions of most proofs without AI assistance"), but Claude Opus 5.5 "found an error in an earlier version of Lemma 3.4".
- Interchange-process mixing (Gil Alon, [2610.22151](https://arxiv.org/abs/2610.22151)): proofs "partially assisted by ChatGPT Pro and
  Astra models, and so were the simulations behind Conjecture 1.5".
- cscK metrics near the canonical class (Frederick Tsz-Ho Fong & Junbang Liu, [2610.21934](https://arxiv.org/abs/2610.21934)): ChatGPT 6
  asked "to produce counterexamples or proofs in special cases"; key estimates human.
- Also: ballistic deposition gaps with a Claude-assisted Lean formalisation of the combinatorial lemmas ([2610.21094](https://arxiv.org/abs/2610.21094));
  triple Schubert coefficients with GPT 5.6 Sol through Codex ([2610.21053](https://arxiv.org/abs/2610.21053)); friends-and-strangers
  graphs, ChatGPT found Example 7.4 ([2610.20648](https://arxiv.org/abs/2610.20648)); BCOV quantisation with ChatGPT 6 Astra
  ([2610.21927](https://arxiv.org/abs/2610.21927)); Bergman spaces with Astra ([2610.21749](https://arxiv.org/abs/2610.21749)); exchange
  networks with a Claude coding agent checked by DeepSeek-V4-Flash and Codex ([2610.22093](https://arxiv.org/abs/2610.22093)).

### Explicitly no AI
- A stronger spectral bound for the clique number, where the Elphick–Wocjan conjecture holds in an example ([2610.21388](https://arxiv.org/abs/2610.21388)):
  "The authors declare that no AI tools were used to conduct this research." Incremental exact vertex connectivity
  ([2610.21531](https://arxiv.org/abs/2610.21531)): results "obtained without the use of artificial intelligence" (Gemini 3.1 Pro for TikZ only).

### Earlier submissions found by the export API (v1 dated 10–14 Oct, scanned 17 Oct)
The 17 Oct sweep's arXiv sections (export API working again) listed papers with v1 dates 10–15 Oct that no catalogue had cited. The
quick run scanned 67 uncited maths and physics PDFs; 15 mention AI. Own entry: Erdős Problem #1117 answered negatively by Yunpeng Li
with ChatGPT and Codex (GPT-6) "used to develop the proof" ([2610.15067](https://arxiv.org/abs/2610.15067); see
2026-10-10-erdos-problem-1117-maximum-modulus-gpt-6-codex). Others:
- Claude credited with proof arguments: Graham & Rotnitzky, local inference for functionals of ill-posed inverse problems
  ([2610.19350](https://arxiv.org/abs/2610.19350), math.ST): "Claude Opus 4.8 developed the main argument underlying the proof of
  Lemma S.3, and Claude Opus 5.5 developed the main argument underlying the proof of strong convergence … in Lemma S.8; those
  arguments were subsequently rewritten by the authors and checked thoroughly for correctness."
- LLM-drafted formal proofs: Troscheit, distinguishing sets and measures from their planar Brownian images, motivated by a question of
  Hochman ([2610.16339](https://arxiv.org/abs/2610.16339)): "LLM models were used heavily to draft much of the Lean code", and some
  Section 9 calculations "were first done by LLMs and later verified using symbolic calculation software"; ideas human. Molnár & Tóth,
  epsilon-hyperdoctrines with a Rocq formalization ([2610.15080](https://arxiv.org/abs/2610.15080)): "The proofs were developed with AI
  assistance under the authors’ mathematical supervision."
- Routine help: Han & Ren, restricted sumsets covering finite abelian groups, determining thresholds posed by Girard, Griffiths and
  Hamidoune ([2610.14496](https://arxiv.org/abs/2610.14496)): "ChatGPT was used to assist with routine calculations and checks, language
  editing, and reorganization"; writing-only statements in [2610.16885](https://arxiv.org/abs/2610.16885) (ChatGPT 5.5),
  [2610.17055](https://arxiv.org/abs/2610.17055), [2610.17078](https://arxiv.org/abs/2610.17078), [2610.15686](https://arxiv.org/abs/2610.15686)
  and [2610.17749](https://arxiv.org/abs/2610.17749) (Doubao and DeepSeek V4 Pro).
- Explicitly no AI for the mathematics: tensor flattenings ([2610.17875](https://arxiv.org/abs/2610.17875): "written without the use of
  large language models"); stereographic MCMC mixing ([2610.18513](https://arxiv.org/abs/2610.18513): proofs "without the assistance of
  generative AI", ChatGPT 5.6 Pro only for reference search and polishing).
- No AI statement found in the other 52 scanned papers, among them claimed resolutions of the non-extinction problem for Brownian
  Fleming–Viot systems (2610.18802), Liu's conjecture on generalized Fibonacci gates (2610.19230) and a Guo–Krattenthaler positivity
  conjecture (2610.19249).

#### Why it matters
One day's listing again holds a dozen named conjectures or published questions where a model contributed the first proof, a key
reduction, a counterexample or the whole result. Two patterns are new here: small companies running multi-model "research
frameworks" (Altifina, Gewu Intelligence Lab) as co-authors, and senior physicists handing whole derivations to a model with a second
model as referee. Several items (6-D Tarski, Hegselmann–Krause) build on AI-assisted papers from earlier weeks.

Sources: [arXiv 2610.21788: Extremals for linear functionals on H^p are outer when 0<p<1 (Brevig & Søvik)](https://arxiv.org/abs/2610.21788); [arXiv 2610.21092: Positivity of Wahl defects and representations from index-one covers (Park & Shin)](https://arxiv.org/abs/2610.21092); [arXiv 2610.21785: The Six-Dimensional Tarski Problem is Ω(log³ n)-Hard (Zólomy)](https://arxiv.org/abs/2610.21785); [arXiv 2610.21415: Ordinary lines, point degrees, and matroid density (Kroeker & Lund)](https://arxiv.org/abs/2610.21415); [arXiv 2610.22345: On exponential sums with multiplicative coefficients (Gou)](https://arxiv.org/abs/2610.22345); [arXiv 2610.20916: Infinite-variation Lévy processes are either strongly eroded or abrupt](https://arxiv.org/abs/2610.20916); [arXiv 2610.22123: An undecidable problem in dynamics (Jason Bell)](https://arxiv.org/abs/2610.22123); [arXiv 2610.20769: Eccentricity Shortest Path: the structural parameter landscape (GPT-5.6/6 Pro)](https://arxiv.org/abs/2610.20769); [arXiv 2610.20649: Cotangent cohomology and higher-order obstructions of matroids (Qihang Wang)](https://arxiv.org/abs/2610.20649); [arXiv 2610.21017: Crossing numbers of simplicial complexes](https://arxiv.org/abs/2610.21017); [arXiv 2610.21070: Exact endpoint asymptotics … uniform empirical measures in two dimensions (Yukich's question)](https://arxiv.org/abs/2610.21070); [arXiv 2610.20737: A constructive solution to the n-representability problem](https://arxiv.org/abs/2610.20737); [arXiv 2610.20802: Two short proofs of the A^A theorem (Grätzer)](https://arxiv.org/abs/2610.20802); [arXiv 2610.22208: Gradient–Hessian query tradeoffs (Colombo auto-research system)](https://arxiv.org/abs/2610.22208); [arXiv 2610.19350: On Local Inference for Functionals of Ill-Posed Inverse Problems (Claude Opus 4.8/5.5 lemma arguments)](https://arxiv.org/abs/2610.19350); [arXiv listing: math, new submissions, Fri 16 Oct 2026](https://arxiv.org/list/math/new)

### 2026-10-17: AI-assisted results, 16–17 Oct 2026 (Zenodo and arXiv leftovers): Ramanujan's prime-counting inequality settled unconditionally with Claude, a Claude Code paper mill on the Hodge conjecture, and LLM agents in materials science

***; Anthropic, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-18-ai-assisted-results-october-17-listing/

Catalogue for 16–17 Oct 2026, following the 16 Oct arXiv catalogue. The best item: Jaideep Sai Padhi (Purdue) proves unconditionally that Ramanujan's inequality π(x)² < (ex/log x)·π(x/e) holds for every x ≥ 38,358,837,683. Until now that threshold was known only under the Riemann hypothesis (Dudek–Platt); the gap e^103 < x < e^3158.442 is now closed, with certified ball arithmetic. "The computations and the drafting of this paper were carried out with substantial assistance from an AI system (Claude, Anthropic)." Also: Deep Bhattacharjee's dozens of Claude Code-made Zenodo records (Hodge conjecture for new Fermat degrees, a Liu Fano-index counterexample, Kourovka 20.71(b)), and four arXiv papers on LLM agents in materials and clinical science. Unrefereed.

- Ramanujan's inequality: Padhi, 'Ramanujan's prime counting inequality holds for all x ≥ 38 358 837 683', Zenodo v2 17 Oct 2026 (v1 about 16 Oct; mirrored on aixiv.online 2610.01760), 25 pp.; Theorem 1.1: the inequality holds for e^58 ≤ x ≤ e^3159, hence N_R = 38 358 837 683 unconditionally
- Ramanujan's inequality, method: only RH verification to height 3·10^12 (Platt–Trudgian) is used, plus a sharper zero-free region evaluated at each height (constant 4.676 instead of 4.896), a repulsion bound for pairs of zeros near σ = 1, and an explicit Barban–Vehov mean square bound of about 1.5 (Bellotti's table had 85–105); all numerics certified in Arb/python-flint
- Ramanujan's inequality, AI statement: 'substantial assistance from an AI system (Claude, Anthropic). The arguments and the code were also reviewed by separate runs of the same system; these reviews are not a substitute for independent human verification.' It also argues that two earlier preprints by De claiming the range 'do not prove (1.1)'
- Same author, earlier: Auluck–Chowla–Gupta unimodality with Claude (arXiv 2610.14084, in the 13 Oct catalogue)
- Claude Code paper mill: Deep Bhattacharjee (with Ushashi Bhattacharya) posted more than a dozen Zenodo records on 15–17 Oct, each with a paper, a Lean 4 file and code in several languages; his GitHub account creelie has 82 repos, many created 10–17 Oct with names such as 'fermats-last-theorem-Machine' and 'Millenium-prize-problem-quant-Closure-'
- Bhattacharjee's claims include the Hodge conjecture for Fermat varieties of degrees 33, 39, 44, 51, 52 and 114 in every dimension (83 pp.; degrees 44, 51, 52 'had not been proved before'); a counterexample to Liu's Fano-index conjecture in dimension four (index 1389); a 63-vertex graph answering Kourovka Notebook Problem 20.71(b) for k = 3; book Ramsey numbers R(B_57, B_58) = 231 and R(B_66, B_67) = 267
- Bhattacharjee's AI statements: 'calculations, in C, Python, Julia, Macaulay2, Lean 4 and shell scripts, with the assistance of Claude Code'; the provenance file of 'Dense Algebraic Loci of Hodge Classes' says the constructions, text, figures, Lean file and programs were done 'with the help of AI assistance (Claude, Anthropic), in sessions in the Claude Code environment' and that 'The AI did not prove … the Hodge conjecture'
- LLM agents in computational materials (arXiv 2610.20779, USC et al.): out-of-the-box agents including 'Opus 4.7 and GPT-5.5' average 54%, 36% and 19% on GW, GW-BSE and GWPT workflow benchmarks; the authors' AI-Donkey framework makes all tested agents produce complete, runnable workflows
- LLM literature mining (arXiv 2610.21050, Adelaide): extraction F1 0.96 against human annotation; corpus of 547 guest-induced transitions in 240 soft porous crystals from 4,252 screened papers; gate-opening frameworks transform at lower pressure than breathing ones (p = 3.8×10⁻³)
- Also: MASON, an MCP toolkit that lets agents build atomistic structures (Nankai, 2610.21015); a dual-LLM consensus pipeline extracting 18 trofinetide-response phenotypes for Rett syndrome from 26,763 clinical notes (Baylor/Texas Children's, 2610.22062)
- Scanned with no AI statement found (17–18 Oct): 25 uncited maths and physics papers from the 13–16 Oct arXiv announcements claiming to settle conjectures, among them Chen's log-concavity conjecture for longest increasing subsequences (2610.18235), lens-space counterexamples to the quotient-round Yamabe formula (2610.18662), the weak cylinder conjecture for polynomial graphs (2610.16803) and the zero-divisor conjecture for medial quandle rings (2610.20306)

#### What happened
arXiv makes no announcements on Friday or Saturday nights, so the 18 Oct sweep had no new listing. Its two arXiv sections held
leftovers from the 13–16 Oct announcements. Most of the disclosure hits were AI-methods papers rather than AI-assisted results.
The 25 uncited maths and physics "conjecture" papers were re-scanned (`pdftotext`, full text) and none had an AI statement. The
new AI-assisted results of 16–17 Oct that we found came from Zenodo.

### Ramanujan's inequality, unconditionally (Padhi, with Claude)
Ramanujan noticed in his notebooks that π(x)² < (ex/log x)·π(x/e) holds for all large x. Assuming the Riemann hypothesis, Dudek
and Platt showed the last integer counterexample is 38,358,837,682. Without RH it was known only for x ≤ e^103 (Axler, Platt–Trudgian,
Johnston) and for x ≥ e^3158.442 (Axler). Padhi closes the range between them, so the RH-conditional threshold is now unconditional.
The proof uses an explicit formula with Logan's kernel (following Büthe). It adds three new explicit inputs about possible zeta
zeros just above the RH-verified height 3·10^12, and every number is certified in ball arithmetic. The acknowledgement says the
computations and drafting were done "with substantial assistance from an AI system (Claude, Anthropic)", reviewed by other Claude
runs, and that "these reviews are not a substitute for independent human verification". No expert reaction was found by 18 Oct.

### A Claude Code paper mill (Bhattacharjee)
Deep Bhattacharjee (affiliation "Formerly, Electro-Gravitational Space Propulsion Laboratory", Bhubaneswar) and Ushashi
Bhattacharya posted more than a dozen Zenodo records on 15–17 Oct alone. Each holds a paper, Lean 4 files and code in four or five
languages, all made in Claude Code sessions by their own statements. The claims range from small (abelian squares in binary
circular words) to large (the Hodge conjecture for Fermat varieties of several new degrees, with a proof the README says "No step
… uses a computer"). None has been refereed or discussed by specialists that we could find. These records are listed here as a
sign of how much AI-made mathematics now reaches Zenodo, not as results we vouch for.

### LLM agents in science (arXiv)
- **Many-body perturbation theory** (2610.20779): frontier agents set up GW, GW-BSE and GWPT workflows correctly only 54%, 36% and
  19% of the time on average. Errors tend to "surface only several stages after they are introduced". An expert-curated framework
  (AI-Donkey) fixes this for every model tested.
- **Anomalous adsorption** (2610.21050): an LLM extraction pipeline builds the first database of guest-induced transitions in
  soft porous crystals. Its analysis suggests describing them by function rather than by structure.
- **MASON** (2610.21015) and a **dual-LLM clinical phenotyping pipeline** (2610.22062) are tools, listed for completeness.

### Evening additions: Axiom Math's Lean repos (18 Oct)
A check of the AxiomMath GitHub organisation found Lean formalizations that no entry cited. Two got their own entries: the full
formal proof of OpenAI's 2D area law, and Lech Mazur's AI-developed Collatz positive-density theorem. The rest are formalizations
of human or AI-assisted papers, each with a comparator-checked challenge file:
- **TernaryPoly** (12 Oct): the Ilin–Ono paper on ternary period polynomials (the Bachmann–van Ittersum result, already in the
  mid-October catalogue), "developed with AxiomProver".
- **HigherDyson2** (15 Oct): the main results of Alfes, Ono and Swaminathan on elliptic corrections for higher Dyson ranks.
- **BondalThomsen** (2 Oct): Reginald Anderson's theorems on exceptional Bondal–Thomsen collections. The rarity theorem "disproves
  the nonvanishing part of Conjecture 6.1" of Ramirez–Zhang–Son–Anderson (arXiv 2410.23290). One published result is taken as a hypothesis.
- **BGP212** (2 Oct): bounded prime gaps H₁ ≤ 212, conditional on a numerical certificate and cited analytic estimates. This is weaker
  than the 186 claimed for GPT-6 Astra in August.

The sweep's arXiv sections (13–16 Oct leftovers) were re-checked in the evening. Every AI-disclosure hit with a maths result was
already catalogued. Nine more uncited "open problem" papers were scanned in full text and none had an AI statement: Sampaio
(Arnold's 1975 corank problem, mod 2), Bravo–Tian–Zhao (Gasull's first problem, cubic rigid systems), Gao–Zhou (finite basis of flat
semirings), Huang–Zhu (Parcet–Randrianantoanina subsequence question), Abin–Gohari (Polyanskiy–Wu subadditivity), Jaffe (coprime
Frobenius problem), Tong–Li (NLS tori), Weitsman (disjoint minimal graphs) and Braun–Machado (real Jacobian counterexamples).

#### Why it matters
Padhi's paper is a second classical explicit-number-theory threshold settled this month by one author working with Claude, after
Auluck–Chowla–Gupta. It is the kind of long, computation-heavy explicit estimate where models are now doing most of the labour.
Bhattacharjee's output shows the other side. One person with Claude Code can produce dozens of Lean-backed papers a week, faster
than anyone can referee them, and this is what Alpöge warned about in the Yang–Mills repo ("10,000-page slop").

Sources: [Zenodo: Ramanujan's prime-counting inequality holds for all x ≥ 38,358,837,683 (Padhi, v2)](https://zenodo.org/records/23440770); [GitHub: jaideepsaipadhi/ramanujans-inequality (certification code)](https://github.com/jaideepsaipadhi/ramanujans-inequality); [Zenodo: On the Hodge conjecture for Fermat varieties (Bhattacharjee)](https://zenodo.org/records/23401546); [Zenodo: Dense Algebraic Loci of Hodge Classes, source and code (with AI provenance file)](https://zenodo.org/records/23439578); [Zenodo: A counterexample to Liu's conjecture on Fano indices in dimension four](https://zenodo.org/records/23387505); [Zenodo: A graph with three card types and five vertex orbits (Kourovka 20.71(b))](https://zenodo.org/records/23384197); [arXiv 2610.20779: Enabling LLM agents for high-fidelity many-body perturbation theory calculations](https://arxiv.org/abs/2610.20779); [arXiv 2610.21050: LLM-assisted systematic study of anomalous adsorption phenomena](https://arxiv.org/abs/2610.21050); [arXiv 2610.21015: MASON, atomistic structure builder operated in natural language](https://arxiv.org/abs/2610.21015); [AxiomMath/HigherDyson2 (Lean: Alfes–Ono–Swaminathan higher Dyson ranks)](https://github.com/AxiomMath/HigherDyson2); [AxiomMath/BondalThomsen (Lean: Anderson, exceptional Bondal–Thomsen collections)](https://github.com/AxiomMath/BondalThomsen); [AxiomMath/BGP212 (Lean: conditional H₁ ≤ 212)](https://github.com/AxiomMath/BGP212); [AxiomMath/TernaryPoly (Lean: Ilin–Ono ternary period polynomials)](https://github.com/AxiomMath/TernaryPoly); [arXiv 2610.22062: Trofinetide phenotypes in Rett syndrome with a dual-LLM consensus pipeline](https://arxiv.org/abs/2610.22062)

### 2026-10-17: China Meteorological Administration puts the Fengqing AI global ensemble forecast system (V1.0) into operation

**; China Meteorological Administration; Science & math; confidence medium; https://postcutoff.com/e/2026-10-17-cma-fengqing-ai-ensemble-forecast-operational/

Pandaily reported on Oct 17, 2026 that the China Meteorological Administration has put Fengqing Ensemble V1.0, an AI global ensemble weather forecast system, into operation after it passed operational review at the National Meteorological Centre, with an 11-day effective forecast range for global 500 hPa height in independent tests. It extends CMA's deterministic Fengqing 1.0 model (operational since Sept 2024) to probabilistic forecasts, following ECMWF's operational AI ensemble.

- Pandaily (Oct 17, 2026; only the article summary was readable for us): 'Fengqing Ensemble V1.0 passed operational review at the National Meteorological Centre, with an 11-day effective forecast range for global 500 hPa height in independent tests'
- Background (People's Daily via Tsinghua University news, 2024): deterministic Fengqing 1.0 went operational on Sept 10, 2024, producing nine global products for 0–15 days, every 6 hours, at 25 km resolution
- Not verified: ensemble size, resolution, developers and comparisons with ECMWF's ensemble; we have not found the CMA's own announcement

#### What happened
China's national weather service added an AI ensemble system to its operational forecasts. Our only source for the new system is a
Pandaily summary (the article body loads by script and could not be read), so confidence is medium.

#### Why it matters
Ensembles give forecasters probabilities, not just one answer. After ECMWF, China's CMA is now running AI ensembles operationally,
which makes AI weather models a standard part of national forecasting.

Sources: [Pandaily: China Meteorological Administration puts Fengqing AI ensemble forecast system into operation (Oct 17, 2026)](https://pandaily.com/cma-fengqing-ai-global-ensemble-weather-forecast-system-v1-0-operational-11-day-skill); [Tsinghua University news (People's Daily): Fengqing 1.0 and Fenglei 1.0 enter operation (2024)](https://www.tsinghua.edu.cn/info/1182/113639.htm)

### 2026-10-18: Bake AI's Épi agent reports 120 record results with GPT-6 Astra, including a narrower n-queens constant and Cohn–Elkies non-sharpness in dimensions 10 and 11

***; Bake AI, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-18-bake-ai-epi-autonomous-research-agent-120-records/

Bake AI's technical report (arXiv 2610.24944, 18 Oct 2026; authors also from Notre Dame, Scale AI, NVIDIA, Ohio State and others) describes Épi, a persistent research agent run "over a few days using OpenAI frontier models (GPT-5.6 Sol and GPT-6-Astra)". The authors claim 120 record results checked by a fixed verifier: an ~84-fold narrower interval for the n-queens constant, exact dual certificates showing the Cohn–Elkies sphere-packing LP bound is not sharp in dimensions 10 and 11, and a computer-assisted proof of a special case of Tuza's conjecture. Unrefereed.

- Agent: persistent research history, hierarchical memory, branching and independent review; experiments run across model sessions and 'a fixed verifier evaluates their outputs'. Models: 'We ran Épi over a few days using OpenAI frontier models (GPT-5.6 Sol and GPT-6-Astra)'
- Claimed: 120 record-setting results, 20 numerical bound records across 12 families and 100 construction records across 8 families (packings etc.), compared with literature and registries as of 16 Sept 2026; the results repo says '106 improved constructions, 15 mathematical results, and their certificates'
- n-queens constant α (Q(n) = (n e^{−α+o(1)})^n): interval narrowed from Nobel et al. (2023) [1.944000752, 1.944001082] to 1.9440010223327 ≤ α ≤ 1.9440010262694 (width 3.30×10⁻⁷ → 3.94×10⁻⁹, ≈84×), each endpoint with a retained witness
- Cohn–Elkies: new dual lower bounds on the LP optimum in dimensions 9, 10, 11 and 13; in dimensions 10 and 11 they exceed the packing upper bounds of Cohn et al. (2022) by ~0.35% and ~1.92%, so 'Even exact optimization of this formulation cannot recover the true packing density' there
- Quartic Feigenbaum attractor dimension: a rigorous interval more than 3.6-million-fold narrower (claimed)
- Tuza's conjecture (1981, τ△ ≤ 2ν△): computer-assisted proof for split graphs with an eight-vertex clique part and at most three triangle-active neighbourhood types, arbitrary multiplicities (a special case, not the conjecture)
- Épi-turbo: on 30 FrontierOR tasks, a 5.4% higher total score than Codex (with GPT-6 Astra) at 64% of its model cost, same model and reasoning effort
- Status: technical report; certificates public on GitHub (BakeLab/Epi-results, created 7 Sept); no independent check found on 24 Oct

#### What happened
Bake AI and co-authors from several universities and labs released a technical report on Épi, an agent built to keep
long investigations going across many model sessions, with memory, branching and independent review, and a fixed
verifier that scores every output. Running on GPT-5.6 Sol and GPT-6 Astra for a few days, it is credited with 120
records against the literature as of mid-September 2026.

The most mathematical claims are an ~84× narrower interval for the n-queens constant, dual certificates showing that
the Cohn–Elkies linear-programming bound for sphere packing cannot be sharp in dimensions 10 and 11, a much
narrower interval for the quartic Feigenbaum attractor dimension, and a computer-assisted proof of Tuza's conjecture for
a restricted class of split graphs.

#### Why it matters
It is one more of the AlphaEvolve-style record-hunting systems, built on OpenAI models by a third party. The n-queens
interval and the Cohn–Elkies separations come with witnesses anyone can check, so they are easy to confirm. The
non-sharpness claims depend on comparing with Cohn et al.'s 2022 upper bounds, and nobody had checked them independently
by 24 Oct. The blog post was not read in detail; facts here come from the arXiv PDF and the GitHub repo description.

Sources: [arXiv 2610.24944: Épi: Towards a Persistent, Self-Improving Agent for Autonomous Research](https://arxiv.org/abs/2610.24944); [Bake AI blog: Épi](https://bakeai.inc/research/articles/epi/); [GitHub BakeLab/Epi-results: constructions, results and certificates](https://github.com/BakeLab/Epi-results)

### 2026-10-18: Guest post on Terence Tao's blog by Francesco Maggi: will AI bring 'a golden age or a dark age' for mathematics? Calls for publicly governed research LLMs

***; UT Austin; Science & math; confidence high; https://postcutoff.com/e/2026-10-18-maggi-golden-or-dark-age-mathematics-tao-blog/

On 18 Oct 2026 Terence Tao published on his blog a guest essay by mathematician Francesco Maggi (author page at UT Austin's math department), "Will it be a golden age or a dark age for mathematics (and science)?". In eight points it argues that LLM proofs need independent scrutiny, that mathematics remains a human culture, that training mathematicians is now "an existential challenge", and that concentrated private control of frontier models is a danger. It calls on universities, societies, funders and foundations to build and govern competitive, publicly governed LLMs.

- Format: guest post on Tao's 'What's new' blog (tags math.HO, opinion, AI); Tao's note: 'This is a guest post by Francesco Maggi.' and 'This blog post was initially written in a different file format and converted using AI. — T.'
- Results discussed: LLM-generated proofs of a finite-time blow-up for Navier–Stokes with forcing, the proposed proof of the quasi-Riemann Hypothesis, and OpenAI's 'Tuesday October 6, 2026, Drop' (about 4,000 problems tried, success on 'some hundreds' after aggregation and selection; the author calls the wording vague)
- Golden age if: researcher access to frontier tools, 'reproducibility and disclosure standards for LLM-generated scholarship' and 'sustained investments in publicly governed LLM infrastructure'. Dark age if output surges while independent human research withers and knowledge concentrates in a few hands: 'The real darkness here would be knowledge becoming alien to human understanding.'
- Training: 'The training of future mathematicians has already become an existential challenge.' 'A strong mathematical result, by itself, can no longer be taken as convincing evidence' of independent ability
- Governance: warns of 'the growing asymmetry between the scientific capabilities' of labs and the community, and says voluntary corporate goodwill is not a sufficient safeguard; universities and other institutions 'should coordinate the development and governance of competitive LLMs'
- Also argues: 'Mathematics is not a finite system of questions'; AI creativity and consciousness claims should be treated as open scientific questions; authorship, credit, peer review and confidentiality need new norms. 9 comments by the evening of Oct 18

#### What happened
Twelve days after OpenAI's mass release of AI-written maths manuscripts and nine days after Tao's "Math 2.0" lecture, Tao gave his
blog, one of the most read in mathematics, to a guest essay by Francesco Maggi. The essay is organised as eight numbered points and
a call to action. It accepts that LLMs are powerful research tools but argues that their proofs need independent checking, that the
field's progress is not a list of problems to tick off, and that cheap proofs could make "mathematization" of new sciences, a human
task, far more common.

Its sharpest points concern institutions. Because a strong result no longer proves a student's own ability, training
mathematicians has become "an existential challenge". Because a few companies control the frontier models, Maggi calls for
universities, professional societies, funding agencies and foundations to build publicly governed, researcher-oriented LLM
infrastructure, with disclosure and reproducibility standards for LLM-generated work. He expects a mix of the two outcomes and says
the "golden age scenario is achievable".

#### Why it matters
It is the most explicit call yet, on the field's most visible platform, for academia to build its own frontier models rather than
rely on the labs. It echoes Tao's lecture (open, interpretable maths models) and the Kyoto vision of a "golden age of science",
but stresses the risk side. Maggi's affiliation is taken from the author link (UT Austin), not from the text.

Sources: [Terence Tao's blog (guest post by Francesco Maggi): Will it be a golden age or a dark age for mathematics (and science)?](https://terrytao.wordpress.com/2026/10/18/will-it-be-a-golden-age-or-a-dark-age-for-mathematics-and-science/)

### 2026-10-19: Erdős–Szemerédi sunflower conjecture proved for every k ≥ 3; the key vector bound (Theorem 2) was found by ChatGPT 6 Astra Pro (Bucić, He, Lee)

*****; University of Vienna, KAIST, IBS, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-19-erdos-szemeredi-sunflower-conjecture-gpt-6-astra-pro/

On 19 Oct 2026 Matija Bucić, Fankang He (University of Vienna) and Hyunwoo Lee (KAIST/IBS) posted arXiv 2610.26686, "Proof of the Erdős-Szemerédi sunflower conjecture". Theorem 1: for every k ≥ 3 and n ≥ 1, a family of subsets of [n] with no k-sunflower has at most 2^((1 − 1/(16k³))·n) members. Until now only k = 3 was known, via Croot–Lev–Pach, Ellenberg–Gijswijt and the Alon–Shpilka–Umans reduction. The proof goes through a new bound on vector sunflowers (Theorem 2). Its "Statement on the use of AI" says: "The proof of Theorem 2 was found by ChatGPT 6 Astra Pro". The authors say they checked everything, that the proof "has been verified in Lean 4.35" and that the formalization "will be made available on the Palomar repository". It is an unrefereed preprint, and the Lean files were not public on 20 Oct.

- Paper: 'Proof of the Erdős-Szemerédi sunflower conjecture', Matija Bucić, Fankang He (Faculty of Mathematics, University of Vienna), Hyunwoo Lee (KAIST and IBS ECOPRO), arXiv 2610.26686 [math.CO], submitted 19 Oct 2026
- Conjecture (abstract): 'In the 1970s, Erdős and Szemerédi conjectured that for any n ≥ k − 1 ≥ 2, any collection of subsets of [n] not containing a k-sunflower has size at most c_k^n for some c_k < 2.' The paper traces the question back to 1970
- Theorem 1: any subset of 2^[n] with no sunflower of k sets has size at most 2^((1 − 1/(16k³))n). Lower bound (Deuber, Erdős, Gunderson, Kostochka, Meyer): k-sunflower-free families of size 2^(n − O(n log log k / log k)), so the result is 'not far from best possible for large k'
- Earlier: Erdős–Szemerédi 2^(n − Ω(√n)); Alweiss–Lovett–Wu–Zhang (2019) and Rao give 2^(n − Ω(n/log n)). k = 3 followed from the 2016 cap-set bounds (Croot–Lev–Pach; Ellenberg–Gijswijt) through Alon–Shpilka–Umans; Naslund and Sawin gave base 3/2^(2/3) ≈ 1.89 using Tao's slice rank
- Method: Theorem 2 bounds the number V_k(D, n) of vectors in [D]^n without a k-sunflower by the polynomial method (dimension counting with low-degree factors and a Chernoff bound); a reduction (Section 4) transfers the vector bound to set families with D = 16k²
- AI statement, verbatim: 'The proof of Theorem 2 was found by ChatGPT 6 Astra Pro, following a short discussion, which started by asking it to generalize the k = 3 case argument in [45], which we needed for our original target of proving the induced variant of the Erdős–Sauer problem. The original argument it produced used the slice-rank terminology due to Tao (as used in [45]); we rewrote it in the polynomial method language … The authors have verified all the results in the paper and take full responsibility for the content. The proof has been verified in Lean 4.35; the verification will be made available on the Palomar repository.' ([45] is Naslund–Sawin)
- Concurrent work cited in the paper: another team proves that the set-sunflower relation on the Boolean cube has maximal slice rank, and 'They discuss extensive use of AI, including multi-agent setups'. The authors say that obstruction concerns set sunflowers, while their tester is for vector sunflowers, so it does not apply
- Consequences listed in the paper: the vector bound extends the Alon–Shpilka–Umans question (connected to fast matrix multiplication) and gives results in arithmetic property testing and on equal subset sums
- Reactions: no Hacker News thread or expert comment found by 20 Oct morning (web search; sweep HN list). A web search instead returned 2023 sources saying Naslund–Sawin had 'proved' the conjecture; that wording refers to the k = 3 case

#### What happened
A sunflower is a family of sets whose pairwise intersections are all the same set. Erdős and Szemerédi asked how large a family
of subsets of an n-element set can be without containing k sets forming a sunflower, and conjectured an exponential saving over
the trivial 2^n. The k = 3 case became a corollary of the 2016 cap-set breakthrough. Bucić, He and Lee had asked ChatGPT 6 Astra
Pro to generalize Naslund and Sawin's k = 3 slice-rank argument for a different problem, the induced Erdős–Sauer problem. The model
produced a bound for vector sunflowers that works for all k, and with the authors' reduction this gives the full conjecture.

The authors rewrote the model's slice-rank argument in polynomial-method language. They say all results were verified, including
in Lean 4.35, but the formal proof had not been posted when the preprint appeared.

#### Why it matters
This is one of the best-known problems in extremal set theory. It sits next to the Erdős–Rado sunflower conjecture, which stays
open, and is tied to cap sets and the group-theoretic approach to matrix multiplication. The authors credit the model with the
proof of the key theorem, not with exposition, which puts it among the strongest AI contributions to combinatorics so far. Until
the Lean files are released or experts review it, the claim stays pending.

Sources: [arXiv 2610.26686: Proof of the Erdős-Szemerédi sunflower conjecture (Bucić, He, Lee)](https://arxiv.org/abs/2610.26686); [Full text (PDF)](https://arxiv.org/pdf/2610.26686)

### 2026-10-19: Integer programming in standard form in N^O(d) time, improving Eisenbrand–Weismantel's N^O(d²): Zhejiang team's ideas, GPT Astra Ultra's proofs (Chen, Mao, Zhang)

****; Zhejiang University, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-19-integer-programming-linear-exponent-gpt-astra-ultra/

arXiv 2610.25955 (19 Oct 2026), "Integer Programing: An Algorithm with Exponent Linear in the Dimension" by Lin Chen, Yuchen Mao and Guochuan Zhang (Zhejiang University), gives a deterministic algorithm for min{cᵀx : Ax = b, ℓ ≤ x ≤ u, x ∈ Zⁿ} with A ∈ Z^(d×n) running in poly(L_in) + f(d)·N^O(d)·(1 + log(c_max + 1))^O(1), where N bounds |A_ij|. The previous best was N^O(d²) (Eisenbrand & Weismantel 2020), and the abstract calls it "a fundamental open problem whether the exponent O(d²) can be improved to one linear in d". The AI statement says the authors designed the route and "GPT Astra Ultra was used to turn the insights of the authors into complete mathematical proofs", and that GPT checked the main results in Lean 4. Unrefereed.

- Paper: arXiv 2610.25955 [cs.DS], 19 Oct 2026, Lin Chen, Yuchen Mao, Guochuan Zhang (Zhejiang University). Builds on their 9 Oct Subset Sum / IP feasibility paper (arXiv 2610.12953)
- Result: deterministic poly(L_in) + f(d) N^O(d) (1 + log(c_max + 1))^O(1) time for standard-form integer programs with d equality constraints and bounded variables; 'Prior to our work, the best known running time was N^O(d²) [Eisenbrand & Weismantel'20] … We answer this question affirmatively'
- Technique (contents): feasibility background 'interior is easy, boundary reduces'; LP pricing to 'separate the unavoidable cost from the repair'; an affordable interior/boundary test with a boundary branch ('few large changes, then a smaller problem') and an interior branch
- AI statement, verbatim: 'The main proof route was designed by the authors, building upon their recent work on subset sum and integer programming feasibility [11]. The authors came up with the basic idea of adapting the algorithmic technique for integer programming feasibility test to optimization, and GPT Astra Ultra was used to turn the insights of the authors into complete mathematical proofs. GPT and Claude were used for drafting the paper. GPT also verified this papers main results using the Lean 4 proof.'
- Caveats: no Lean repository is linked in the PDF; no independent check or expert reaction found by 20 Oct morning; the companion Subset Sum claim is also unreviewed

#### What happened
Eisenbrand and Weismantel's 2020 proximity bound gave integer programs with few constraints an N^O(d²) algorithm. Chen, Mao and Zhang
carry over the "interior is easy, boundary reduces" approach from their Subset Sum feasibility paper to optimization, and get an
exponent linear in d. The division of labour matches their earlier paper: the authors supplied the plan, the model wrote the proofs.

#### Why it matters
If it holds, the result changes the best known running time for a core problem of combinatorial optimization. It is also the
second claim in ten days in which one group says a GPT model turned its sketches into complete proofs of a long-standing
algorithms question. That puts the weight on independent checking.

Sources: [arXiv 2610.25955: Integer Programing: An Algorithm with Exponent Linear in the Dimension](https://arxiv.org/abs/2610.25955); [arXiv 2610.12953: Subset Sum and Integer Programming Feasibility Test in Near-Optimal Time (same authors)](https://arxiv.org/abs/2610.12953)

### 2026-10-19: Kaplansky's quasitrace conjecture reported disproved by an unreleased OpenAI model; experts reproduce it and refute the monotone completeness conjecture

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-19-quasitrace-monotone-completeness-counterexamples-openai/

Jananan Arulseelan and James E. Hanson write that "an unreleased OpenAI model produced a claimed counterexample" to the quasitrace (Kaplansky) conjecture for AW*-algebras in an unpublished manuscript (Sept 2026). They "fully reproduce the disproof" by a modified construction and use it, with continuous model theory, to refute the monotone completeness conjecture (finite case). They called these "the two major open problems in the theory of AW*-algebras". GPT 6 Astra wrote a preliminary write-up of their own proof.

- Paper: arXiv 2610.27184, 'Counterexamples to the monotone completeness conjecture' (Arulseelan, Hanson), posted 19 Oct 2026
- Quote: 'The two major open problems in the theory of AW*-algebras were the Kaplansky conjecture and the monotone completeness conjecture. Recently, an unreleased OpenAI model produced a claimed counterexample to the former in an unpublished manuscript (September 2026)'
- Abstract: 'We fully reproduce the disproof of the quasitrace conjecture by a modified construction and arguments.' Main theorem: a finite AW*-factor with canonical quasitrace τ and elements y1, y2, y3 with Σ yi*yi = 1 showing uniform failure of traciality
- Limits: 'Our construction does not appear suited to properly infinite AW* algebras, so that case of the monotone completeness conjecture remains open'
- AI disclosure: 'The idea for the proof was developed by the authors. It was then completed with a preliminary write-up given by GPT 6 Astra. The arguments were significantly reworked and fixed by the authors … The bundle construction here adapts a construction from an unpublished manuscript produced by an unreleased OpenAI model.'
- Background: Haagerup (2014) proved quasitraces on exact C*-algebras are traces; Kaplansky's 1951 'Projections in Banach algebras' is the origin; the same authors had earlier proved the weaker normality conjecture (cf. arXiv 2610.17435, listed in the 13 Oct roundup)
- Not verified by us: the OpenAI quasitrace manuscript itself is not public; a web search on 21 Oct found no announcement or coverage

#### What happened
In 1951 Irving Kaplansky introduced AW*-algebras, an algebraic generalization of von Neumann algebras. Two questions about them
stayed open for decades. The first asks whether the quasitrace of a finite AW*-factor must be a trace, which is equivalent to
asking whether every 2-quasitrace on a unital C*-algebra is a trace. The second asks whether every AW*-algebra is monotone
complete. Haagerup proved the first for exact C*-algebras in 2014.

On 19 Oct 2026 Jananan Arulseelan and James E. Hanson posted a counterexample to the monotone completeness conjecture for finite
AW*-algebras. They start from "the quasitrace manuscript", an unpublished September 2026 manuscript in which an unreleased OpenAI
model produced a claimed counterexample to the quasitrace conjecture. They reproduce that disproof by a modified construction and
combine it with model-theoretic results (quasitracial ultraproducts, countable saturation, separable reflection). In their AI
disclosure they say GPT 6 Astra completed a preliminary write-up of their own proof idea, which they then "significantly reworked
and fixed".

OpenAI has not published the quasitrace manuscript, and it does not appear among the files we can check. Everything here comes
from the Arulseelan–Hanson paper.

#### Why it matters
If it holds, an AI system has settled one of the oldest problems in operator algebras, and specialists rebuilt the result within
weeks and pushed it to the companion conjecture. It also shows that results from unreleased OpenAI models are reaching
mathematicians privately before any public announcement.

Sources: [arXiv 2610.27184: Counterexamples to the monotone completeness conjecture (Arulseelan, Hanson)](https://arxiv.org/abs/2610.27184)

### 2026-10-19: AI-assisted results in the 19 Oct 2026 arXiv listing: experts rework OpenAI's release papers (Deligne–Drinfeld, Hénon–Lane–Emden, Navier–Stokes), Wright's harmonic-analysis problems disproved with Codex, an Aaronson–Ambainis lower bound, and DeepMind's Busy Beaver agent

***; OpenAI, Anthropic, Google DeepMind, Harmonic, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-19-ai-assisted-results-october-19-listing/

Catalogue of the 19 Oct 2026 arXiv announcement (papers submitted 15–16 Oct). Four results have their own entries: Brennan's conjecture (Smirnov's group checks OpenAI's proof and shortens it), Smith–Toda V(4) for p ≥ 11 (Angeltveit after OpenAI), the lonely runner bound (1+c)/2n with ChatGPT 6 Pro, and the asymptotic maximum number of Hamiltonian cycles in tournaments (proofs found by GPT-5.6). This page lists the rest. Hirose gives an alternative proof of the Deligne–Drinfeld conjecture, "for which a proof was previously presented in the OpenAI Math Release". Li–Lu–Zhang extend OpenAI's Hénon–Lane–Emden argument to fractional orders. Cen–Hong–Wu disprove Wright's Problems 14 and 15 with ChatGPT and Codex. Weikang Li proves an Ω(√(N log N)) lower bound for 2D quantum spatial search, answering Aaronson and Ambainis, with GPT-6 Astra and Opus 5.5. Google DeepMind's Gemini agent finds Lean-verified accelerations for eleven BB(6) machines. Humpherys proves strong shock-layer stability with ChatGPT and Claude (Lean). Davies et al. repair the string-graph separator theorem, with Lean proofs written by Aristotle.

- Own entries: Brennan (2610.23853), Smith–Toda V(4) (2610.23215), lonely runner (2610.23790), tournaments (2610.23755)
- Deligne–Drinfeld, second human proof: Minoru Hirose, 'Confluence relations, Hoffman basis over Z_(2), and the Deligne–Drinfeld conjecture' (2610.23719): every multiple zeta value is a Z_(2)-combination of the Hoffman basis (Hirose–Sato conjecture); with Brown and Furusho this 'gives an alternative proof of the Deligne–Drinfeld conjecture, for which a proof was previously presented in the OpenAI Math Release' (family 008). AI: 'The author used ChatGPT (GPT-6 Astra Pro) for mathematical discussions and assistance with the exposition'
- Fractional Hénon–Lane–Emden (Kui Li, Haoyang Lu, Zhitao Zhang, 2610.22802; not in the sweep's AI list, found by our full-text scan): subcritical nonexistence for orders 0 < s < 1; 'The localized kernel identity and the comparison of level-set intervals in OpenAI [26] and Mei-Wei [27] provide the starting point'; acknowledgement: 'We used AI assistance in parts of the proof of the subcritical fractional Hénon-Lane-Emden conjecture.' OpenAI family 370
- Navier–Stokes follow-up in physics: Antti Niemi (Nordita), 'Magnetic flux ring splitting near a Navier–Stokes singularity' (2610.22510), simulates resistive MHD driven by a flow reproducing 'the geometry and scaling of the collapsing core' of OpenAI's forced singular flow; code and analysis 'written with the assistance of large language models (both Anthropic Claude and OpenAI models)'
- Wright's Problems 14 and 15 disproved (Xi Cen, Guixiang Hong, Xinfeng Wu, 2610.23563): a compactly supported singular measure with Fourier decay of every order < 1/2 whose dyadic maximal operator has infinite weak-L¹ and H¹→weak-L¹ norms, refuting the proposed endpoint bounds from James Wright's 2017 harmonic-analysis problem list; 'the authors used ChatGPT and Codex to discuss possible proof strategies, assist in drafting and checking parts of the mathematical arguments'
- Quantum spatial search (Weikang Li, Tsinghua IIIS, 2610.23560): a local quantum walk needs Ω(√(N log N)) steps to find a marked vertex on an L×L lattice, 'resolving the long-standing open question posed by Aaronson and Ambainis in 2003', so Tulsi's algorithm is optimal; proof via an electrostatics correspondence; 'We acknowledge the use of GPT-6 Astra and Opus-5.5 to assist with theoretical calculations'; code github.com/LWKJJONAK/QuSearch
- Busy Beaver agent (Google DeepMind: Dos Santos, Veeriah, Havrilla, Bolt, Zuzic, Singh, Zahavy; 2610.22603): a Gemini-based pipeline (pattern detection, LLM conjecturing, informal proof, Lean 4 proof) 'autonomously discovered and formally verified novel Collatz-like accelerations for eleven machines in BB(6)', provisionally classified as Cryptids; follows DeepMind's cryptids repo of 9 Oct
- Strong shock layers (Jeffrey Humpherys, 2610.22660): analytic proof of 1D spectral stability of sufficiently strong ideal-gas Navier–Stokes shock layers on a region including air and monatomic gases, verified in Lean; 'ChatGPT (OpenAI) and Claude (Anthropic) contributed substantially to this work, ChatGPT to the development of the energy estimates and the exposition, and Claude to the Lean formalization'
- String-graph separators repaired (James Davies, Paweł Rzążewski, Jack Spalding-Jamieson, Bartosz Walczak; 2610.23792): O(√m) balanced separators for string graphs, region intersection graphs over minor-closed families and induced-minor-free graphs, after a 2024 flaw broke Lee's proof and the works built on it; main results 'obtained entirely by the human authors'; Lean formalization 'almost entirely constructed using the proprietary LLM-based tool Aristotle', the improved bound formalized and first obtained with GPT 5.6 Sol
- Hornich–Hlawka (Seokbin Lee, 2610.23785): answers three of five Marinescu–Niculescu problems; 'An initial version of the proof of Theorem 4.3 was generated by ChatGPT-5.6 Sol'; parts of Theorem 1.4 and Lemma 6.7 'discovered by ChatGPT-6 Astra'
- Krzyż-type extremal problems (Instanes, Llinares, Vukotić; 2610.23817): ChatGPT 5.6 Sol simplified Lemma 19 and suggested Example 10; ChatGPT 6 Astra supplied Lemma 20, removing an extra assumption from Theorem 2(a); Claude Opus 5.5 proofread and suggested Remark 14
- Smaller roles: energy-critical NLS upper cone (Ma, Miao, Yang, Zhang, Zheng; 2610.22447; ChatGPT 5.6 Sol 'used to assist in the proof of Lemma 4.1'); Brlek–Reutenauer conjecture proved (D'Alessandro, Papi; 2610.22663; AI only for literature checks); online geometric matching (2610.23130; ChatGPT used for Section 4 results); Erdős–Pósa for induced even cycles (2610.22967; language editing only)
- Harmonic analysis follow-up (evening scan): Korobenko (Reed College) and Sawyer (McMaster), 'Probabilistic Fourier extension and the Schrödinger equation' (2610.22634, submitted 15 Oct): 'Just recently, OpenAI has claimed to prove the diagonal inequality p = q in the case of d = 3 dimensions'; this led us to OpenAI's uncovered restriction, Bochner–Riesz and local smoothing families (own entry). No AI-use statement
- Number theory built on OpenAI manuscripts (found in the evening scan of an earlier listing): Yuanda Xu and Hejian Sang, 'A matching lower bound for the sextic large sieve and structured prime moments' (2610.17675, v2 14 Oct) match de Faveri's upper bound using 'the uniform unmarked character moments of OpenAI's September 2026 manuscript' (cubic Gauss sums, 25 Sept) and cite the quasi-RH paper; disclosure: 'The main framework of the present argument is due to the author. Generative AI was used interactively in mathematical exploration and the development and refinement of parts of the argument'
- Methods and tools: 'Proof Perplexity' (Das, Banerjee, Cambridge; 2610.22612) scores Claude's Riemann-zeta zeros proof against Mathlib; QMBL quantum many-body leaderboard, 1,434 energies, aggregation 'made cheap by AI agents' (2610.22387); Lean rate–distortion library (2610.22404); LLM-assisted CHC solving (2610.23771)
- Scanned in full text with no AI statement (19 Oct): Kosniowski's conjecture for four fixed points (2610.23855), Beauville–Voisin for Debarre–Voisin fourfolds (2610.23376), Mbekhta's conjecture (2610.23229), matroid-tangle counterexamples (2610.23059), Niu's tandem-queue conjecture (2610.22837), metric Poissonian pair correlation (2610.22836, 2610.22833), local antimagic leaf counterexamples (2610.22783), Lovász number of random circulants (2610.22971), and Reid's square achievement game (2610.22410; Lean-formalized, answers Erickson's question, no AI statement)

#### What happened
Monday's arXiv announcement (19 Oct, submissions of 15–16 Oct) was the first big batch since Thursday. Its main theme is that
mathematicians are now publishing papers that check, simplify and extend results from OpenAI's 6 Oct release. Two of them
(Brennan, Smith–Toda) have their own entries. Three more follow up OpenAI release papers:

- **Deligne–Drinfeld.** After Thomas Willwacher's simplified version of OpenAI's proof (arXiv 2610.15592), Minoru Hirose gives an
  independent route through his Z_(2)-integrality of the Hoffman basis. He discussed the mathematics with GPT-6 Astra Pro.
- **Hénon–Lane–Emden.** Li, Lu and Zhang start from OpenAI's localized kernel identity (family 370) and extend nonexistence to
  fractional orders. Their one-line AI acknowledgement sits at the end of the paper, which is why the sweep's abstract filter missed it.
- **Navier–Stokes.** Niemi tests the physical meaning of OpenAI's forced singular flow with magnetic reconnection simulations.

Two other strong results are human-led with substantial AI help. Cen, Hong and Wu disprove two endpoint bounds proposed in
James Wright's problem list, with ChatGPT and Codex. Weikang Li's lower bound for quantum walk search on the 2D grid answers a 2003
question of Aaronson and Ambainis, with GPT-6 Astra and Opus 5.5 assisting with calculations. Google DeepMind's Busy Beaver paper
is the most autonomous item: a Gemini agent finds and Lean-proves compact descriptions of eleven unresolved BB(6) machines.

No Hacker News thread or high-reach X post on any of these papers was found on the morning of 19 Oct (HN Algolia search; X not
searchable through fxtwitter without post ids).

#### Why it matters
Two weeks after the release, verification of OpenAI's release papers has moved from blog posts and forum threads to arXiv papers by
specialists. So far the pattern is the same each time: no errors found, the arguments are longer than they need to be, and the
experts quickly strengthen the results.

Sources: [arXiv 2610.23719: Confluence relations, Hoffman basis over Z(2), and the Deligne–Drinfeld conjecture (Hirose)](https://arxiv.org/abs/2610.23719); [arXiv 2610.22802: The fractional Hénon–Lane–Emden conjecture (Li, Lu, Zhang)](https://arxiv.org/abs/2610.22802); [arXiv 2610.22510: Magnetic flux ring splitting near a Navier–Stokes singularity (Niemi)](https://arxiv.org/abs/2610.22510); [arXiv 2610.23563: Disproof of Wright's Problems 14 and 15 on lacunary maximal functions](https://arxiv.org/abs/2610.23563); [arXiv 2610.23560: Electrostatic Limits on Quantum Spatial Search (Weikang Li)](https://arxiv.org/abs/2610.23560); [arXiv 2610.22603: Steps Towards Autonomous AI Discovery on the Busy Beaver Frontier (Google DeepMind)](https://arxiv.org/abs/2610.22603); [arXiv 2610.22660: Spectral stability for strong ideal-gas shock layers (Humpherys)](https://arxiv.org/abs/2610.22660); [arXiv 2610.23792: Revisiting separators in string graphs (Davies, Rzążewski, Spalding-Jamieson, Walczak)](https://arxiv.org/abs/2610.23792); [arXiv 2610.23785: Hornich–Hlawka inequality, constructions and reductions (Lee)](https://arxiv.org/abs/2610.23785); [arXiv 2610.23817: Extremal problems related to the Krzyż conjecture](https://arxiv.org/abs/2610.23817); [arXiv 2610.22612: Proof Perplexity (Das, Banerjee)](https://arxiv.org/abs/2610.22612); [arXiv 2610.22447: Energy-critical NLS, higher-kinetic-energy regime](https://arxiv.org/abs/2610.22447); [arXiv 2610.22663: Rational decomposition of polyhedra and a conjecture of Brlek and Reutenauer](https://arxiv.org/abs/2610.22663); [arXiv 2610.22387: QMBL, the Quantum Many-Body Leaderboard](https://arxiv.org/abs/2610.22387); [arXiv 2610.22634: Probabilistic Fourier extension and the Schrödinger equation (Korobenko, Sawyer)](https://arxiv.org/abs/2610.22634); [arXiv 2610.17675: A matching lower bound for the sextic large sieve and structured prime moments (Xu, Sang)](https://arxiv.org/abs/2610.17675)

### 2026-10-19: AI-assisted maths on Zenodo, 17–19 Oct 2026: a claimed proof of the 1985 Arad–Herzog conjecture, Messerschmidt's conjecture confirmed with Codex, and a Claude-run study on AI-written science

***; OpenAI, Anthropic, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-19-ai-assisted-results-zenodo-october-17-19/

With no new arXiv announcement over the weekend, a new Zenodo step in the news sweep listed about 50 Zenodo records from 17–19 Oct 2026 that matched conjecture/AI keywords. Most were junk: grand unified theories, crank Riemann hypothesis proofs and off-topic surveys. Nine records were serious mathematics or science with an explicit AI statement in the PDF or record. The largest claim is "A proof of the Arad–Herzog conjecture" by Aran S. Ziegler (Oct 19): every product of two nonidentity conjugacy classes in a nonabelian finite simple group contains at least two classes, a question open since 1985. The author "used agentic AI tools developed by OpenAI and Anthropic to carry out their research strategy and assist with Lean formalisation". The Lean check is relative to 856 named literature inputs encoded as axioms, and the manuscript calls itself a private, unreleased review version. Martino Lupini (Bologna) used OpenAI Codex to confirm Messerschmidt's 2018 conjecture on Lipschitz positive decompositions. Smaller items include a Codex-found planar counterexample to a matching-complex conjecture, a partial proof of the Cubelike Graph Core Conjecture with six GPT and Claude models, and a Claude-assisted stability theorem for Burgers vortices. A Claude Opus 5.5 agent ran a full preregistered epidemiology study as a meta-experiment. All unrefereed.

- Arad–Herzog conjecture (Arad & Herzog, Products of Conjugacy Classes in Groups, LNM 1112, 1985): Aran S. Ziegler, 'A proof of the Arad–Herzog conjecture', Zenodo 23450488 (Oct 19, 2026; manuscript dated 18 Oct, 'Private review version 3 … unreleased'; ~66 pp. plus a review-support zip). The paper finishes the remaining families on top of Fisman–Arad (1987, alternating groups), Moori–Tong-Viet (2011), Guralnick–Malle–Tiep (2013, projective linear and more) and Guralnick–Navarro (2016, square case). It also identifies an exceptional family of products with exactly two classes in odd-dimensional orthogonal groups over odd fields
- Arad–Herzog AI statement: 'The author used agentic AI tools developed by OpenAI and Anthropic to carry out their research strategy and assist with Lean formalisation and manuscript preparation. The author takes full responsibility for the content.'
- Arad–Herzog Lean caveat (§12): the compiled declaration AH.FinalLiterature.global_literature_relative states the conjecture for all finite simple nonabelian groups, but its dependency closure contains 859 axioms. Besides the 3 standard Lean axioms, these are '587 binary ordinary imports and 269 other named mathematical inputs' (published classification and character-theoretic results taken as axioms). The 'revised verification artifacts are currently held in the private project archive'. The author is not otherwise known in the field, and there was no expert reaction as of Oct 19
- Messerschmidt's 2018 conjecture confirmed: Martino Lupini (Università di Bologna, ERC grant DAT), 'Lipschitz positive decompositions of von Neumann preduals', Zenodo 23440415 (Oct 17). The selfadjoint predual of a von Neumann algebra has a Lipschitz positive decomposition iff the algebra is subhomogeneous, so infinite-dimensional selfadjoint trace class answers the Lipschitz decomposition problem negatively. The finite-dimensional constants satisfy c·log log N ≤ c_N ≤ C·log N. Disclosure: 'OpenAI Codex was used in October 2026 … for mathematical exploration, proof development, source retrieval, manuscript drafting … and to assist in checking proofs'
- Lupini, 'Common null complements for polynomial images of finite sums of perfect sets', Zenodo 23440154 (Oct 17): for sums of D+2 compact perfect sets, one null F_σ set B gives P(C)+B = ℝ for every nonconstant polynomial of degree ≤ D, so Jung and Lai's Conjecture 4.8 holds for such sums (it stays open in general). Codex was used 'for mathematical exploration' and 'Separate Codex collaborators developed and reviewed arguments'
- Matching complexes: Zihuan Xu, 'A Planar Counterexample to the Sphere-Wedge Conjecture for Matching Complexes', Zenodo 23437631 (Oct 17). The matching complex of the planar pentagonal bipyramid collapses to a torus, which disproves the conjecture that matching complexes of planar graphs are contractible or wedges of spheres; it is the smallest possible counterexample, unique among 7-vertex planar graphs. 'An OpenAI Codex assistant was used in developing the arguments, drafting the manuscript, and writing the verification programs'
- Cubelike Graph Core Conjecture (Mančinska–Pivotto–Roberson–Royle), partial: Qihang Wang, 'Integral Cayley graphs and abelian rigidity in cores of cubelike graphs', Zenodo 23445385 (Oct 18). It proves the conjecture for cores with a normal abelian Cayley representation and classifies the abelian groups on which every integral Cayley graph is cubelike. Models named: 'OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, Anthropic Claude Fable 5.1, OpenAI GPT-6.1 Sol, OpenAI GPT-6 Astra, and Anthropic Claude Opus 5.5'; acknowledges Gewu Intelligence Lab (the same author and lab as arXiv 2610.20649 in the 16 Oct catalogue)
- Fluid dynamics: Adilkhan Salkimbayev, 'Linear stability of asymmetric Burgers vortices at large circulation beyond the half-strain threshold', Zenodo 23448800 (Oct 18). Maekawa's vortex is linearly stable for every asymmetry λ ≤ 0.9399 at large circulation (previously known only for λ < 1/2), with interval-arithmetic certificates (python-flint). 'The work was carried out with substantial help from generative AI (Anthropic's Claude)'
- Quantum chemistry: Jonas Matuzas, 'Single-centre expansions of three-centre 1s Slater integrals: convergence rates and certified error bounds', Zenodo 23448184 (Oct 18): 'OpenAI ChatGPT-6 Pro was used for derivations, computations, and drafting; Claude Opus 5.5 (Anthropic) was used for independent re-derivation and audit'
- Minor: Lien-Hung Su, congruences for restricted-cycle and Pell-weighted permutations (OEIS A193437/A193438/A244432), Zenodo 23445724, ChatGPT used 'for exploratory mathematical discussion'
- AI-run science as experiment: Diego Halabi, 'Do cats rot your teeth?', Zenodo 23458160 (v2 Oct 18, preregistration v1 doi 10.5281/zenodo.23435708, OSF RVH7J). It is a preregistered NHANES 2005–2006 study 'produced almost entirely by an AI agent (Claude, model claude-opus-5-5, Anthropic, run in Claude Code)' and released with the agent's redacted session transcripts. The primary result was inconclusive (prevalence ratio 1.12, 95% CI 0.93–1.36). The preprint shows how 4 of 45 preregistered estimates 'could support a positive paper without breaking the registration' and argues that 'verification, not writing, is now the bottleneck of scientific communication'
- Pentagonal Pizza Conjecture disproved: John W. G. Seamons, 'Counterexamples to the Pentagonal Pizza Conjectures', Zenodo 23462615 (dated Oct 20): explicit strictly convex pentagons refute both the area and perimeter versions of Berele and Catoiu's conjecture (American Mathematical Monthly 129(5), 2022); the area witness has alternating slice areas 3/16 and 253/400; exact certificates plus a standard-library Python checker. AI statement: 'OpenAI Sol and Astra were used to assist with computational search strategy, code development, and verification workflows. AI-generated output was not treated as independent evidence.' Not peer-reviewed, no Lean
- Same author, 'Two Exact Witnesses for Perpendicular Quadrisection', Zenodo 23462455: the integer pentagon (0,−20),(14,0),(12,9),(−12,9),(−14,0) has at least seven quadrisections by perpendicular lines, refuting Eberhart's proposed bound of five for convex pentagons (and the upper-bound clause of his odd-polygon conjecture), plus a rational triangle answering Eberhart's existence question; AI statement names 'OpenAI Sol and Astra, and Claude Opus 5.5'
- Krawczyk's Conjecture 3.4 proved (claim): Zihuan Xu (Capital Normal University), 'Substitutive points in factors of growing substitution shifts', Zenodo 23437635: a point of a factor of a growing substitution shift is substitutive iff it is the image of a point with T^m φ^r(x) = x. AI declaration: 'OpenAI Codex was used during 16–17 October 2026 for literature retrieval, exploration and checking of proof arguments, preparation of auxiliary finite-verification programs, and drafting and revision of the manuscript. This assistance included substantive mathematical development as well as writing.' Unrefereed, written in two days
- No AI statement found in the PDF (not counted): Anonymous, two-host dormancy model counterexample to Blath and Tóbiás's conjecture (Zenodo 23448514; 'computer assisted, not proof-assistant formalisation'); Paul Bilokon, Goldbach problems for twin/cousin/sexy prime summands (23463971)
- Zenodo noise (sweep of 19 Oct): of ~50 Zenodo hits, 9 were serious records with AI statements (above), about 6 were serious-looking records with no AI statement found or not scanned (e.g. Weak Viterbo pentagonal certificate and (ℤ/4ℤ)^d sumsets: no statement in the PDF; substitutive points: not scanned), and the rest were crank physics/RH 'proofs', off-topic social-science surveys or matches with no AI involvement

#### What happened
The weekend of 17–19 Oct 2026 had no arXiv announcement, and the sweep's new Zenodo step became the main source of AI-assisted
results. Each candidate record's PDF was downloaded and searched for an AI statement; the quotes above come from those PDFs or the
record descriptions.

**Arad–Herzog.** In 1985 Zvi Arad and Marcel Herzog asked whether a product of two nonidentity conjugacy classes in a nonabelian
finite simple group must contain at least two conjugacy classes. The question came out of Szép's conjecture on products of centralisers. Alternating groups (Fisman–Arad 1987),
sporadic groups and some low-rank cases (Moori–Tong-Viet 2011), projective linear groups and wide classical and exceptional ranges
(Guralnick–Malle–Tiep 2013) and the square case (Guralnick–Navarro 2016) were already known. Ziegler's ~66-page manuscript claims the remaining families
and assembles the full statement over all finite simple groups. Readers should weigh three caveats: (1) the author has no
visible track record in the field and the record calls itself a private, unreleased review version; (2) the Lean development proves
the theorem *relative to* 856 named inputs, mostly published classification and character-table results imported as axioms, and its
final verification artifacts are not public; (3) the author says agentic OpenAI and Anthropic tools carried out "their research
strategy", which suggests a largely agent-driven project. If experts confirm it, it is a notable AI-assisted resolution in finite group
theory. Until then it is an unverified claim.

**Codex as collaborator.** Martino Lupini (Università di Bologna, holder of the ERC grant "Definable Algebraic Topology") posted two Codex-assisted
papers on Oct 17. One confirms Messerschmidt's 2018 conjecture with a complete characterisation (subhomogeneous algebras) and
near-optimal growth bounds. In the other, "separate Codex collaborators developed and reviewed arguments". Zihuan Xu's
matching-complex counterexample follows the same single-author, Codex-assisted pattern. His same-day paper on substitutive points (Krawczyk's
Conjecture 3.4, Zenodo 23437635) was not scanned for an AI statement.

**AI-run study.** Diego Halabi's "Do cats rot your teeth?" is a meta-experiment. A Claude Opus 5.5 agent in Claude Code ran a deliberately implausible
epidemiology question from preregistration to preprint, and the record logs the time taken, human interventions, agent errors and the share of agent-written text.
The point is that a by-the-book AI-written paper could be spun positive from 4 of 45 preregistered estimates.

#### Why it matters
Zenodo has become a channel for AI-assisted mathematics that skips arXiv moderation. That makes it the place to catch claims early, such as
Ramanujan's inequality on Oct 16–17 and Arad–Herzog now. Many of these claims carry heavy caveats. Literature-relative Lean proofs of the Arad–Herzog kind check the assembly of a proof, not the classification inputs.

Sources: [Xu: Substitutive points in factors of growing substitution shifts (Zenodo 23437635)](https://zenodo.org/records/23437635); [Seamons: Counterexamples to the Pentagonal Pizza Conjectures (Zenodo 23462615)](https://zenodo.org/records/23462615); [Seamons: Two Exact Witnesses for Perpendicular Quadrisection (Zenodo 23462455)](https://zenodo.org/records/23462455); [Zenodo 23450488: A proof of the Arad–Herzog conjecture (Aran S. Ziegler, Oct 19)](https://zenodo.org/records/23450488); [Zenodo 23440415: Lipschitz positive decompositions of von Neumann preduals (Lupini)](https://zenodo.org/records/23440415); [Zenodo 23440154: Common null complements for polynomial images of finite sums of perfect sets (Lupini)](https://zenodo.org/records/23440154); [Zenodo 23437631: A planar counterexample to the sphere-wedge conjecture for matching complexes (Xu)](https://zenodo.org/records/23437631); [Zenodo 23445385: Integral Cayley graphs and abelian rigidity in cores of cubelike graphs (Wang)](https://zenodo.org/records/23445385); [Zenodo 23448800: Linear stability of asymmetric Burgers vortices beyond the half-strain threshold (Salkimbayev)](https://zenodo.org/records/23448800); [Zenodo 23448184: Single-centre expansions of three-centre 1s Slater integrals (Matuzas)](https://zenodo.org/records/23448184); [Zenodo 23445724: Congruences for restricted-cycle and Pell-weighted permutations (Su)](https://zenodo.org/records/23445724); [Zenodo 23458160: Do cats rot your teeth? Study produced by an AI agent (Halabi)](https://zenodo.org/records/23458160); [Guralnick, Malle & Tiep (2013): Products of conjugacy classes in finite and algebraic simple groups (arXiv 1202.2627)](https://arxiv.org/abs/1202.2627)

### 2026-10-19: Commutative counterexamples to Arveson's hyperrigidity conjecture: proof developed with GPT-6 Astra, finite-dimensional example proposed by it (Bilich, Scherer)

***; University of Haifa, Technion, OpenAI, Anthropic; Science & math; confidence medium; https://postcutoff.com/e/2026-10-19-arveson-hyperrigidity-commutative-counterexamples-gpt-6-astra/

arXiv 2610.26556 (19 Oct 2026) by Boris Bilich (Haifa) and Marcel Scherer (Technion) constructs operator systems in commutative C*-algebras for which every irreducible representation of the C*-envelope is a boundary representation, yet hyperrigidity fails. The examples are continuous affine functions on an infinite-dimensional weighted ℓp ball, a dilation of commuting operators, and a finite-dimensional system from a truncated epigraph of a strictly convex function (affine dimension d ≤ 129). Earlier counterexamples (Bilich–Dor-On; Scherer; Tian) were noncommutative. The AI declaration says the initial proof "was developed in a long conversation with ChatGPT using the GPT-6 Astra model" and the finite-dimensional counterexample "was proposed by ChatGPT".

- Paper: 'Commutative Counterexamples to Arveson's Hyperrigidity Conjecture', Boris Bilich (University of Haifa) and Marcel Scherer (Technion), arXiv 2610.26556 [math.OA], 19 Oct 2026
- Context (paper): the conjecture 'was disproved by the first author and Dor-On'; Scherer found a finite-dimensional counterexample and Tian a three-dimensional one, all in noncommutative algebras, where a dilation comes from an irreducible representation. 'This mechanism is unavailable for commutative C*-algebras'. Davidson–Kennedy had restated the commutative case as a question on Choquet versus dilation order (Question 8.4)
- Examples: (1) A(K) for an infinite-dimensional weighted ℓp ball (originally ℓ^(4/3) with |x_k| ≤ B/k); (2) commuting operators with a nontrivial dilation; (3) a finite-dimensional operator system from a compact truncated epigraph; such a counterexample needs affine dimension d ≥ 3; the authors conjecture the minimum is 4; the construction gives d ≤ 129
- AI declaration, verbatim: 'The weighted ℓp ball counterexample was discovered by the second author. … Its initial proof was developed in a long conversation with ChatGPT using the GPT-6 Astra model and spanned nearly thirty pages, with extensive use of probabilistic arguments. The finite-dimensional counterexample was proposed by ChatGPT using the GPT-6 Astra model, with the weighted ball example as its starting point. OpenAI's Codex was used … for mathematical exploration, the development and simplification of proofs, symbolic and numerical calculations, literature searches, and English-language editing.'
- Agent experiment: 'After more than 40 hours of exploration by GPT-6.1 Sol and Claude Opus 5.5 agents, the smallest proposed counterexample had dimension d = 11. However, the construction is very complicated, and we have not yet verified its correctness. We therefore do not include it in this paper.'

#### What happened
Hyperrigidity asks whether a generating set of a C*-algebra determines every representation among completely positive maps.
Arveson conjectured it follows from all irreducible representations being boundary representations. Bilich and Dor-On had already disproved it for
noncommutative algebras. Commutative algebras resisted, because their irreducible representations are one-dimensional. Scherer
found the weighted-ball example; GPT-6 Astra built its initial proof and then proposed the finite-dimensional version.

#### Why it matters
It settles the main remaining case of a central conjecture in noncommutative Choquet theory. The authors also report a negative
detail: a long multi-agent search produced a smaller candidate they could not verify, so they left it out.

Sources: [arXiv 2610.26556: Commutative Counterexamples to Arveson's Hyperrigidity Conjecture (Bilich, Scherer)](https://arxiv.org/abs/2610.26556)

### 2026-10-19: FAHR-Face: Mass General Brigham foundation model estimates biological age and cancer survival from a face photo (Lancet Digital Health)

***; Mass General Brigham; Science & math; confidence medium; https://postcutoff.com/e/2026-10-19-fahr-face-cancer-survival-face-photos/

A study published on Oct 19, 2026 in Lancet Digital Health presents FAHR-Face, a foundation model trained on about 40 million face images and adapted to estimate biological age and survival of cancer patients from a single photo. Of the 122 highest-risk patients in a 3,820-patient test group, 64 died within 90 days of their photo. Researchers say it is not ready for clinical use; an ethicist warned about insurers and employers.

- Name: FAHR-Face (foundation AI models for health recognition using face photographs); team led by radiation oncologist Ray Mak, first author Fridolin Haugg (Mass General Brigham)
- Pre-training: ~40 million face images without a specific task, then adapted to biological age and cancer survival
- Survival test: 3,820 cancer patients; 64 of the 122 with the highest risk scores died within 90 days of the photo; age findings used >38,000 cancer patients
- Adding face-age and face-survival scores to a clinical prognostic model predicted survival better than either alone
- Model attention: age on the nasolabial folds and forehead; death risk on the bridge of the nose and below the eyes (not explained)
- Robust to expressions, makeup and lighting; rhinoplasty made people look older after surgery
- Limits: trained mainly on a majority-white Boston-area population; patent pending; Mak: 'There's still a lot of development ahead and clinical testing'
- Matthew Liao (NYU, not involved): 'If it gets it wrong, we might be sort of not providing care to someone who actually needs it'

#### What happened
Mass General Brigham's team pre-trained a general face model and then adapted it to two health tasks. The figures above come from
Scientific American's report; the Guardian article and the paper were not readable for us.

#### Why it matters
A camera-only prognostic signal is cheap to collect and could help clinicians, but it also raises the risk of face photos being used
to judge people's health by insurers or employers, as outside experts warned.

Sources: [Scientific American: Can AI predict whether you'll die from cancer from just an image of your face?](https://www.scientificamerican.com/article/can-ai-predict-whether-youll-die-from-cancer-from-just-an-image-of-your-face/); [The Guardian: New AI tool can predict cancer survival by a single patient selfie, study suggests](https://www.theguardian.com/technology/2026/oct/19/ai-tool-predict-cancer-survival-patient-selfie)

### 2026-10-19: Gregory Chudnovsky, with ChatGPT, claims the lemniscate period has irrationality exponent 2, with a Lean 4 proof built on OpenAI's π-exponent library

***; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-19-lemniscate-irrationality-exponent-chudnovsky-chatgpt/

arXiv 2610.26539 (19 Oct 2026), "The irrationality exponent of the lemniscate period: A determinant proof with a Lean 4 formalization" by Gregory V. Chudnovsky, proves that for every ν > 2 and all large q, |Ω − p/q| > q^(−ν), so Ω = 2∫₀¹(1−t⁴)^(−1/2) dt has irrationality exponent 2. The proof uses elliptic separated-weight interpolation on the curve y² = x³ − x and a determinant bound on a complex-multiplication grid, an elliptic analogue of the method in OpenAI's π-exponent manuscript. The title-page footnote says the paper was "Developed in collaboration with OpenAI ChatGPT, including mathematical exposition and formalization assistance". The ancillary Lean 4 sources are said to prove the endpoint "without additional hypotheses" and use a vendored copy of OpenAI's library (namespace OAI…PiExponent). Unrefereed.

- Paper: arXiv 2610.26539 [math.NT], Gregory V. Chudnovsky, dated 10 Oct 2026, submitted 19 Oct; 33 pages; comment: 'Lean 4 proof sources and reproduction instructions included as ancillary files'
- Theorem (abstract): 'for every real ν > 2, there is Q ≥ 2 such that q^(−ν) < |Ω − p/q| for every integer p and every integer q ≥ Q. Consequently Ω is irrational and its irrationality exponent is two.'
- Method: 'elliptic separated-weight interpolation, arithmetic denominator estimates, and an analytic determinant bound on a square complex-multiplication grid', using E: y² = x³ − x
- AI disclosure, verbatim (title-page footnote): 'Developed in collaboration with OpenAI ChatGPT, including mathematical exposition and formalization assistance. The accompanying Lean source files contain the formal proof.' Reference [5]: 'G. V. Chudnovsky, with OpenAI ChatGPT assistance, Lemniscate-period formal proof snapshot'
- Lean: 'The accompanying Lean 4 development proves the endpoint without additional hypotheses'; Appendix A says no additional axiom, sorryAx or unfinished proof is present, only propext, Classical.choice and Quot.sound. 'The generic graded-jet, cohomological, and collision-counting infrastructure is supplied by the vendored OAI library [4]' (openai/math snapshot fd4aeeb); arXiv ancillary files sit under anc/lean/OAI/NumberTheory/PiExponent/…
- Status: the formal statement was written by the author with ChatGPT's help, and the shared infrastructure comes from OpenAI's unrefereed π-exponent development; no independent review found by 20 Oct

#### What happened
OpenAI's 6 October release claimed that π has irrationality exponent 2, with a Lean proof. Two follow-ups already carried the
method over to logarithms of rationals. Gregory Chudnovsky, a long-standing figure in transcendence theory, now gives an
elliptic-curve version for the lemniscate period. He worked with ChatGPT and built his Lean development on a vendored copy of
OpenAI's π-exponent library.

On 22 Oct Gregory and David Chudnovsky generalized the result ([arXiv 2610.31112](https://arxiv.org/abs/2610.31112)): every nonzero real period of a CM elliptic curve over ℚ (for a specified rational invariant differential) has irrationality exponent 2, and for the lemniscate period Ω they give the uniform bound μ(Ωⁿ) = O(n log(2n)) for all integers n ≥ 1. "Lean verifies the new CM determinant argument and rational normalization relative to explicit classical uniformization, CM-lifting and theta inputs; the lemniscate-power endpoint is also verified", so the classical inputs are assumed, not formalized. Disclosure: "Prepared with OpenAI ChatGPT assistance in mathematical exploration, exposition, formalization development, and internal review."

#### Why it matters
Before 2026, exponent exactly 2 was proved for very few classical constants. A Lean proof with a short, readable statement,
from a well-known specialist, is the strongest signal yet that OpenAI's interpolation-determinant method works and carries over.
It is still not an independent check of OpenAI's shared infrastructure.

Sources: [arXiv 2610.26539: The irrationality exponent of the lemniscate period (Chudnovsky)](https://arxiv.org/abs/2610.26539); [openai/math (π irrationality-exponent family 017 and Lean library)](https://github.com/openai/math); [arXiv 2610.31112: Irrationality exponents of real CM elliptic periods and powers of the lemniscate period (G. and D. Chudnovsky)](https://arxiv.org/abs/2610.31112)

### 2026-10-19: Neumann–Zagier 1985 conjecture on tree lattices settled after an AI proof with ChatGPT 5.5 Pro and Opus 4.8

***; OpenAI, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-10-19-neumann-zagier-conjecture-tree-lattices-ai-proof/

Joshua Evan Greene, Ahmad Issa and Duncan McCoy prove that a positive definite tree lattice has an irreducible vertex element, which settles a 1985 conjecture of Neumann and Zagier. Greene and McCoy had submitted their partial result to the First Proof AI benchmark. Issa then proved the theorem with ChatGPT 5.5 Pro and Opus 4.8, and the three authors simplified the AI argument.

- Paper: arXiv 2610.27095 'Tree lattices and the Neumann–Zagier conjecture' (Greene, Issa, McCoy), 19 Oct 2026
- Result: if a tree lattice L(T) is positive definite, some vertex element is irreducible; corollary: if L(T) ≅ Z^n then T has a vertex of weight one (Neumann–Zagier 1985)
- AI methodology: 'In 2026, JEG and DM submitted this result to the First Proof AI benchmark test. Soon thereafter, JEG and DM were contacted by AI (the second author), who had used AI (artificial intelligence) to prove Theorem 1.1.'
- 'The AI proof contained two main innovations: the use of the Perron-Frobenius theorem to identify a vertex, and an inductive proof akin to Lemma 3.1 to establish its irreducibility. While the use of Perron-Frobenius appeared to be a stroke of genius, it turned out to be a red herring'; the inductive lemma 'persisted as an essential linchpin'
- 'The initial AI proof of Theorem 1.1 used ChatGPT 5.5 Pro and Opus 4.8 … We chucked AI efforts to write this paper in favor of our own, although we engaged it for proofreading.'
- Prior human progress: Greene and McCoy proved the single-bad-vertex case in 2020 and strengthened it in 2023

#### What happened
A tree with integer weights on its vertices presents an integral "tree lattice". In 1985 Walter Neumann and Don Zagier conjectured that a tree lattice isometric to the standard lattice Z^n must
contain a vertex of weight one. Greene and McCoy proved special cases in 2020 and 2023. In 2026 they submitted the open case to
the First Proof benchmark of research problems for AI.

Ahmad Issa then contacted them with a proof he had obtained using ChatGPT 5.5 Pro and Claude Opus 4.8. The authors' AI
methodology section describes how they digested it. The model's Perron–Frobenius idea "appeared to be a stroke of genius" but
was a red herring. Its inductive irreducibility lemma stayed essential, and they strengthened the theorem. The paper itself was
written by hand, with AI used only for proofreading.

#### Why it matters
A problem posted as a benchmark task was solved with frontier models and then confirmed and improved by the experts who posed
it. The paper also describes candidly what the AI contributed.

Sources: [arXiv 2610.27095: Tree lattices and the Neumann–Zagier conjecture](https://arxiv.org/abs/2610.27095)

### 2026-10-20: Double-cap conjecture (Witsenhausen 1974, Kalai) claimed in all dimensions, with a separate AI-built Lean proof

****; Universidade Federal do Ceará, Universidade de São Paulo, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-20-double-cap-conjecture-witsenhausen-lean-ai/

Ibiapina, Gomes, Pio, Belo and Ferreira (Brazil) claim that caps minimize the orthogonal neighbourhood in RP^(n−1), which yields the double-cap conjecture: for n ≥ 3, a measurable subset of the sphere S^(n−1) with no orthogonal pair has measure at most that of two opposite caps of radius π/4. This answers Witsenhausen's 1974 problem. AI "played an assistive role" in the paper, and a separate Lean 4 proof by a different argument was built "almost entirely using AI-assisted tools" (Codex agent per its metadata).

- Claim (arXiv 2610.27523): 'a measurable subset of S^(n−1) containing no orthogonal pair has measure at most that of two opposite open caps of radius π/4', for n ≥ 3, with rigidity up to null sets; also the measurable fractional chromatic number of the spherical orthogonality graph
- History: Witsenhausen (Amer. Math. Monthly 1974) proved α_n ≤ 1/n; the double-cap conjecture α_n = δ_n is attributed to Kalai; the best earlier bound on S² was α_3 < 0.313
- AI statement (abstract): 'AI played an assistive role in preparing this paper. The formalization and verification workflow for a separate Lean 4 proof of the main inequality and double-cap result, based on a different argument, were carried out almost entirely using AI-assisted tools.'
- Paper's caveat: the Lean development 'does not constitute a verification of the proof or the additional applications presented here'
- LeanShadow repo: 138 modules, sorry-free, standard axioms; Comparator matched 12 main results (incl. double_cap_conjecture, s2_double_cap_bound 1 − √2/2) against a reference challenge; Lean and nanoda kernels; formalization.yaml lists automation 'agent, framework: Codex' for packaging, adapters and tooling; review status 'self-assessed'

#### What happened
The double-cap conjecture asks for the largest measurable set on a sphere that contains no two orthogonal points. Witsenhausen
posed the problem in 1974, and Gil Kalai's survey on Borsuk's problem made it widely known. The conjectured answer is two
opposite caps of angular radius π/4.

On 20 Oct 2026 five mathematicians from Fortaleza and São Paulo posted a proof for all n ≥ 3. It runs through a sharp
"orthogonal neighbourhood" inequality in real projective space, proved with the Funk transform, rotation comparisons and
second-variation arguments. A separate Lean 4 development (LeanShadow, public since 11 Sept 2026) proves the same main inequality
and the double-cap result by a different argument. The authors say they built it "almost entirely using AI-assisted tools", and
its metadata names a Codex agent.

The paper is unrefereed, and the Lean proof was self-assessed. No expert
reaction was found on 21 Oct.

#### Why it matters
If confirmed, a 52-year-old geometry problem falls in every dimension, and a machine-checked proof of the statement already
exists. Experts can check that Lean statement against the classical formulation faster than a long analytic paper. This is
another case of AI tools mainly turning a human result into a formally verified one.

Sources: [arXiv 2610.27523: An orthogonality inequality in RP^(n−1) and the double-cap conjecture](https://arxiv.org/abs/2610.27523); [GitHub: allenibiapina/LeanShadow (Lean 4 proof)](https://github.com/allenibiapina/LeanShadow)

### 2026-10-20: MDS conjecture settled in even characteristic: two independent proofs, one checked in Lean with Claude's help, the other written with ChatGPT

****; South China University of Technology, Guangzhou University, OpenAI, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean/

Two independent claims settle the linear MDS conjecture (from Segre's 1955 question on arcs) for every finite field of even order: an [n, k] MDS code over F_q, q even, has n ≤ q + 1, except n ≤ q + 2 for k = 3 and k = q − 1. Pranava Kumar registered a Lean 4 proof on the Palomar registry on 15 Oct (Claude helped translate it into Lean; Claude Opus 5.5 ported and packaged it); Bocong Chen and Jing Huang posted a different proof on arXiv on 20 Oct, written with ChatGPT's help on proof strategies and calculations.

- Statement (Chen–Huang, arXiv 2610.28208): 'every nontrivial linear [n, k] maximum distance separable (MDS) code over a finite field Fq of characteristic two satisfies n ≤ q + 1, except in dimensions k = 3 and k = q − 1, where n ≤ q + 2. This resolves the linear MDS conjecture in even characteristic.'
- History: Segre's 1955 problems on arcs in finite projective spaces; Ball (JEMS 2012) proved the conjecture for k ≤ p, settling prime fields; odd characteristic remains open in general
- Chen–Huang AI statement: 'The authors used ChatGPT to assist with exploring proof strategies, checking intermediate calculations, and editing the exposition. The authors are responsible for all mathematical arguments and for the final exposition.'
- Kumar's Lean proof: Palomar record PALOMAR-2026-10-15-000014 (registered 15 Oct 2026, trust level 'high'), theorem EvenOrderMDS.maximum_length_of_MDS_code over Mathlib, standard axioms only, checked by Comparator against Challenge.lean and by the Lean kernel plus the NanoDa and con-ron kernels
- Kumar repo authorship note: 'The mathematical argument was devised by Pranava Kumar … Claude (Anthropic) assisted him in translating the proof into Lean 4' and 'The present repository was assembled with the help of an AI agent (Claude Opus 5.5, in Claude Code)', which wrote Challenge.lean and Solution.lean; 'No independent human expert has yet reviewed the mathematics.'
- Chen–Huang note the overlap: Kumar's report 'also claims to establish Theorem 1.1'; both use a Frobenius obstruction, a ternary colon ideal and Gorenstein Hilbert-function unimodality, with different local constructions

#### What happened
The MDS conjecture asks how long a maximum distance separable code (one that meets the Singleton bound) can be. Equivalently, it
asks for the largest arc in a finite projective space, a question Beniamino Segre posed in 1955. The expected answer is q + 1,
with exceptions in even characteristic. Simeon Ball proved it over prime fields in 2012.

On 15 Oct 2026 Pranava Kumar registered a Lean 4 proof of the full even-characteristic case on Palomar, the registry of
Lean-verified mathematics. The repository's README says the argument is Kumar's: Claude helped him translate it into Lean, and
Claude Opus 5.5 in Claude Code later ported it, wrote the Challenge and Solution files, the README and the report. It also says
that no independent human expert has yet reviewed the mathematics. Palomar lists the record with trust level "high".

On 20 Oct Bocong Chen (South China University of Technology) and Jing Huang (Guangzhou University) posted a 33-page proof on
arXiv. Their AI statement says ChatGPT helped explore proof strategies, check intermediate calculations and edit the text. They
cite Kumar's report as an independent claim and compare the two arguments.

No expert reaction or press coverage was found on 21 Oct.

#### Why it matters
If it holds, this is the biggest advance on the conjecture since Ball's prime-field theorem. It is also an early case of a major
classical result arriving with a kernel-checked Lean proof from the start, built with an AI coding agent. Two caveats remain. The
Lean statement was written by the AI agent; it was audited against the standard formulations but uses its own definitions of
minimum distance and MDS, not Mathlib's. And neither proof has been refereed yet.

Sources: [arXiv 2610.28208: On the MDS conjecture in even characteristic (Chen, Huang)](https://arxiv.org/abs/2610.28208); [Palomar registry API: PALOMAR-2026-10-15-000014 (Kumar)](https://data.palomar-registry.org/api/v1/results?id=PALOMAR-2026-10-15-000014); [GitHub: PranavaKCode/EvenOrderMDS (Lean 4 proof)](https://github.com/PranavaKCode/EvenOrderMDS)

### 2026-10-20: AI-assisted results in the 20 Oct 2026 arXiv listing: poset dimension proofs found by GPT-6 Astra, Hitchin domination from ChatGPT 5.6 Sol's ideas, the section conjecture for generic curves, Yau's geometric conjecture, Erdős–Gyárfás r = 5, a Claude-built Gray-code reduction, and a Lean harness solving 6/6 IMO 2026

***; OpenAI, Anthropic, DeepSeek, AWS, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-20-ai-assisted-results-october-20-listing/

Catalogue of AI-disclosing papers in the 20 Oct 2026 arXiv announcement (mostly submitted 16–19 Oct) and late-found ones from 13–17 Oct. Own entries: the Erdős–Szemerédi sunflower conjecture (Theorem 2 found by ChatGPT 6 Astra Pro), Erdős–Hajnal for all paths (Nguyen with GPT-6 Astra; DeepMind's model in parallel), a non-sofic hyperbolic group (Sandhu–Thom), integer programming in N^O(d) (GPT Astra Ultra), the equiregular sub-Riemannian Sard counterexample (Claude), commutative counterexamples to Arveson's hyperrigidity conjecture (GPT-6 Astra), and Chudnovsky's lemniscate irrationality exponent (ChatGPT, Lean). Expert follow-ups were added to the Riemann-zeta two-thirds entry (Arias de Reyna) and the Jacobian conjecture entry (de Bondt). The rest is listed here.

- Own entries: sunflower (2610.26686), Erdős–Hajnal paths (2610.24887), non-sofic hyperbolic group (2610.24836), integer programming (2610.25955), Sard (2610.24080), hyperrigidity (2610.26556), lemniscate (2610.26539). Enriched: zeta two-thirds (Arias de Reyna, 2610.24612), Jacobian conjecture (de Bondt, 2610.24889)
- Poset dimension (Jędrzej Hodor, Piotr Micek, William T. Trotter; 2610.24774): for every d ≥ 3 there is α < 1 with posets excluding the standard example S_d having dimension O_d(n^α); for d = 3, O(n^(2/3) log n) and a construction of dimension Ω(n^(1/3)/log n). AI statement: 'The proofs were found by OpenAI's ChatGPT-6 Astra. The authors digested the initially generated output and made an effort to present the argument in a readable and optimized form.'
- Hitchin representations (Subhojoy Gupta; 2610.24606): every Hitchin representation dominates one in Hit_n(S)′ in the Hilbert length spectrum, 'this confirms a conjecture of Lee-Zhang and, independently, Tholozan who had proved this for n = 3'. 'The key ideas in this note, specifically the convexity of the log-spectral radius function in Corollary 3.2 and the use of the Brouwer fixed-point theorem in the proof of Theorem 1.1, are due to ChatGPT 5.6 Sol.' Written entirely by the author
- Grothendieck's section conjecture for the generic n-pointed genus-g curve (g ≥ 3, n ≥ 1) over function fields of M_g,n over k/Q (Ma Luo, Tatsunari Watanabe; 2610.24986). 'The authors used OpenAI's ChatGPT to assist with literature searches, aid their mathematical review, refine notation, and improve the exposition of their original mathematical ideas and arguments.' Supporting AI role only
- Yau Geometric Conjecture for isolated hypersurface singularities (Siyuan Fu, Stephen S.-T. Yau, Huaiqing Zuo; 2610.24636), 'a proof of the long-standing Yau Geometric Conjecture'. 'AI tools (ChatGPT and DeepSeek) were used … to help with checking calculations and arguments, developing details of proofs using standard techniques, and improving the exposition'
- Erdős–Gyárfás balanced colourings, r = 5 (Hai Xia, Stefan Szeider, TU Wien; 2610.24484): K_26 has no balanced 5-colouring, 'the first new case settled in over two decades'; SAT-based (CaDiCaL), 'We used generative AI tools to implement the Lean formalizations'; 'Code and proofs are checked by tests and the Lean kernel'
- Single-track Gray codes (Chris Dzoba; 2610.24758): the Degni–Drisko (2007) necklace conjecture implies optimal codes for every prime, formalized in Lean; with computation, optimal codes for every odd prime n ≤ 47 (open since 1996; previously n ≤ 19). 'Claude (Anthropic) wrote and ran the searches and verifiers, found the closing pivots of Theorem A by computer search, wrote the proofs and the Lean formalization, and drafted the text … Separate instances refereed the proof and the formalization, and Codex (OpenAI) wrote two further checkers'
- Crossing numbers (Balogh, Layman, Lidický, Pfender, Salazar, Spiro, Volec; 2610.25204): flag algebras give cr(K_n,n) ≥ 0.96936·Z(n,n) + o(n⁴) toward Zarankiewicz's conjecture, rectilinear ≥ 0.987·Z(n,n). 'Both Conjectures 1 and 7 were claimed to be true by OpenAI [26, 27]. ChatGPT-6 was used to assist with writing Section 3'; ChatGPT-6 also found an integer-overflow bug in their search program
- Strong Erdős–Frankl–Füredi cover-free conjecture (Yanan Hu, Leilei Zhang; 2610.24442): T(r) ≥ (r+1)² for all r, with equality when r+1 is a prime power. 'At an early stage of this work, AI tools were used to assist in developing proofs.'
- Siemons–Wagner property and Bradley's conjecture (Zhilin Zhang; 2610.24502): classification of 2-transitive groups with property ⋆3, settling Bradley's thesis conjecture. 'When this work had long been stalled, it was GPT6-Astra that opened a new line of thought and helped break the impasse'; Astra also wrote and debugged the GAP census scripts
- Super-localization conjecture (Moritz Hauck, Chupeng Ma, Daniel Peterseim; 2610.25230): short proof of the exp(−c n^(1/(d−1))) Kolmogorov n-width decay conjectured by Ma (FoCM 2026). 'ChatGPT was used to develop the central arguments in the proof of Theorem 1. In particular, it identified the energy-determinant argument of Li and Wang [9] and suggested adapting it'
- Michaelis–Menten networks (Eduardo D. Sontag; 2610.24351): global stability, stated as an open problem by Blanchini–Breda–Giordano–Liessi, holds without connectedness or non-degeneracy hypotheses; 'The author worked closely with several AI assistants to brainstorm ideas, search the literature, develop proofs, check arguments, and write and edit the paper'
- Counterexamples suggested or generated by AI: wave observability under GCC with a smooth time-periodic potential (Liao, Lü, Zhang; 2610.25327: an AI tool was used 'in particular, to suggest the counterexample presented in this paper'); heavy-ball non-acceleration in 1D (ETH seminar 'AI for Mathematics and Optimization'; 2610.25104: 'Example 3.2 was generated by AI … The counterexample in Section 5 was generated by AI and verified by the authors'); Nesterov gradient restart never occurring (Bao, Chen, Li, Sun; 2610.26546: 'GPT-6 was used as an auxiliary tool to help develop the counterexample construction'); Reich's fixed-point problem answered negatively (Luong Van Nguyen; 2610.25156: ChatGPT for 'assistance in conceptualizing the example'); Costantino–Frigerio–Martelli–Petronio geometric triangulation conjecture refuted with a 2,688-tetrahedron triangulation (Ke Feng, Aijin Lin; 2610.25217; ChatGPT helped explore 'proof strategies and calculations')
- Analysis and operator theory: Baranov–Bessonov–Kapustin conjecture proved (Bellavita, Virtanen; 2610.26232: 'suggestions generated by ChatGPT-6 contributed to the formulation and organization of the argument that appears in Section 4'); CAT(0) maximal monotone vs surjectivity, negative and partial answers (Chaipunya; 2610.25080; Claude Opus 5.5 drafted and revised 'parts of the text and proofs'); Schoen conjecture for Weil–Petersson circle maps (Martin, Yao; 2610.25580: work predates generative AI, but Claude and GPT, asked 'what have we missed', pointed to Riestenberg–Smillie and Ripoll–Tomi, which 'overcame some gaps in our argument'); translation-invariant pure states with exotic half-chain factors (Liu, Yi, Perimeter; 2610.24154; GPT-5.6 Sol and Claude Opus 5.5 'used extensively')
- Number theory and combinatorics: sum–product holds for fixed-field lattice products, the constructions used to disprove it over R (Lemley, Chew; 2610.26737: models 'used substantively during the development of proofs, in some cases proposing key arguments'); Huppert's ρ–σ conjecture, ρ(n) ≤ 4n − 1 + π(2n) and a gap in Bellotti–Keller–Trudgian fixed (Jabara, Keller; 2610.25571: Claude 'noticed a couple of inaccuracies and fixed them'); Terao's freeness conjecture for 15 and 16 lines (Anghel, Stan; 2610.25868; Claude for literature search, programming and drafting); Fuglede's conjecture in F₂⁷ (Kadowaki; 2610.24022; 'AI-assisted tools were used in the research'); size Ramsey numbers of two-component star forests, a case of Erdős Problem #561 (Chunlin You; 2610.25958; ChatGPT 'to assist in exploring proof strategies'); non-uniform Baranyai factorizations (Gao, Li; 2610.24695); zero forcing of P(n,3), P(n,4), proving Krishnan's Conjecture 5 (2610.24059; AI for 'Lean proof development'); Erdős–Ginzburg–Ziv for finite groups (Qu; 2610.26122; AI for 'proofs in the sections on finite simple groups')
- Algorithms and control: promise colouring in adversarial streams, 3- vs 6-colourable in O(n^(3/2) log(n/ε) log n) bits (Thoumi; 2610.24644: 'Theorem 10 and Appendix A were entirely obtained using GPT Astra; the author … takes no credit, but full responsibility'); strategy recovery for mean-payoff games as hard as solving them (Dorfman, Zwick; 2610.25188; 'Interactions with ChatGPT 5.6 Pro influenced the content of Section 5.2, Section 6 and Section 7'); Witsenhausen's counterexample, exact boundary of linear optimality and non-definability in o-minimal structures (Lu, Li; 2610.25853; AI expanded the authors' arguments into derivations); Coppersmith–Winograd as tensor-contraction aggregation (Solomonik, Bläser; 2610.26563; Gemini, ChatGPT and Claude helped develop embeddings and corrections)
- Methods: Maestro (AWS AI Labs, UIUC, Georgia Tech; 2610.25920), an open Lean harness on top of Claude Code or Codex: with Claude Opus 5 it proves 34 of 48 theorems of the June 2026 arXivLean release (19 for the bare agent), 669/672 PutnamBench, 59/60 Lean-IMO-Bench and 6/6 IMO 2026. Safal Aryal's proof-theoretic study of AI-generated Lean corpora such as OpenAI's (2610.25357; code written by Claude Code). A Lean-verified curved simplex Hilbert transform with no AI in the mathematics but 'AI coding agents' for the formalization (Durcik, Fraccaroli, Roos; 2610.24850)
- Minor roles (editing, proofreading, figures): center conjecture for cyclotomic Hecke algebras (Hu, Mathas, Shi, Webster; 2610.25101); Ramanujan conjecture and RH for unitary groups over function fields (Lomelí; 2610.23985; Claude for editing); Streets–Tian on homogeneous quotients (Cao, Chen; 2610.25229); Landau score iterations (2610.25780); asymptotic dimension and fat minors (2610.22495; figures)
- Scanned in full text, no AI statement (20 Oct): Ward's divisibility conjecture bound (2610.26200), hyperplane conjecture (2610.26186), rational hit conjecture (2610.26028), factorial separation conjecture (2610.25661), Property C/D inverse zero-sum conjectures (2610.25271), conformal normal metrics counterexample (2610.24541), real Jacobian conjecture vector fields (2610.19360), Abel limit cycles (2610.19340), mutual f-information (2610.21231), NLS tori (2610.22072), four-qubit contextuality degree 315 (2610.26465) and about fifteen others; the sweep's ML and physics papers with 'LLM' in the text are out of scope here

#### What happened
Tuesday's arXiv announcement (20 Oct; submissions of 16–19 Oct), together with papers from 13–17 Oct that the sweep had not listed
before, had 94 papers that disclose or mention AI use. We read the full text of every uncited mathematics paper among them,
plus 28 uncited "conjecture/open problem" papers without a disclosure. Seven results got their own entries. Two more are expert
follow-ups to famous AI results. Juan Arias de Reyna re-proves Claude's two-thirds theorem on zeta zeros for a textbook. Michiel
de Bondt stabilizes Alpöge's Jacobian counterexample down to cubic maps in dimension 12 and to Zhao's Vanishing Conjecture.

In the papers listed here, AI plays roles of three kinds:
- **The AI found the proof**: Hodor–Micek–Trotter on poset dimension ("The proofs were found by OpenAI's ChatGPT-6 Astra"),
  Thoumi's streaming theorem, and Dzoba's Gray codes (Claude did the search, proofs and Lean under the author's direction).
- **AI ideas at a key step**: Gupta's Hitchin note (ChatGPT 5.6 Sol supplied the convexity and fixed-point ideas), Hauck–Ma–Peterseim
  (ChatGPT found the argument to adapt), Zhang's Bradley conjecture (Astra "helped break the impasse"), and five papers whose
  counterexample was suggested or generated by AI.
- **Supporting roles**: literature, checking, Lean and editing, including in a few famous-name papers: the section conjecture for
  generic curves, the Yau geometric conjecture with S.-T. Yau, and Balogh et al. on Zarankiewicz's crossing-number conjecture.

No Hacker News thread or high-reach X post on any of these papers was found on the morning of 20 Oct (web search; the sweep's HN
and X sections list none).

#### Why it matters
The day's papers show AI credited with the decisive step on problems that specialists had studied for decades: sunflowers,
Erdős–Hajnal and soficity. Authors are also writing more precise disclosures, often citing the Leiden Declaration and naming the
lemmas a model proved. Two labs' models (OpenAI's Astra and a DeepMind internal model) reached the same Erdős–Hajnal case
independently.

Sources: [arXiv 2610.24774: On dimension of posets excluding a standard example (Hodor, Micek, Trotter)](https://arxiv.org/abs/2610.24774); [arXiv 2610.24606: Length-convexity and domination for Hitchin representations (Gupta)](https://arxiv.org/abs/2610.24606); [arXiv 2610.24986: The section conjecture for generic pointed curves (Luo, Watanabe)](https://arxiv.org/abs/2610.24986); [arXiv 2610.24636: The Yau Geometric Conjecture for isolated hypersurface singularities (Fu, Yau, Zuo)](https://arxiv.org/abs/2610.24636); [arXiv 2610.24484: Erdős–Gyárfás balanced five-colorings (Xia, Szeider)](https://arxiv.org/abs/2610.24484); [arXiv 2610.24758: Optimal single-track Gray codes from Gray-ordered necklaces (Dzoba)](https://arxiv.org/abs/2610.24758); [arXiv 2610.25204: Crossing numbers of complete bipartite graphs (Balogh et al.)](https://arxiv.org/abs/2610.25204); [arXiv 2610.24442: A proof of the strong Erdős–Frankl–Füredi cover-free conjecture (Hu, Zhang)](https://arxiv.org/abs/2610.24442); [arXiv 2610.24502: The Siemons–Wagner property and Bradley's conjecture (Zhang)](https://arxiv.org/abs/2610.24502); [arXiv 2610.25230: Local approximation of A-harmonic functions and the super-localization conjecture](https://arxiv.org/abs/2610.25230); [arXiv 2610.24351: Global stability of Michaelis–Menten networks (Sontag)](https://arxiv.org/abs/2610.24351); [arXiv 2610.25327: Observability of wave equations with time- and space-dependent potentials](https://arxiv.org/abs/2610.25327); [arXiv 2610.25104: Non-acceleration of the heavy-ball method in 1D](https://arxiv.org/abs/2610.25104); [arXiv 2610.26546: On the nonoccurrence of gradient restart in Nesterov's method](https://arxiv.org/abs/2610.26546); [arXiv 2610.25156: A counterexample to Reich's fixed point problem](https://arxiv.org/abs/2610.25156); [arXiv 2610.25217: A counterexample to the geometric triangulation conjecture](https://arxiv.org/abs/2610.25217); [arXiv 2610.26232: A proof of the Baranov–Bessonov–Kapustin conjecture](https://arxiv.org/abs/2610.26232); [arXiv 2610.26737: The sum–product conjecture holds for fixed-field lattice product constructions](https://arxiv.org/abs/2610.26737); [arXiv 2610.25571: On Huppert's ρ–σ conjecture (Jabara, Keller)](https://arxiv.org/abs/2610.25571); [arXiv 2610.25868: Terao's freeness conjecture for arrangements of 15 and 16 lines](https://arxiv.org/abs/2610.25868); [arXiv 2610.24644: Improved space bounds for promise graph coloring in adversarial streams](https://arxiv.org/abs/2610.24644); [arXiv 2610.25920: Maestro: agentic formal proof development by coordination (AWS)](https://arxiv.org/abs/2610.25920); [arXiv 2610.25580: Weil–Petersson circle homeomorphisms and the Schoen conjecture (Martin, Yao)](https://arxiv.org/abs/2610.25580)

### 2026-10-20: Borodin–Kostochka conjecture claimed for maximum degree ≥ 4583 (down from 5.3 million), with GPT-6 Astra assisting

***; Hong Kong University of Science and Technology, OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-20-borodin-kostochka-threshold-4583-gpt-6-astra/

A preprint posted on Zenodo on Oct 20, 2026 by Lizhong Chen (HKUST) claims the Borodin–Kostochka conjecture (1977) for every graph of maximum degree Δ ≥ 4583: χ ≤ max{ω, Δ − 1}. The best published threshold was 5.3 × 10⁶ (Liu, Sun, Wang and Zeng, 2026). Its "Declaration of AI use" says OpenAI GPT-6 Astra assisted with the resampling estimates, moment-bound optimisation, local-lemma parameter choices and small-deficit constructions. Unrefereed; the conjecture stays open for 9 ≤ Δ < 4583.

- Conjecture (Borodin & Kostochka, JCTB 1977): every graph with Δ ≥ 9 has χ ≤ max{ω, Δ − 1}, one colour fewer than Brooks's theorem when there is no clique of size Δ
- Earlier: Reed proved it for sufficiently large Δ (JCTB 1999); Liu, Sun, Wang and Zeng gave the explicit threshold 5.3 × 10⁶ (arXiv 2603.16670, v3 May 2026); Chen's own SSRN working manuscript had 2.9 × 10⁶
- New claim: Δ ≥ 4583, about a thousandfold lower; method: clique/sparse decomposition, random partial colouring with pre-selected families of witness triples, two applications of the local lemma, finite exponential-moment bounds
- The numerical part checks finitely many scalar inequalities in exact rational arithmetic (no graph enumeration); programs, parameters and reference output accompany the paper
- Disclosure (verbatim): 'OpenAI GPT-6 Astra assisted with the resampling estimates in Section 6.4, moment-bound optimisation in Sections 7.3 and 8.1, weighted selection and local-lemma parameter choices in Sections 5 and 9, and the small-deficit constructions in Appendix A.3.'
- Single author; ~22,800-word PDF; no Lean formalisation; no expert reaction found on Oct 20

#### What happened
The paper does not settle the conjecture, but it claims to move the explicit bound from millions to a few thousand. The AI's role is stated precisely in the paper's own declaration (above). It was posted on
Zenodo, not arXiv, and found through the sweep's Zenodo step; its PDF was read for the disclosure.

#### Why it matters
It is another case of a frontier model credited with the technical estimate-tuning in a probabilistic combinatorics proof on a
named 1977 conjecture. Like other single-author Zenodo claims, it needs expert checking before it counts.

Sources: [Zenodo: The Borodin–Kostochka conjecture for maximum degree at least 4583 (L. Chen)](https://zenodo.org/records/23487163); [arXiv 2603.16670: Liu, Sun, Wang, Zeng, On the Borodin–Kostochka conjecture for graphs with large maximum degree](https://arxiv.org/abs/2603.16670v3)

### 2026-10-20: Anthropic: an internal Claude model mostly autonomously designed protein binders against scorpion neurotoxins; 17 of 211 hit in two labs

***; Anthropic, Citrini Research; Science & math; confidence high; https://postcutoff.com/e/2026-10-20-claude-scorpion-toxin-binders/

On Oct 20, 2026 Anthropic's Science Blog reported that an internal Claude research model, working mostly autonomously for 64 hours, designed protein binders against five scorpion neurotoxins. It produced 318,484 designs and picked 211 for testing. Two labs found that 17 bound their target at 1 µM or tighter; the best two bound the Mexican scorpion toxin Cn2 at single-digit nanomolar affinity. Neutralization was not tested, and Anthropic says it has not produced an antivenom.

- Partners: Anthropic's Science team with Citrini Research; a test of whether Claude can help motivated non-experts tackle a health problem. Model: 'an internal research model, operating within Claude Science' (not named)
- Autonomy: 64 hours, four follow-up prompts, about 24 hours unattended after the first prompt. Claude chose toxin structures (PDB, AlphaFold DB), five hotspot residues per toxin, thresholds and multi-toxin strategies, and emailed sequences to Twist Bioscience itself
- Scale: 12 design tools, 318,484 designs, 553,052 design–toxin pairs co-folded and scored, ~10,000 GPU hours; 180 designs chosen by Claude plus 31 added later with its help = 211
- Targets: Cn2 (Mexico), Ts1 (Brazil), AaH I and AaH II (North Africa), LqhIII (Middle East), aimed at each toxin's sodium-channel binding site
- Validation: Adaptyv Bio (cell-free) and Twist Bioscience (human cells, Fc fusions), surface plasmon resonance at both. Criteria: hit in both labs, KD ≤ 1 µM, no binding to off-target CLEC12A. 17/211 passed (Cn2 7, AaH II 4, AaH I 4, LqhIII 2); single-toxin designs 11/60 (~18%). No multi-toxin design passed for more than one toxin
- Leads: Cn2-B1 and Cn2-B2, single-digit nM at both labs; competition experiments suggest they bind at or near the site of the LR antibody, reported to neutralize Cn2 in mice
- Caveats (Anthropic): tested only against recombinant toxins, not venom-purified ones; no in vitro or in vivo neutralization; 'we do not claim to have produced an antivenom'. All 211 designs and data released on Hugging Face
- Reviewers quoted: Susana Vázquez Torres (CNIO) called it a promising first step; Timothy P. Jenkins (DTU) said binding 'is no longer the main bottleneck'
- Context: >1.2M scorpion stings and >3,250 deaths a year, mostly children; Anthropic's X post had ~1.38M views (fxtwitter, Oct 21)

#### What happened
Anthropic and Citrini Research let an internal Claude model run a protein-design campaign against five scorpion toxins with little human
input, then had the chosen designs made and tested by two commercial labs. About 8% of all designs, and about 18% of single-toxin designs,
bound their targets, a hit rate the authors call typical for protein design. The attempt to make one binder that catches several toxins
failed.

#### Why it matters
It is one of the first published cases of a frontier lab's model running a wet-lab-validated design campaign mostly by itself, with data
released for checking. It also shows the limits: binding is not neutralization, and the hardest goal (a broad binder) was not reached.
Anthropic chose a project with clear medical value, while its CEO also points to bio misuse risk as a reason for strict biology safeguards.

Sources: [Anthropic: How Claude designed protein binders against scorpion neurotoxins](https://www.anthropic.com/research/claude-scorpion-toxin-binder-design); [Anthropic on X](https://x.com/AnthropicAI/status/2112579251555721573)

### 2026-10-20: Gale's 1953 conjecture fails in 3D: a 101-point set found with LLM code evolution and GPT-6 Astra beats the sphere's Borsuk partition ratio

***; OpenAI, Anthropic; Science & math; confidence high; https://postcutoff.com/e/2026-10-20-gale-conjecture-r3-counterexample-gpt-6-astra/

Gale (1953) conjectured that among sets of unit diameter in R³, the sphere is the hardest to cut into four parts of small diameter, with ratio √((3+√3)/6) ≈ 0.88807. Vsevolod Voronov (arXiv 2610.28828, 20 Oct 2026) gives a 101-point set V in which every partition into four parts has a part of diameter greater than 0.88928·diam V, verified in exact integer arithmetic. It is the first counterexample in dimension below 9. The search algorithms were developed "with the help of GPT-6 Astra … some of which were proposed by the AI itself", after earlier GigaEvo LLM code-evolution runs fell short.

- Result: |V| = 101, β₄(V) > 0.88928 > β₄(S²) = 0.88807…; verification: compute diam V, take pairs at distance > 0.88928·D, check the distance graph has no proper 4-colouring
- Before: counterexamples to Gale's conjecture came only from Borsuk counterexamples, none in dimension below 9 (OpenAI's 9-dimensional Borsuk counterexample of 23 Sep 2026 was the lowest)
- Also: similar examples for dimensions 4 ≤ n ≤ 6; code and configurations at github.com/vsvor/Gale-conjecture
- AI disclosure: 'In April 2026, we carried out extensive code-evolution experiments with the GigaEvo framework using DeepSeek 4, Qwen 3 Max, Gemini 3, and Claude Opus 4.6; the best ratio obtained in R3 was about 0.872. In September 2026, we returned to the problem and, with the help of GPT-6 Astra, developed new algorithms and heuristics, some of which were proposed by the AI itself. GPT-6 was also used to prepare the GitHub repository and to write the document describing the algorithms; Claude Opus 5.5 was used for proofreading and language editing.'
- Author's caveat: 'At present, there is no full formalization in Lean 4 or an independent review of this paper.'

#### What happened
Borsuk asked whether every bounded set in Rⁿ splits into n+1 parts of smaller diameter. The quantitative version asks how small the largest part can be made. In R³ with four parts, the sphere gives √((3+√3)/6) ≈ 0.88807, and David Gale conjectured in 1953 that no set does worse.

Voronov's paper ([arXiv 2610.28828](https://arxiv.org/abs/2610.28828)) gives a 101-point set that does worse, 0.88928. The set was found numerically by global search from random starts and verified exactly, by checking that a distance graph has no proper 4-colouring. Earlier LLM code-evolution experiments with GigaEvo (April 2026) reached only about 0.872. The successful algorithms were developed in September with GPT-6 Astra, which also proposed some of them.

#### Why it matters
This is a small, explicit, machine-checkable counterexample to a 73-year-old conjecture, found by human–AI algorithm design. It follows AI-assisted progress on the Borsuk problem itself (dimension 63 in May, OpenAI's dimension 9 in September).

Sources: [arXiv 2610.28828: Voronov, A counterexample to Gale's conjecture in R³](https://arxiv.org/abs/2610.28828); [GitHub: vsvor/Gale-conjecture (code and configurations)](https://github.com/vsvor/Gale-conjecture)

### 2026-10-21: Nagata's 1959 conjecture: algebraic geometer Jenia Tevelev writes out OpenAI's 'remarkably simple' proof and calls the release 'a stunning achievement'

*****; OpenAI, UMass Amherst; Science & math; confidence medium; https://postcutoff.com/e/2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition/

OpenAI's 6 Oct 2026 math release (family 039, manuscript dated 23 Sep) claims a proof of Nagata's conjecture: for r ≥ 10 very general points in the plane, every curve of degree d with multiplicities m_i satisfies Σ m_i < d√r. On 21 Oct Jenia Tevelev (UMass Amherst) posted arXiv 2610.30050, an expanded account of his seminar talk on "OpenAI's remarkably simple proof", which reorganizes and simplifies the argument and proves the conjecture in full (Theorem A). He calls the manuscripts "a stunning achievement" but says releasing them "almost entirely unedited" "harms our profession". Unrefereed.

- Problem: Nagata (Amer. J. Math. 81, 1959), arising from his counterexample to Hilbert's 14th problem; Nagata proved it only when r is a perfect square
- Claim: OpenAI family 039, 'Nagata's conjecture and maximal Seshadri constants': strict inequality Σ m_i < d√r for every nonzero effective plane curve through r ≥ 10 very general complex points, plus maximal multipoint Seshadri constants (L^n/r)^(1/n) for every smooth polarized variety and all large r
- Exposition: Jenia Tevelev, 'On OpenAI's proof of the Nagata conjecture', arXiv 2610.30050 [math.AG], 21 Oct 2026; expanded account of his talk at UMass Amherst
- Method (abstract): reduction to the normal cone and collision of fat points into one monomial scheme reduce the problem to interpolation on the normal ruled surface of a smooth plane cubic; 'the novelty is that the study of the asymptotics of theta functions as the cubic develops a node, together with interpolation duality, reduces the problem further to interpolation on a triangle of lattice points, which is straightforward'
- Tevelev: 'I have reorganized and simplified the argument by working directly on the normal ruled surface of a smooth plane cubic'; he follows the complex-geometry manuscript and has 'not studied in detail' the algebraic companion manuscripts on Seshadri constants
- Tevelev: 'At the same time, I think these manuscripts are a stunning achievement. The model managed to connect the dots and solve the problems that many of the experts thought were out of reach. I think a wholesale re-evaluation of the mathematical corpus is currently underway, with the old underbrush of mathematical conjectures rapidly being burned in the AI fire.'
- Criticism: 'As I read the released manuscripts, I wondered whether I might be their first human reader. They seemed almost entirely unedited.' He proposes that AI companies fund an independent, NSF-style foundation for mathematics research
- AI disclosure (Tevelev's notes): 'These notes were edited and illustrated with ChatGPT Astra.'
- Extra result in the notes: the same calculation determines the nef and pseudoeffective cones of a weighted blow-up of P² at one point for very general local parameters when the ratio of weights exceeds nine

#### What happened
[OpenAI's 6 Oct release](/e/2026-10-06-openai-math-release-722-manuscripts/) listed, as family 039, a proof of Nagata's conjecture in its strict, nonhomogeneous form, together with three companion manuscripts on maximal multipoint Seshadri constants (surfaces, higher dimensions, positive characteristic). According to the release, the manuscripts were written by an OpenAI model with no human author.

On 21 Oct Jenia Tevelev (UMass Amherst) posted [arXiv 2610.30050](https://arxiv.org/abs/2610.30050), notes expanding a talk he gave at UMass Amherst. He claims no originality: he reorganizes the argument, works directly on the normal ruled surface of a smooth plane cubic, and proves the conjecture in full (Theorem A). He also derives the nef and pseudoeffective cones of certain weighted blow-ups of P².

The notes end with a commentary. Tevelev calls the manuscripts "a stunning achievement" and says that "the old underbrush of mathematical conjectures" is "rapidly being burned in the AI fire". He also says the release, apparently unedited, leaves the community a large unpaid editorial job that "harms our profession, especially the motivation and training of young people", and he proposes an independent research foundation funded by AI companies.

No other expert reactions were found on 22 Oct (one web search; X not searched).

#### Why it matters
Nagata's conjecture is one of the best-known open problems in the geometry of plane curves. With this note it joins Thompson's group F among the 6 Oct claims that specialists have reconstructed by hand. The note's closing section is also an unusually direct statement, from a working expert, of both the achievement and the cost to the field.

Sources: [arXiv 2610.30050: Tevelev, On OpenAI's proof of the Nagata conjecture](https://arxiv.org/abs/2610.30050); [OpenAI: Nagata's conjecture for plane curves (23 Sep manuscript)](https://github.com/openai/math/tree/main/preprints/Nagatas-Conjecture-for-Plane-Curves-September-23-2026); [openai/math CONTENTS.md (family 039)](https://github.com/openai/math/blob/main/CONTENTS.md)

### 2026-10-21: Barnette's 1969 Hamiltonicity conjecture: graph theorist Zhentao Li publishes a digested, partly re-proved version of OpenAI's proof

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-21-barnette-conjecture-openai-proof-zhentao-li-exposition/

OpenAI's 6 Oct 2026 math release (family 180, manuscript "Paired states and Hamiltonian cycles in cubic bipartite planar graphs", 24 Sep) claims Barnette's conjecture, with a Lean formalization listed in its catalogue. On 21 Oct Zhentao Li posted arXiv 2610.30490, "a digested exposition of OpenAI's generated proof", which keeps the proof "essentially unchanged" but replaces the complex-valued exponential sum with two sign-reversing involutions and proves a stronger statement: G − e has a Hamiltonian cycle for every Barnette graph G and edge e. Unrefereed.

- Conjecture: David Barnette (1969): every cubic 3-connected bipartite planar graph has a Hamiltonian cycle; a weakening of Tait's disproved conjecture
- OpenAI claim (CONTENTS.md, family 180): 'Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle', with a Lean link
- Li's note (arXiv 2610.30490, math.CO, 21 Oct): Theorem 3, for every Barnette graph G and every edge e0, G − e0 has a Hamiltonian cycle; main case cyclically 4-edge-connected, lower connectivity in Appendix C
- Proof outline (Li): find a perfect matching of the face–black-vertex incidence graph minus N[b0] whose complement is a spanning tree; take the dual tree on white vertices; trace it into a spanning tree of G; orient towards w0 and delete e0 and the out-edges of white vertices; the remaining 2-factor is a Hamiltonian cycle. Existence of the matching is shown by counting with sign-reversing involutions
- Li: 'The proof presented here is essentially unchanged, but replaces some parts of the generated proof with combinatorial arguments. In particular, we replace the complex-valued exponential sum with direct bijections (namely, two sign-reversing involutions).'
- AI disclosure (verbatim): 'LLMs were used to read and understand the OpenAI preprint, to revise and format this note and to generate figures from their description.'

#### What happened
Barnette's conjecture is one of the best-known open problems on Hamiltonian cycles in planar graphs. OpenAI's 6 Oct release listed a proof as family 180, with a Lean formalization in its catalogue.

On 21 Oct Zhentao Li posted [arXiv 2610.30490](https://arxiv.org/abs/2610.30490), a 14-page "digested exposition". He keeps OpenAI's strategy: a perfect matching in an auxiliary face–vertex graph whose complement is a spanning tree, which is traced into a spanning tree of G and turned into a Hamiltonian cycle. He replaces the generated proof's complex-valued exponential sum with two sign-reversing involutions and proves the stronger statement that G − e is Hamiltonian for every edge e. He used LLMs to read the OpenAI preprint and to format the note.

No other expert reactions were found on 23 Oct (X not searched).

#### Why it matters
With Thompson's group F and Nagata's conjecture, Barnette's conjecture is now the third headline claim of OpenAI's release that an outside specialist has written out by hand. Here the claim also has a machine-checked proof from OpenAI, so the human exposition mainly adds readability and a second, partly independent argument.

Sources: [arXiv 2610.30490: Zhentao Li, A note on the proof of Barnette's conjecture](https://arxiv.org/abs/2610.30490); [OpenAI: Paired states and Hamiltonian cycles in cubic bipartite planar graphs (24 Sep manuscript)](https://github.com/openai/math/blob/main/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026/paper.pdf); [openai/math CONTENTS.md (family 180, Lean link)](https://github.com/openai/math/blob/main/CONTENTS.md)

### 2026-10-21: Hochster's 1973 small Cohen–Macaulay conjecture: an OpenAI manuscript's counterexample is surveyed, re-proved and extended to every dimension by Cristian Anghel

****; OpenAI, Simion Stoilow Institute of Mathematics of the Romanian Academy; Science & math; confidence medium; https://postcutoff.com/e/2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey/

OpenAI's 6 Oct 2026 math release (family 195, manuscript dated 23 Sep) constructs a three-dimensional complete Noetherian normal local domain over C with no nonzero finitely generated maximal Cohen–Macaulay module, disproving the domain form of Hochster's 1973 small Cohen–Macaulay conjecture. On 21 Oct Cristian Anghel posted arXiv 2610.30027, a survey of the five September–October texts behind this (his two preprints, Lopez, Liang Chen and the OpenAI manuscript). It gives a complete proof of the OpenAI theorem and extends it to every dimension m ≥ 2. Anghel used ChatGPT and Claude to draft those proofs. Unrefereed.

- Claim: OpenAI family 195, 'A counterexample to the small Cohen–Macaulay module conjecture'; per the release notes the manuscript is the output of an OpenAI model with no human author; the catalogue links a Lean scope note (lean/docs/195.md)
- Chain (Anghel's survey): no Ulrich bundles on certain smooth surfaces (Anghel, arXiv 2609.02718, 2 Sep), answering the Eisenbud–Schreyer–Weyman question of 2003 negatively; extended to all dimensions by Angelo Felice Lopez (2609.06678); no aCM bundles / graded MCM modules (Anghel, 2609.15589); local counterexamples to Hochster's conjecture if K_Y ≡ 4H (Liang Chen, 2609.24142, 'proposed') or σ > 2H² (OpenAI)
- Survey, §11: '[OAI] claims a complete proof. Its argument is short … and uses only standard tools. Section 10 gives a complete proof of its main theorem, in every dimension.' Chen's argument is 'a proposed argument circulated for verification'
- New in the survey: the three obstructions become one 'variance identity', which explains the factor 2 between the graded bound σ ≤ H² and the local bound σ ≤ 2H²; products of Hesse surfaces with projective spaces give complete local domains with an isolated singularity, in every dimension ≥ 3, without small Cohen–Macaulay modules
- OpenAI's manuscript credits Anghel's surface inequality and complete-quadrangle construction and 'acknowledges that its existential claim overlaps with [Ch]' (survey §11)
- AI disclosure (survey): 'ChatGPT and Claude were used for exploratory computations, literature search and editorial assistance, including drafting of the proofs in Sections 9 and 10. The author is solely responsible for the final text.'
- Status (survey): 'none of the five texts has yet appeared in a refereed publication'
- Open (survey §12): mixed characteristic, where the conjecture was originally motivated by Serre's positivity of intersection multiplicities; reductions mod p of these rings do have small CM modules

#### What happened
In three weeks of September 2026 a numerical obstruction for vector bundles on polarized surfaces moved from projective geometry to local commutative algebra. Cristian Anghel showed that some smooth surfaces carry no Ulrich bundles (2 Sep), answering a 2003 question of Eisenbud, Schreyer and Weyman. Angelo Felice Lopez extended this to every dimension. Anghel then ruled out arithmetically Cohen–Macaulay bundles (14 Sep), and Liang Chen proposed a local-cohomology argument against Hochster's conjecture.

The decisive local step came from a manuscript dated 23 Sep and released on 6 Oct as family 195 of [OpenAI's catalogue](/e/2026-10-06-openai-math-release-722-manuscripts/). It constructs a complete local domain of dimension 3 with no small Cohen–Macaulay module, under the hypothesis σ(Y) > 2H².

Anghel's 21 Oct survey ([arXiv 2610.30027](https://arxiv.org/abs/2610.30027)) calls the OpenAI argument "short" and built on "only standard tools". It gives a complete proof of the main theorem, generalized to every dimension, and unifies the three obstructions in one identity. He used ChatGPT and Claude to draft those proofs.

No expert reactions beyond the survey were found on 22 Oct (one web search; X not searched).

#### Why it matters
Hochster's small Cohen–Macaulay conjecture has been one of the guiding open problems of commutative algebra for five decades. A counterexample in equal characteristic zero changes the field's expectations. Mixed characteristic, where the conjecture came from, stays open. The result also shows a pattern: human preprints and an AI manuscript built on each other within weeks.

Sources: [arXiv 2610.30027: Anghel, The recent counterexamples to the Eisenbud–Schreyer–Weyman Ulrich problem and to Hochster's small Cohen–Macaulay conjecture: a survey](https://arxiv.org/abs/2610.30027); [OpenAI: A Complete Local Domain without a Small Cohen–Macaulay Module (23 Sep manuscript)](https://github.com/openai/math/tree/main/preprints/A-Complete-Local-Domain-Without-a-Small-Cohen-Macaulay-Module-September-23-2026); [openai/math CONTENTS.md (family 195)](https://github.com/openai/math/blob/main/CONTENTS.md); [arXiv 2609.02718: Anghel, Smooth counterexamples to the Eisenbud–Schreyer–Weyman Ulrich existence problem](https://arxiv.org/abs/2609.02718); [arXiv 2609.24142: Liang Chen, A local cohomology obstruction to small Cohen–Macaulay modules](https://arxiv.org/abs/2609.24142)

### 2026-10-21: AI-assisted results in the 21 Oct 2026 arXiv listing: a ChatGPT 6.1 Sol counterexample to Frankl–Ota–Tokushige, klt Fano proofs from ChatGPT, Vince's conjecture with GPT Astra, Pólya for spherical shells, a Claude-built 55-addition 3×3 scheme and a Claude Opus 5.5 lemma in reverse mathematics

***; OpenAI, Anthropic, Google DeepMind, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-21-ai-assisted-results-october-21-listing/

Catalogue of AI-disclosing papers in the 21 Oct 2026 arXiv announcement (submitted 19–20 Oct). Own entries: MDS conjecture in even characteristic (Chen–Huang with ChatGPT; Kumar's Lean proof with Claude), double-cap conjecture (Lean proof built with AI tools), counterexamples to the quasitrace and monotone completeness conjectures (from an unreleased OpenAI model's manuscript), and the Neumann–Zagier conjecture (first proved with ChatGPT 5.5 Pro and Opus 4.8). Enriched: Hadamard matrices below 2000 (independent verification, 16 new orders with Claude). The rest is listed here.

- Own entries: MDS even characteristic (2610.28208), double cap (2610.27523), quasitrace/monotone completeness (2610.27184), Neumann–Zagier (2610.27095). Enriched: Hadamard below 2000 (2610.27175)
- Frankl–Ota–Tokushige conjecture b(t) = 2t refuted (Balázs Patkós; 2610.28029): 'The counterexample was found by ChatGPT 6.1Sol, the lower bound proof is due to the human author'; 'ChatGPT 6.1Sol produced a lower bound with a better leading' term as well, and the author improved the multiplicative constant
- Nef complexity of klt Fano varieties (Kiwamu Watanabe; 2610.27676): no value of c_X strictly between 0 and 1, full classification for c_X < 2 and the full spectrum. 'Most proofs in this paper were developed from arguments generated by OpenAI's ChatGPT through discussions initiated and guided by the author'; 'many of the central technical ideas were also proposed by the model' (GPT-5.6 Sol, later GPT-6 Sol for checking)
- Vince's conjecture on the average order of connected induced subgraphs proved (Jungang Chen, Hongxia Ma; 2610.27553): 'The Danus multi-agent workflow with GPT Astra was used to assist in developing some of the proofs. GPT Sol 6.1 was used to revise and polish the language'
- Pólya's conjecture for Dirichlet eigenvalues of spherical shells and Aharonov–Bohm annuli (Yutian Li; 2610.27707): 'the author used ChatGPT (OpenAI) to assist with mathematical exploration and proof development, checking intermediate arguments and calculations, literature searches, and drafting'
- Concentration conjecture for 3D Dirac–Coulomb operators (Habib Ammari, Jiayu Qiu; 2610.28193): 'They used ChatGPT 6 Astra to assist in working out details, particularly the small-separation analysis in Sections 5–7. The authors then substantially simplified the arguments and rewrote the manuscript.'
- Kim–Vu sandwich conjecture via sprinkling for random regular graphs (Julien Portier; 2610.28395): final proofs are the author's, but 'the author first produced a completely different proof of Theorem 1.2 which heavily relied on ChatGPT assistance', with ChatGPT proving d_TV(µ_d, ν_d) = O(d^(−1/2))
- Reverse mathematics (Ludovic Patey; 2610.27643), first-order part of the Ordered Variable Word theorem: 'Claude Opus 5.5 (Anthropic) was used to prove the Π⁰₁ extension lemma (Lemma 3.13) which is a core lemma in the proof. The proof was completely rewritten by the author'
- 3×3 matrix multiplication with 23 products and 55 additions over any ring, one below the record of 56, correctness in Lean 4 (Greg Sidebottom; 2610.27451): 'extended interactive collaboration with Claude (Anthropic; the Fable 5 and Opus 4.8 models), which implemented the repository's search tooling, exact minimizers, the Lean formalization, and much of this text under the author's direction and review'
- Hohenberg–Kohn conjecture for total currents refuted (Laestadius, Csirik; 2610.27294): 'OpenAI Codex and Anthropic's Claude were used … The proofs in this manuscript were developed using these tools. Lean 4 was also used to check formalized versions of the main results', with stated background results as assumptions
- Follow-ups to OpenAI's 6 Oct maths release: free entropy χ vs χ* (David Jekel; 2610.27243) presents 'a similar example' to the one 'OpenAI recently generated', using no AI himself apart from a Claude-generated assessment of missing attributions shared by Jonathan Shi; weighted cube inequality in all dimensions (Yuchen Bi; 2610.28390) adapts 'No. 336 of OpenAI's October 6, 2026 mathematics release' after 'several discussions with ChatGPT (GPT-6 Astra)'; single generation theorem for simple unital C*-algebras (Xin Ma; 2610.28478) assumes 'the finite factor generation theorem obtained by OpenAI' (GPT 6 Astra for references and checking)
- Other substantive AI roles: rational polynomial solvability reduces to stratifiability of semilinear sets (Michaël Cadilhac; 2610.27482: 'The ideas and writing in this paper were developed in conjunction with ChatGPT-6 Astra'); Brandt conjecture and sharp binding number of K_k-free graphs (Jifu Lin, Heng Yang; 2610.28536: ChatGPT 'exploring proof strategies'); bipanchromatic colourings, counterexamples and bounds (Novaković, Jojić; 2610.27133: 'ChatGPT was used to assist with proof development'); measures of maximal entropy for meromorphic maps (Muhan Luo, Qi Zhou; 2610.28617: follows [LZ26], 'whose main idea was suggested by GPT-6 Astra'); optimal global well-posedness for semilinear heat equations (Diebou, Phuc; 2610.27365: 'ChatGPT Plus GPT-5.6 Sol was used to show the continuity in time of T(u) in the proof Theorem 4.11')
- Irrationality: Gregory Chudnovsky's degree-dependent irrationality exponents of logarithmic numbers (2610.28079), 'Developed with OpenAI ChatGPT, including mathematical exposition and formalization assistance', Lean statement review through 'Checkpoint136' (see the lemniscate entry); Technion's 'New Directions in Irrationality' (Kaminer group; 2610.27799) says its proofs were found 'independently of LLM-generated mathematical arguments' and cites Lai's LLM-generated proof as 'a direct competition'
- Minor roles (checking, editing, figures): Janson–Sós conjecture on balanced subgraph counts proved (Wang, Wang, Yue; 2610.28124: LLM for 'routine calculations and checks'); sharp L^p-Poincaré gradient stability (2610.27281); periodic Anderson–Bernoulli model with infinitely many gaps (Fillman, Wood; 2610.27160: theorem proved without AI, AI suggested simplifications); vanishing uniform Turán densities (Lin, Ma, Zhou; 2610.28574: polishing only)
- Benchmark: ReuF2F (2610.27046), 244 formalized undergraduate research problems for coding agents; the best, Gemini 4 Argon at high effort, reaches pass@1 42.42% and pass@2 44.67%, solving 109 of 244 in at least one of two attempts
- Scanned, no AI statement or 'no LLM tools': Gersho's conjecture, local BCC crystallization in 3D quantization (Aydın, Iacobelli; 2610.28547: 'No LLM tools have been used'); fractional Michael–Simon counterexample (2610.27973); Kohayakawa–Prömel–Rödl conjecture (2610.27478); Tóth–Varolin counterexample (2610.27660); Hamel–Nadirashvili for Fisher-KPP (2610.27896); Merino–Welsh matroid counterexamples (2610.27706); Alperin weight conjecture elements (2610.27988); Ho–Nguyen conjecture for Gaussian mixtures (2610.27265)

#### What happened
Wednesday's arXiv announcement (21 Oct; submissions of 19–20 Oct) had about 40 new papers that disclose or mention AI use. We read
the full text of the uncited mathematics papers among them and skipped generic ML papers. Four results got their own entries:
- the MDS conjecture in even characteristic, with two independent proofs, one Lean-checked with Claude's help;
- Witsenhausen's double-cap problem, with a separate Lean proof built with AI tools;
- counterexamples to the quasitrace and monotone completeness conjectures, from an unreleased OpenAI model's manuscript;
- the Neumann–Zagier conjecture, first proved with ChatGPT 5.5 Pro and Opus 4.8.

A four-author survey also independently verified the twelve Hadamard matrices below 2000 and, with Claude, built sixteen more.

In the papers listed here, AI plays roles of three kinds:
- **The AI found the key object or proof**: Patkós's counterexample (ChatGPT 6.1 Sol), Watanabe's klt Fano classification
  ("most proofs … developed from arguments generated by OpenAI's ChatGPT"), and Patey's core lemma (Claude Opus 5.5).
- **AI developed parts of the argument**: Vince's conjecture (GPT Astra workflow), Pólya for shells, Dirac–Coulomb concentration,
  Hohenberg–Kohn and Cadilhac.
- **Follow-ups to OpenAI's 6 Oct release**: free entropy, the cube inequality, and the C*-algebra generation theorem.

The previous weekday's leftovers (13–17 Oct ML and physics papers that mention LLMs) are out of scope here.

No Hacker News thread or high-reach X post on these papers was found on 21 Oct.

#### Why it matters
Two classical problems (Segre's MDS conjecture in even characteristic, Witsenhausen's double cap) arrived with Lean proofs built
by AI agents on the day the preprints appeared. Mathematicians also keep building directly on OpenAI's model-generated
manuscripts, including one that has not been published.

Sources: [arXiv 2610.28029: Intersecting families with large covering number (Patkós)](https://arxiv.org/abs/2610.28029); [arXiv 2610.27676: Nef complexity of klt Fano varieties (Watanabe)](https://arxiv.org/abs/2610.27676); [arXiv 2610.27553: A proof of Vince's conjecture (Chen, Ma)](https://arxiv.org/abs/2610.27553); [arXiv 2610.27707: Pólya's conjecture for spherical shells and Aharonov–Bohm annuli (Li)](https://arxiv.org/abs/2610.27707); [arXiv 2610.28193: The concentration conjecture for Dirac–Coulomb operators (Ammari, Qiu)](https://arxiv.org/abs/2610.28193); [arXiv 2610.28395: Sprinkling for random regular graphs and Kim–Vu's sandwich conjecture (Portier)](https://arxiv.org/abs/2610.28395); [arXiv 2610.27643: The first-order part of the Ordered Variable Word theorem (Patey)](https://arxiv.org/abs/2610.27643); [arXiv 2610.27451: A 55-addition rank-23 scheme for 3×3 matrix multiplication (Sidebottom)](https://arxiv.org/abs/2610.27451); [arXiv 2610.27294: Counterexamples to the total-current Hohenberg–Kohn conjecture](https://arxiv.org/abs/2610.27294); [arXiv 2610.27243: On counterexamples to the unification of free entropy (Jekel)](https://arxiv.org/abs/2610.27243); [arXiv 2610.28390: A note on the cube inequality (Bi)](https://arxiv.org/abs/2610.28390); [arXiv 2610.28478: A single generation theorem for simple unital C*-algebras (Ma)](https://arxiv.org/abs/2610.28478); [arXiv 2610.27482: Rational polynomial solvability and stratifiability (Cadilhac)](https://arxiv.org/abs/2610.27482); [arXiv 2610.28536: Binding number of K_k-free graphs and the Brandt conjecture](https://arxiv.org/abs/2610.28536); [arXiv 2610.28079: Irrationality exponents of logarithmic numbers (Chudnovsky)](https://arxiv.org/abs/2610.28079); [arXiv 2610.27799: New Directions in Irrationality (Technion)](https://arxiv.org/abs/2610.27799); [arXiv 2610.27046: ReuF2F benchmark](https://arxiv.org/abs/2610.27046); [arXiv 2610.28124: A proof of the Janson–Sós conjecture](https://arxiv.org/abs/2610.28124)

### 2026-10-21: Tristan Buckmaster's guest essay on Terence Tao's blog: AI companies 'strip mine' mathematicians' data; calls for opt-in training, zero-retention deals and open-weight infrastructure

***; NYU; Science & math; confidence high; https://postcutoff.com/e/2026-10-21-buckmaster-mathematics-subject-for-the-young/

On Oct 21, 2026 Terence Tao's blog published a guest essay by NYU mathematician Tristan Buckmaster arguing that mathematics is at a "Deep Blue–Kasparov moment" and will be "unrecognizable" within a few years. He accuses AI companies of a conflict of interest in competing with their own customers using their data, criticises OpenAI's Oct 6 manuscript release and its "three hours" claim, and proposes opt-in training, zero-data-retention institutional agreements, open-weight infrastructure and a new credit system.

- Expands on Buckmaster's Oct 19 talk at the Institute for Advanced Study; Tao notes the post was converted from another format using AI, and Buckmaster discloses LLM help with editing
- On cost: OpenAI's Navier–Stokes proof 'reportedly required the coordination of 10,000 agents over 88 hours, at an estimated cost in the millions'
- Charge: 'AI companies, who strip mine our data, including unpublished work, private notes and research grant applications, have an egregious conflict of interest in competing against their own customers with unlimited resources and internal models unavailable to the public'
- On the Oct 6 release: manuscripts 'mostly unreadable and in some cases false'; the average-three-hours claim was 'essentially meaningless' because failed attempts and earlier agent work were not counted; no due diligence on building on customers' unreleased work, 'some of whom were graduate students and postdocs who were literally days away from releasing their own work'
- Says OpenAI's Millennium-problem push was, by OpenAI's own account, spurred by rumours that a rival had solved two of them; after Navier–Stokes, teased further Millennium problems 'yet they achieved none'
- Proposals: collective enterprise agreements with zero-data-retention and audit clauses for all academics; opt-in (not opt-out) training on user data; ban on absorbing a whole session when a user rates one reply; independently funded open-weight infrastructure; credit for exposition, shared tools and detailed AI disclosures
- 'Chess did not end in 1997. Mathematics will not end in 2026.'

#### What happened
Buckmaster, whose Euler blow-up result with Levent Alpöge came out the night before OpenAI's Navier–Stokes announcement and who was in
a priority dispute with OpenAI, set out the case that AI labs' data practices threaten academic mathematicians. Publication on Tao's
blog gives it wide reach in the field.

#### Why it matters
It is the most detailed public statement from a leading mathematician involved in the 2026 AI-maths results about privacy, credit and
the conduct of labs, with concrete policy demands. His claims about OpenAI are his own and have not been answered in the sources here.

Sources: [Terence Tao's blog: Mathematics is a subject for the young (guest post by Tristan Buckmaster)](https://terrytao.wordpress.com/2026/10/21/mathematics-is-a-subject-for-the-young/)

### 2026-10-21: Thousands of GPT-Rosalind agents analyse millions of mouse brain cells in 32 minutes and propose a cholesterol-transport lead for Perry syndrome

***; OpenAI, PerturbAI, Scripps Research; Science & math; confidence medium; https://postcutoff.com/e/2026-10-21-gpt-rosalind-agents-perry-syndrome-lead/

OpenAI, PerturbAI and Scripps Research said on Oct 21, 2026 that thousands of GPT-Rosalind agents, working in parallel, linked gene-expression data from millions of mouse brain cells with the literature in 32 minutes. They proposed that disrupting DCTN1 (the Perry syndrome gene) raises cholesterol-synthesis genes because cholesterol is not reaching where neurons need it, a hypothesis not yet tested in the lab. The partners are expanding the work to look for drug targets.

- OpenAI (X, 1.26M views): 'a team of thousands of GPT-Rosalind agents worked in parallel, connecting gene-expression results from millions of mouse brain cells with scientific literature in just 32 minutes'
- Finding: in mouse-brain data, disrupting DCTN1 increased expression of cholesterol-synthesis genes; GPT-Rosalind suggested cholesterol may not be reaching where neurons need it
- Status: a hypothesis 'to test'; no experimental validation reported in the post
- Partners: PerturbAI and Scripps Research; collaboration being expanded to identify new drug targets
- Hacker News: 146 points

#### What happened
OpenAI showed GPT-Rosalind used as a swarm: thousands of agents split a single-cell perturbation dataset and the literature, then
converged on one mechanistic hypothesis. The OpenAI page returned HTTP 403 to our fetcher, so details come from OpenAI's X post;
confidence is medium until the write-up is read.

#### Why it matters
It is a large public example of massively parallel agent science in biology. The value depends on lab follow-up, which has not
happened yet.

Sources: [OpenAI: Using thousands of AI agents to uncover new leads in biological data](https://openai.com/index/scaling-ai-research-for-complex-biological-data); [OpenAI on X (1.26M views)](https://x.com/OpenAI/status/2112899854569762936); [Hacker News discussion](https://news.ycombinator.com/item?id=50180752)

### 2026-10-22: Eleven probabilists construct the Critical 2D KPZ process, a solution of the critical two-dimensional KPZ equation, 'through extensive interactions' with ChatGPT 5.5, 5.6 and 6

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-22-critical-2d-kpz-process-chatgpt/

arXiv 2610.31614 (22 Oct 2026) by Berger, Caravenna, Clark, Dunlap, Gu, Liu, Nakajima, Sun, Tsai, Xie and Zygouras constructs the "Critical 2D KPZ Process", a non-trivial distribution-valued solution of the ill-defined two-dimensional KPZ equation at criticality, as the limit of centred log partition functions of 2D directed polymers and a measurable function of the Critical 2D Stochastic Heat Flow. It is non-Gaussian. The authors say the results "were obtained through extensive interactions with generative AI (ChatGPT 5.5, 5.6 and 6)" and call the paper a "community project" representing more than a decade's collective effort. Unrefereed.

- Result: centred log partition functions of 2D Gaussian directed polymers at criticality converge in law to a unique limit, which is a measurable function of the Critical 2D Stochastic Heat Flow via a renormalised Cole–Hopf transform; the limit is non-Gaussian, its covariance has a log-log blow-up near the diagonal, and Gaussian fluctuations appear when averaging on shrinking scales
- Authors include the groups behind the Critical 2D Stochastic Heat Flow (Caravenna, Sun, Zygouras) and KPZ specialists (Yu Gu, Li-Cheng Tsai, Alexander Dunlap, Quentin Berger, Shuta Nakajima)
- AI statement (verbatim): 'The results presented here were obtained through extensive interactions with generative AI (ChatGPT 5.5, 5.6 and 6) … The proof builds on key recent and older advances in our field, combined in novel ways with substantial AI input.'
- Appendix C: AI 'contributed to the tightness argument, the proof of convergence to a unique limit, and its non-Gaussianity'. From a Gaussian convexity theorem supplied by the authors, 'AI derived a covariance bound' giving tightness; an AI 'sparse switch-off construction' 'initially appeared to close the remaining estimate, but a subsequent audit identified a gap' and was abandoned; asked for simpler arguments, AI 'proposed a simpler use of conditional Gaussian convexity' that completed the convergence proof; AI 'developed a proof using Gaussian rigidity' for non-Gaussianity
- Division of labour (Appendix C): 'We supplied structural ideas, assessed the proposed arguments, and guided the choice and revision of approaches. AI developed these suggestions and generated further proof ideas that were not contained in the original prompts. Some proposals were fruitful; others contained errors or introduced unnecessary complications.'

#### What happened
The KPZ equation in two dimensions has no classical meaning, and at the critical disorder strength the usual renormalisation tools fail. Over the past decade Caravenna, Sun and Zygouras built the Critical 2D Stochastic Heat Flow, the natural candidate for the exponential of a solution. The new paper ([arXiv 2610.31614](https://arxiv.org/abs/2610.31614), 22 Oct) takes its logarithm: the centred log partition functions of critical 2D directed polymers converge to a unique limit, the "Critical 2D KPZ Process", which is non-Gaussian.

Eleven authors sign it. Appendix C describes how the work was done: the humans proposed starting points (Gu–Tsai's block factorisation, Chen's Gaussian convexity theorem, an earlier decomposition), and ChatGPT 5.5, 5.6 and 6 turned them into the tightness proof, the convergence argument and the non-Gaussianity proof. One AI-developed construction was abandoned after an audit found a gap.

No expert reactions were found on 23 Oct (X not searched).

#### Why it matters
This is one of the clearest cases so far of a leading research group in a mainstream field crediting a general chatbot with the core technical steps of a major theorem, while documenting a failed AI attempt too. It is still an unrefereed preprint.

Sources: [arXiv 2610.31614: The Critical 2D KPZ Process](https://arxiv.org/abs/2610.31614)

### 2026-10-22: Barry Simon's L² conjecture (Problem 8 of his 2000 list) disproved: Becker and Yang build a counterexample largely found with ChatGPT

****; OpenAI; Science & math; confidence medium; https://postcutoff.com/e/2026-10-22-simon-l2-conjecture-disproved-chatgpt/

On 22 Oct 2026 Simon Becker and Mengxuan Yang posted arXiv 2610.31637, a counterexample to Barry Simon's multidimensional L² conjecture (Problem 8 in "Schrödinger operators in the twenty-first century", 2000). For every d ≥ 2 they construct a smooth, bounded, nonnegative "lens-type" potential with arbitrarily small sup norm and weighted L² integral such that −Δ + V has only pure point spectrum on (0, ∞), so not even singular continuous spectrum survives. The construction was developed in conversations with ChatGPT and then rewritten by the authors. Unrefereed preprint.

- Conjecture (Simon 2000, Problem 8; Simon 2019, Conjecture 20.2): if ∫|x|^{1−d}|V(x)|² dx < ∞ then −Δ + V has absolutely continuous spectrum of infinite multiplicity at almost every positive energy. The 1D analogue is the Deift–Killip theorem (1999)
- Theorem 1.1: for d ≥ 2, 0 ≤ σ < 1/3 and δ > 0 there is a smooth bounded nonnegative V with ‖V‖∞ + I_{d,0}(V) + I_{d,σ}(V) < δ such that σ(H_V) = σ_ess(H_V) = [0, ∞) and the continuous spectral projection is zero; the positive eigenvalues are dense
- Construction: potentials on widely separated annuli, each a weak angular 'lens' that focuses waves into narrow channels plus bounded periodic 'reflecting barriers' whose spectral gaps cover a prescribed energy window; pure point spectrum via Combes–Hislop spectral averaging; 3D by cylindrical extension, all d by tensor sums
- AI disclosure (verbatim): 'Our interest in this conjecture was renewed when it remained among the few of Simon's problems left untouched by OpenAI … We developed the first construction of the potential in discussions with ChatGPT, obtaining a proof of absence of absolutely continuous spectrum in a bounded interval … When prompted with this question, ChatGPT showed that a rather intricate adjustment of the parameters of the potential … could yield point spectrum in a bounded interval through a mechanism very similar to the one used in the final proof and already including the spectral-averaging argument. Very much to our surprise, ChatGPT was then able to extend this result to the full spectrum by pushing the parameters of the construction further. We thoroughly rewrote the resulting argument and manuscript'
- ChatGPT model version not stated. Simon Becker earlier used GPT-6.1 Sol for counterexamples to Fried's conjecture (October 2026)

#### What happened
In 2000 Barry Simon published fifteen open problems for Schrödinger operators. Problem 8 asked whether the condition ∫|x|^{1−d}|V|² < ∞, which by the Deift–Killip theorem guarantees absolutely continuous spectrum in one dimension, does the same in higher dimensions. Simon restated it as Conjecture 20.2 in 2019.

Simon Becker and Mengxuan Yang ([arXiv 2610.31637](https://arxiv.org/abs/2610.31637), 22 Oct) show it is false in every dimension d ≥ 2. Their potential is a sum of annular blocks: a weak "lens" bends waves into narrow channels that end at bounded periodic barriers whose spectral gaps cover a given energy window. Spectral averaging then gives pure point spectrum, dense in [0, ∞), so the operator has no continuous spectrum at all, even under the stronger weighted condition with |x|^σ, σ < 1/3.

The authors write that they returned to the problem because it "remained among the few of Simon's problems left untouched by OpenAI". ChatGPT produced the first construction, then the mechanism used in the final proof, and finally the extension to all energies. The authors rewrote the argument and take full responsibility for it.

No expert reactions were found on 23 Oct (X and blogs not searched).

#### Why it matters
Simon's 2000 list is one of the standard problem lists of mathematical physics. This is a clean negative answer to one of its problems, with the AI credited for the decisive steps, not just for checking. It also shows researchers choosing targets by what OpenAI's 6 Oct release did *not* cover.

Sources: [arXiv 2610.31637: Becker, Yang, Lenses make many points: pure point spectrum disproves Simon's L² conjecture](https://arxiv.org/abs/2610.31637)

### 2026-10-22: AI-assisted results in the 22 Oct 2026 arXiv listing: the gradient conjecture at infinity with GPT-6 Astra, Claude-built zero-knowledge PCPs, nucleon algorithms with GPT Sol 6.1 and Fable 5.1, Kelly's Condorcet conjectures

***; OpenAI, Anthropic, Google DeepMind, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-22-ai-assisted-results-october-22-listing/

Catalogue of AI-disclosing papers in the 22 Oct 2026 arXiv announcement (submitted 20–21 Oct). Own entries: Tevelev's re-proof of OpenAI's Nagata proof, Anghel's survey and generalization of OpenAI's counterexample to Hochster's small Cohen–Macaulay conjecture, and Voronov's GPT-6 Astra-assisted counterexample to Gale's conjecture in R³. Gouéré's exposition of the Kozma–Nitzan proof was added to the dying-percolation entry. The rest is listed here.

- Own entries: Nagata (2610.30050), Hochster small CM (2610.30027), Gale in R³ (2610.28828). Enriched: dying percolation (Gouéré, 2610.30063)
- Gradient conjecture at infinity and the definable Thom gradient conjecture (Thang Do, Arnulf Jentzen, Benno Kuckuck; 2610.29211): every gradient trajectory of a definable C¹ potential has finite spherical length. 'The authors gratefully acknowledge extensive assistance from OpenAI's GPT-5.6 Sol and GPT-6 Astra/Sol … used via ChatGPT/Codex as research collaborators for developing mathematical arguments and ideas, exploring proof strategies, testing arguments, identifying gaps and possible counterexamples'; all proofs 'independently checked by the authors'
- Quasilinear-size constant-query zero-knowledge PCPs (Inhyuk Park, Chenglin Fan; 2610.29066), improving the Õ(T+b²) length of Zeilberger–O'Connor–Gur–Spooner (2026): 'We used Anthropic's Claude extensively, under the direction of the authors, to assist with developing the constructions and the proofs, with checking the proofs, with the computations reported in Appendix B, and with drafting this paper. Its use materially affected every section of the paper'
- Nucleon of matching games (Haris Aziz, Gergely Csáji; 2610.29731), resolving three open problems from a recent survey (polynomial algorithm for matching games; NP-hardness for b-matching and partitioned games): 'The authors acknowledge the assistance of OpenAI's GPT Sol 6.1 and Anthropic's Fable 5.1 in working out the proof details of the combinatorial algorithm.'
- Kelly's two monotonicity conjectures on Condorcet winner probabilities for odd electorates (Seick Kim, Hocheol Shin; 2610.29904): 'Large language models (LLMs) were used to explore proof ideas and assist with computational checks, literature searches, and language polishing.'
- Functional limit theorems for self-interacting random walks (Kosygina, Marêché, Mountford, Peterson; 2610.28935), solving a problem open since Tóth's mid-90s work: 'The literature search for these results with large language models (Gemini, Google; Claude, Anthropic) produced their initial proof drafts instead of references. We verified, corrected, and rewrote these proofs' (appendix lemmas only)
- Rayleigh's conjecture for clamped plates, a simpler planar proof of Nadirashvili's theorem plus nonlinear extensions (Dorin Bucur, Richard Laugesen; 2610.29840): 'The authors used OpenAI's ChatGPT through Codex to explore the theorems and conjectures, check various endpoint arguments, locate and verify references, and prepare this manuscript and its figures.'
- Supporting roles (checking, exploring, drafting): conjugacy problem for virtually soluble groups of finite Prüfer rank (2610.29563, ChatGPT 'for checking mathematical arguments, exploring proof strategies'); Hong–Krahn–Szegő sharp stability (2610.28757); gradient absorption prevents blow-up (2610.28913, ChatGPT); low-degree filtrations in Grassmannian cohomology (2610.28780, Codex for 'exploration of some of the proof strategies and calculations'); pivot-based distance oracles in sparse graphs (2610.29624, ChatGPT and Claude); local Gan–Gross–Prasad epsilon dichotomy (2610.29483: 'The draft of this paper was produced from handwritten notes with GPT 5.6, and GPT Astra was used for proofreading')
- No substantive AI role: Velani's conjecture on dyadic approximation in the Cantor set (Sam Chow, Han Yu; 2610.28839: 'This work was completed entirely without AI', apart from a figure and search results); Ward's conjecture on Griesmer codes (2610.29275), Kemper–Malle conjecture (2610.29849), Schäffer's conjecture for even k ≤ 42 (2610.28768), eventual stability counterexample (2610.28688): no AI statement in the sweep's scan

#### What happened
Three results from Thursday's announcement have their own entries:
- Jenia Tevelev's re-proof of OpenAI's Nagata proof, which he calls "a stunning achievement";
- Cristian Anghel's survey of the counterexamples to Hochster's small Cohen–Macaulay conjecture, which gives a complete proof of OpenAI's theorem and extends it to every dimension;
- Vsevolod Voronov's 101-point counterexample to Gale's conjecture, found with algorithms developed with GPT-6 Astra.

Jean-Baptiste Gouéré's exposition of OpenAI's Kozma–Nitzan proof was added to the dying-percolation entry.

In the papers listed here, AI plays two kinds of role:
- **AI as an extensive collaborator**: the gradient conjecture at infinity (GPT-5.6 Sol and GPT-6 Astra/Sol), quasilinear zero-knowledge PCPs (Claude "materially affected every section"), the nucleon algorithm (GPT Sol 6.1 and Fable 5.1 worked out proof details), and appendix lemmas that Gemini and Claude drafted when asked for references.
- **Support**: exploring ideas, checking, drafting and editing (Kelly's conjectures, Rayleigh's conjecture, Gan–Gross–Prasad and others).

Older items still listed in the sweep (14–19 Oct ML, physics and materials papers that only mention LLMs) are out of scope here.

#### Why it matters
Mathematicians in the fields concerned are now writing up OpenAI's 6 Oct claims (Nagata, Hochster, Kozma–Nitzan), and they speak openly about the editorial burden this places on them. Meanwhile, disclosures of AI as a central collaborator on long-standing conjectures have become routine.

Sources: [arXiv 2610.29211: On the gradient conjecture at infinity (Do, Jentzen, Kuckuck)](https://arxiv.org/abs/2610.29211); [arXiv 2610.29066: Quasilinear-size constant-query zero-knowledge PCPs (Park, Fan)](https://arxiv.org/abs/2610.29066); [arXiv 2610.29731: Computing the nucleon of matching games (Aziz, Csáji)](https://arxiv.org/abs/2610.29731); [arXiv 2610.29904: Kelly's conjectures on Condorcet winner probabilities (Kim, Shin)](https://arxiv.org/abs/2610.29904); [arXiv 2610.28935: Functional limit theorems for self-interacting random walks](https://arxiv.org/abs/2610.28935); [arXiv 2610.29840: Rayleigh's conjecture for vibrating plates (Bucur, Laugesen)](https://arxiv.org/abs/2610.29840); [arXiv 2610.29563: Conjugacy problem for virtually soluble groups of finite Prüfer rank](https://arxiv.org/abs/2610.29563); [arXiv 2610.28757: Sharp quantitative stability of the Hong–Krahn–Szegő inequality](https://arxiv.org/abs/2610.28757); [arXiv 2610.28913: Strong gradient absorption and initial decay prevent blow-up](https://arxiv.org/abs/2610.28913); [arXiv 2610.28780: Low-degree filtrations in Grassmannian cohomology](https://arxiv.org/abs/2610.28780); [arXiv 2610.29624: Tight space bounds for pivot-based distance oracles](https://arxiv.org/abs/2610.29624); [arXiv 2610.29483: Local Gan–Gross–Prasad conjecture: epsilon dichotomy](https://arxiv.org/abs/2610.29483); [arXiv 2610.28839: On a conjecture of Velani (Chow, Yu)](https://arxiv.org/abs/2610.28839)

### 2026-10-22: Science publishes Microsoft's MAI-DxO 'Sequential diagnosis with language models': 83% on complex NEJM cases vs 36% for generalist doctors; Suleyman chases 'medical superintelligence'

***; Microsoft; Science & math; confidence high; https://postcutoff.com/e/2026-10-22-microsoft-mai-dxo-science-sequential-diagnosis/

On Oct 22, 2026 Science published Microsoft AI's "Sequential diagnosis with language models", the peer-reviewed version of the MAI Diagnostic Orchestrator (MAI-DxO), in which several model agents act like a panel of doctors that question the patient, order tests and debate a diagnosis. On simulated complex New England Journal of Medicine cases it was right 83% of the time, against 36% for primary-care doctors and general internists working alone; GPT-5.5 went from 72% to nearly 87% with the orchestrator. Mustafa Suleyman framed it as a step toward "medical superintelligence"; a real-world study at Beth Israel Deaconess is planned. All results are simulations.

- Paper: 'Sequential diagnosis with language models', Science, Oct 22, 2026 (DOI 10.1126/science.aeh7930)
- Accuracy on complex NEJM cases (TechCrunch): MAI-DxO 83%; primary-care doctors and general internists working alone with reference materials 36%. Specialists and physician teams, who normally handle such cases, were not compared
- Model-agnostic: tested with OpenAI, Google, Anthropic, Meta and DeepSeek models; GPT-5 alone 78.9% at ~$8,400 per case, with DxO ~4 points higher and ~$3,000 cheaper; GPT-5.5 72% → nearly 87% at slightly lower cost; Gemini 2.5 Flash 46.4% → ~69%
- Limits: on emergency-department cases DxO added little (GPT-5.5 ~85% with or without it); most of the gain comes from the base models, and gains shrink as models near the ceiling; all tests are simulations, not real patients
- Next steps (Suleyman): a prospective study with Beth Israel Deaconess Medical Center (consenting patients also consult DxO; its recommendations compared with doctors'); a healthcare foundation model with Mayo Clinic due next year; he predicts a breakthrough in forecasting how medical records unfold by 2027

#### What happened
Science published Microsoft AI's diagnostic-orchestrator paper on Oct 22. The system turns a language model into a virtual panel that
asks questions, orders tests with costs attached and argues toward a diagnosis on cases from the New England Journal of Medicine.
Microsoft AI CEO Mustafa Suleyman used the publication to restate the goal of "medical superintelligence".

#### Why it matters
The result is now peer-reviewed, and it is model-agnostic: the orchestrator lifts every frontier model tested. The paper's own data
also shows its limits: most of the gain comes from better base models, it adds little on emergency cases, and the doctor comparison
excludes specialists. The Beth Israel study will be the first test on real patients.

Sources: [Science: Sequential diagnosis with language models (Oct 22, 2026)](https://www.science.org/doi/abs/10.1126/science.aeh7930); [TechCrunch: Microsoft's Mustafa Suleyman says its AI outperforms doctors on hard cases as he chases 'medical superintelligence'](https://techcrunch.com/2026/10/22/microsofts-mustafa-suleyman-says-its-ai-outperforms-doctors-on-hard-cases-as-he-chases-medical-superintelligence/); [The Verge: Microsoft says its AI tool is better than doctors at diagnosing conditions](https://www.theverge.com/ai-artificial-intelligence/1016690/suleyman-microsoft-diagnostic-orchestrator-science-medical-superintelligence)

### 2026-10-22: ChatGPT 6 Astra outlines the argument that removes the Hodge-conjecture assumption from Nakayama's local height asymptotics

***; OpenAI; Science & math; confidence high; https://postcutoff.com/e/2026-10-22-nakayama-local-height-without-hodge-conjecture-chatgpt-astra/

Yuta Nakayama's preprint (arXiv 2610.30574, 22 Oct 2026) proves his earlier asymptotic formula for the Archimedean local Beilinson–Bloch height in a degenerating family without assuming the Hodge conjecture (and drops other minor assumptions). He states that "The argument was outlined by ChatGPT 6 Astra", which also found a similar p-adic theorem of Scholl; his advisor S.-W. Zhang had suggested removing the Hodge conjecture "using AI". It does not prove the Hodge conjecture.

- Result: for a family X → S of smooth projective varieties over a Riemann surface degenerating at s, and cycles W, Z of complementary codimension that are disjoint in the generic fibre and homologically trivial in a general fibre, h(t) − ⟨W, Z⟩_{a,s} log|t| = O(1), where h is the Archimedean part of the Beilinson–Bloch height pairing and ⟨W, Z⟩_{a,s} Beilinson's local intersection index. The author's earlier paper [Nak25] needed the Hodge conjecture for this
- Method: uses 'the cohomological nature of Beĭlinson's local intersection index rather than arguing motivically' (Hain's blended extensions); the paper is 5 pages
- Disclosure: 'AI usage. The argument was outlined by ChatGPT 6 Astra. The author worked out the remaining minor details and wrote the manuscript. ChatGPT 6 Astra proofread the manuscript.'
- Introduction: 'We remove unnecessary assumptions in this paper with the help of AI. It identified a similar p-adic theorem by Scholl [Sch94, Theorem 5.3].' Acknowledgments: 'The author thanks his advisor S.-W. Zhang for suggesting that the author should remove the Hodge conjecture using AI.'
- Not the Hodge conjecture itself: the conjecture was an assumption of one theorem, now shown to be unnecessary
- Status: single-author unrefereed arXiv preprint; no formal proof; no expert reaction found on 24 Oct

#### What happened
Yuta Nakayama posted a five-page note on arXiv (math.AG, 22 Oct 2026) that removes the Hodge conjecture from the
hypotheses of an asymptotic formula for the Archimedean part of the Beilinson–Bloch height pairing, which he proved in
an earlier paper. In that paper the Hodge conjecture was "the most substantial assumption", used to modify a cycle
so that a line bundle of Bloch could be defined. The new argument is cohomological, via Hain's blended extensions of
variations of Hodge structure.

The AI statement is unusually direct: "The argument was outlined by ChatGPT 6 Astra. The author worked out the
remaining minor details and wrote the manuscript." The model also found a similar p-adic theorem of Scholl (1994), and
the acknowledgments say his advisor S.-W. Zhang suggested he "should remove the Hodge conjecture using AI".

#### Why it matters
It is another case of a model doing the key step of an arithmetic-geometry argument, with the human doing details and
writing. It also shows senior mathematicians now telling students to try AI to remove hypotheses. The title is easy to
misread: the Hodge conjecture itself is not proved, it is only no longer needed for this one result.

Sources: [arXiv 2610.30574: Asymptotics of local height pairing without Hodge conjecture (Yuta Nakayama)](https://arxiv.org/abs/2610.30574); [PDF (AI usage statement on p. 2)](https://arxiv.org/pdf/2610.30574)

### 2026-10-23: AI-assisted results in the 23 Oct 2026 arXiv listing: DeepMind's Gemini 4 Argon loop on Lehmer's problem, Demmel's Claude Code-built MRRR SVD solver, a Claude–GPT agent harness proof with Lean, GPT-6 Astra on k-Hessian and Steklov problems

***; Google DeepMind, OpenAI, Anthropic, Harmonic, various mathematicians; Science & math; confidence medium; https://postcutoff.com/e/2026-10-23-ai-assisted-results-october-23-listing/

Catalogue of AI-disclosing papers in the 23 Oct 2026 arXiv announcement (submitted 21–22 Oct), plus leftovers. Own entries: Becker–Yang's disproof of Simon's L² conjecture, the Critical 2D KPZ process, and Zhentao Li's exposition of OpenAI's Barnette proof. Enriched: Lekili–Sheridan's sign-error diagnosis (Hodge entry), Banks's number-field Siegel-zero paper, the Chudnovskys' CM-period generalization, Bellotti–Kerr's Gauss sphere bound (quasi-RH entry). The rest is listed here.

- Own entries: Simon's L² conjecture (2610.31637), Critical 2D KPZ (2610.31614), Barnette exposition (2610.30490). Enriched: Hodge (Lekili–Sheridan, 2610.30976), Landau–Siegel (Banks, 2610.31402), lemniscate (Chudnovsky, 2610.31112), quasi-RH (Bellotti–Kerr, 2610.29636)
- Lehmer's problem (Yuri Chervonyi, Sergei Gukov, Thang Luong; Google DeepMind; 2610.30277): a 'human–AI discovery loop' with Gemini in the Google Antigravity agent environment derives a five-parameter 'Generalized Boyd algorithm' giving irreducible polynomials with Mahler measure below Boyd's limit point α1 ≈ 1.255434 for every target degree; across even degrees up to 2000 it improves six published values and holds 99.6% of best-known records. 'Throughout this work, we used internal versions of Gemini, including Gemini 4 Argon.' Does not settle Lehmer's question
- MRRR for the bidiagonal SVD (James Demmel, Inderjit Dhillon, Duvvuri, Julien Langou, Osni Marques, Ryan Schneider; 2610.30227): DBDSVDMR3, an O(kn) MRRR-based bidiagonal SVD solver built from LAPACK 3.0's DSTEGR, after two earlier attempts were abandoned as unreliable. 'Our implementation … was obtained with the help of Claude Code'; an expert 'did not write the Fortran but fixed the mathematical constraints through prompts'; 'developed and analyzed with substantial assistance from large-language-model agents (Anthropic Claude) operating in evaluator-gated loops'
- Global convergence of detailed balanced mass-action systems (Lv, Ni, Wang, He; BitMiracle AI Lab; 2610.30808): unique positive equilibrium per class and convergence of every positive trajectory, with a ~9,300-line Lean 4 formalization (7,900 lines adapted from OpenAI's persistence formalization). 'The proofs, the Lean development, and the manuscript were produced in ClawsGO Science, an agent harness … Claude Opus 5.5 made the research decisions, GPT 6 Astra carried them out, and the Lean kernel checked every formal step.' Builds on OpenAI's claimed persistence theorem for weakly reversible systems
- k-Hessian eigenfunctions (Guohuan Qiu, Jin Yan; 2610.31445): smoothness up to the boundary for 2 ≤ k < n on uniformly convex domains, 'an open problem in the regularity theory of degenerate fully nonlinear elliptic equations'. 'Through extensive discussions with GPT-6 Astra over about a month, we resolved the analytic difficulties … Most of the blow-up techniques used in the proof were suggested by GPT.'
- Steklov eigenfunctions on curvilinear polygons (Jeffrey Galkowski, Ruoyu P. T. Wang; 2610.31387): optimal lower bounds. 'The proof in this paper was obtained through an interaction with ChatGPT 6 Astra … ChatGPT suggested the ideas for a proof based on a regularized trace and the singularity in the kernel of the DtN map near a corner'; the authors then simplified it with a conformal approach
- Port-based teleportation (Anurag Anshu, Quynh T. Nguyen; 2610.30761): an elementary polylog-depth PBT circuit and a simple proof of unitary parallelization. 'In a few prompts, ChatGPT Astra came up with the protocol in this note, in particular the remarkable single reflection teleportation protocol.'
- Pólya's conjecture for Neumann eigenvalues of spherical shells (Yutian Li; 2610.31464), every dimension, radius ratio and energy: ChatGPT and Claude used 'to assist with mathematical exploration and proof development, checking intermediate arguments and calculations, literature searches, and drafting'
- Littlewood's L^α-flat theorem (el Houcein el Abdalaoui; 2610.30797): argues that OpenAI's ultraflat-polynomial counterexample (family 076) uses hyper-lacunary coefficients that violate an implicit assumption of Littlewood's 1961 theorem, so it 'offers only a limited refutation'
- Fuglede's conjecture in F₂⁸ and F₁₁³ (Ferguson, Iosevich; 2610.30712): 'The authors acknowledge the use of OpenAI's ChatGPT for assistance in writing the computer programs used in this work and for collaboration in drafting'
- Degenerations of Jordan algebras (Bovdi, Kaygorodov, Popov, Saydaliyev, Volkov; 2610.31179): all degenerations of 4-dim Jordan and 5-dim nilpotent Jordan algebras, 'formally verified in the proof assistant Lean 4'; 'Aristotle (https://aristotle.harmonic.fun/) acted as a non-human research assistant, assisting in error verification and proof simplification'
- Supporting roles: Cheng–Murray conjecture on deep holes of Reed–Solomon codes, all remaining non-exceptional cases (Weijun Fang, 2610.31531; GPT-6 for 'checking some of the main proofs'); Lipschitz retractions of finite-subset spaces settling open problems (Kovalev, 2610.31555; ChatGPT and Claude for 'mathematical exploration'); Li–Zhao conjecture on partitions (2610.30700, ChatGPT 'exploratory discussions'); 1-extension and rigidity (Heaton, Winter, 2610.31212: 'an iterative process that included important ideas from both humans and LLM'); Linearity Conjecture for blocking sets, p^e ≠ 3, 4 (Peter Sziklai, 2610.30503: 'assisted by AI tools, in parts of mathematical exploration, computations and drafting'); 2D inhomogeneous Euler well-posedness (Suleiman, 2610.30096: quasiconformal approach 'found in conversation with Gemini Pro'); Gamma conjectures for Milnor hypersurfaces (2610.31150, ChatGPT for proofreading and minor checks)
- Conway's thrackle conjecture: new upper bound ⌊4(n−1)/3⌋ edges (Dániel Gábor Simon, 2610.31589); a computational result 'also double-checked by an AI agent', otherwise human
- Repository only (no paper or post found on 23 Oct): AxiomMath/RiemannJensen (created 23 Oct), a Lean formalization that 'almost all zeros of the Jensen polynomials of the Riemann and Dedekind xi functions are distinct negative reals': for d ≥ 2 the degree-d Riemann Jensen polynomial has at most C√(d log d) nonreal roots, and similarly for Dedekind zeta of every number field; checked against a Mathlib-only challenge file with the Lean comparator. The README does not say how much was AI-generated (Axiom builds AxiomProver)
- Software and physics: GEM-PIC-RZ laser-plasma PIC code (Alexander Pukhov, 2610.31622): 'Claude generated the code, tests, benchmark scripts, and documentation, while the author directed the work and did not write or edit the source code'; Lean-QuantumAlg framework for AI-produced quantum algorithms (2610.30216)
- No substantive AI role or no statement: {2,4}-distance code conjecture for q = 3, 4 (2610.30615, verification code only); Pfaffian recognition in coRP, open since Vazirani–Yannakakis 1989 (2610.29817, ChatGPT for language only); TSP local-search gaps (2610.31513, 'without the use of AI tools'); no perfect randomized composition (2610.31638), Bugeaud–Evertse complexity exponent 1/4 (2610.30937), Breiman's conjecture, slowly varying case (2610.29551): no AI statement found by the sweep

#### What happened
Three results from Friday's announcement have their own entries:
- Simon Becker and Mengxuan Yang's counterexample to Barry Simon's L² conjecture, whose key mechanism ChatGPT found;
- the Critical 2D KPZ process, built by eleven probabilists "through extensive interactions" with ChatGPT 5.5, 5.6 and 6;
- Zhentao Li's hand-written, partly re-proved version of OpenAI's proof of Barnette's conjecture.

Four papers were added to existing entries: Lekili and Sheridan's explanation of the sign error that sank OpenAI's three withdrawn Hodge papers, Banks's extension of the Siegel-zero argument to number fields (initial argument by Claude), the Chudnovskys' generalization to CM elliptic periods, and Bellotti–Kerr's Gauss sphere bound, which uses OpenAI's quasi-GRH claim.

The next tier is in the key facts. Two stand out. Google DeepMind researchers, including Sergei Gukov, ran Gemini (including Gemini 4 Argon) in the Antigravity agent environment on Lehmer's Mahler-measure problem and set new records up to degree 2000. James Demmel's group used Claude Code to finish an MRRR bidiagonal SVD solver, something LAPACK developers had tried and abandoned twice. A small lab's "ClawsGO Science" harness, with Claude Opus 5.5 directing and GPT-6 Astra executing, produced a fully Lean-checked convergence proof for detailed-balanced reaction networks on top of OpenAI's persistence formalization.

#### Why it matters
The pattern of the past weeks continues: frontier chatbots are credited with core proof ideas in analysis and spectral theory (k-Hessian, Steklov, Simon's problem), and OpenAI's 6 Oct claims are now being both used (quasi-GRH, persistence) and audited (Hodge, Littlewood, Barnette) by outside experts.

Sources: [arXiv 2610.30277: Breaking the high-degree barrier in Lehmer's problem (Google DeepMind)](https://arxiv.org/abs/2610.30277); [arXiv 2610.30227: DBDSVDMR3, an MRRR-based bidiagonal SVD solver (Demmel et al.)](https://arxiv.org/abs/2610.30227); [arXiv 2610.30808: Global convergence of detailed balanced mass action systems (ClawsGO Science)](https://arxiv.org/abs/2610.30808); [arXiv 2610.31445: Boundary regularity of k-Hessian eigenfunctions](https://arxiv.org/abs/2610.31445); [arXiv 2610.31387: Lower bounds on Steklov eigenfunctions for curvilinear polygons](https://arxiv.org/abs/2610.31387); [arXiv 2610.30761: A simple proof of unitary parallelization via port-based teleportation](https://arxiv.org/abs/2610.30761); [arXiv 2610.31464: Pólya's conjecture for Neumann eigenvalues of spherical shells](https://arxiv.org/abs/2610.31464); [arXiv 2610.30797: OpenAI partially disproves Littlewood's 1961 L^α-flat theorem](https://arxiv.org/abs/2610.30797); [arXiv 2610.30712: Fuglede's conjecture in F₂⁸ and F₁₁³](https://arxiv.org/abs/2610.30712); [arXiv 2610.31179: Degenerations of Jordan algebras (Lean, Aristotle)](https://arxiv.org/abs/2610.31179); [arXiv 2610.31531: The Cheng–Murray conjecture on deep holes of Reed–Solomon codes](https://arxiv.org/abs/2610.31531); [arXiv 2610.31555: Lipschitz clustering in normed and barycentric spaces](https://arxiv.org/abs/2610.31555); [arXiv 2610.30700: Arithmetic-progression biases in random partitions (Li–Zhao conjecture)](https://arxiv.org/abs/2610.30700); [arXiv 2610.31212: An algebraic proof of 1-extension](https://arxiv.org/abs/2610.31212); [arXiv 2610.30503: The Linearity Conjecture (Sziklai)](https://arxiv.org/abs/2610.30503); [arXiv 2610.30096: 2D inhomogeneous incompressible Euler well-posedness](https://arxiv.org/abs/2610.30096); [arXiv 2610.31150: Gamma conjectures for Milnor hypersurfaces](https://arxiv.org/abs/2610.31150); [arXiv 2610.31589: Conway's thrackle conjecture, upper bound 4n/3](https://arxiv.org/abs/2610.31589); [arXiv 2610.31622: GEM-PIC-RZ (Claude-written PIC code)](https://arxiv.org/abs/2610.31622); [arXiv 2610.30216: Lean-QuantumAlg](https://arxiv.org/abs/2610.30216); [arXiv 2610.30615: {2,4}-distance code conjecture, ternary and quaternary cases](https://arxiv.org/abs/2610.30615); [arXiv 2610.29817: Pfaffian recognition is in coRP](https://arxiv.org/abs/2610.29817); [AxiomMath/RiemannJensen (Lean, GitHub)](https://github.com/AxiomMath/RiemannJensen)

### 2026-10-24: AI-assisted results in the 24 Oct 2026 sweep: ChatGPT 6 Astra removes a Hodge-conjecture assumption, Bake AI's Épi agent, Google's lab-in-the-loop aptamer agents, agentic turbulence closures

**; OpenAI, Google Research, Google DeepMind, Anthropic, Bake AI, various researchers; Science & math; confidence medium; https://postcutoff.com/e/2026-10-24-ai-assisted-results-october-24-listing/

Catalogue of AI-disclosing preprints from the 16–22 Oct 2026 arXiv listings not cited by earlier roundups, read on 24 Oct. Own entries: Yuta Nakayama's removal of the Hodge-conjecture assumption from his local height asymptotics ("The argument was outlined by ChatGPT 6 Astra") and Bake AI's Épi agent (120 claimed records). Listed here: Google's Gemini-driven lab-in-the-loop aptamer design, the DEScover agentic turbulence-closure framework, Lean Finder V2, and supporting-role disclosures.

- Own entries: Nakayama, local height pairing without the Hodge conjecture (2610.30574); Épi autonomous research agent (2610.24944)
- Lab-in-the-loop aptamers (Google Research, Stanford, Google Cloud, Google DeepMind; Cutts, …, Hassidim, Matias, Rao, Pawlosky; 2610.24806): agentic pipelines built on 'Google Gemini Pro and Flash models' designed aptamer binders against α-thrombin and myoglobin over three design–test–improve wet-lab rounds (~1,000 candidates per round, ~4,400 sequenced constructs). On-target hit rates on α-thrombin rose from 0.0% to 26.0% to 42.1%; on myoglobin 10.2% → 3.7% → 33.5%. Agent-engineered literature motifs reached apparent K_D 24.6–133 nM, while strictly de novo scaffolds stayed above 1 µM. The agents found cross-reactivity in the myoglobin screen (specificity fell from 75.0% to 19.1%) and restored it to 72.4%
- DEScover (Jiawei Chen, Xiang I.A. Yang, Ping He; Iowa State, Penn State; 2610.29850, under consideration at J. Fluid Mech.): four LLM agents (discovery, review, code implementation, evaluation) propose and test subgrid length scales for detached eddy simulation from a one-sentence modelling idea. On a flat plate, backward-facing step, free shear layer and iced airfoil, the best candidate beats the standard maximum-cell-size length scale and matches or slightly beats the expert shear-layer-adapted one; ~25 minutes per discovery run excluding CFD
- Lean Finder V2 (Simon Fraser, Rice, Pittsburgh; 2610.30843): Mathlib semantic search trained on adversarially generated short, ambiguous queries; ≥11% relative improvement on MathlibQR; plugged into Claude Code it raises proof success on FATE-H to 39.3% from the next-best 36.0%; frontier LLM judges (Claude Opus 5, GPT-5.6 Sol) rank it most helpful 18+% more often on real user queries
- Ashbaugh–Benguria stability (Long-Han Huang, Yuxuan Zhou, Wenming Zou; 2610.31223): sharp quantitative stability on bounded convex domains, optimal exponents 5/2 (Fraenkel asymmetry) and (n+4)/2 (Hausdorff distance); the quadratic estimate proposed as Open problem 7.65 in the cited survey fails. 'ChatGPT was used to assist in discussing certain mathematical details, checking mathematical arguments and references, and preparing the manuscript'
- Local dyadic conjecture (Alghasi, Guenin, Tunçel; Waterloo; 2610.29197): a local version of Seymour's dyadic conjecture for ideal clutters, with an important special case proved; 'The final draft of this manuscript was reviewed by Anthropic (Claude Fable 5.1, October 2026) and the authors edited the draft in response to such a review'
- AI-assisted addendum to a 2018 thesis (Haitham Masaud Amar; 2610.29637): definitions and theorems on cooperation and competition in trip planning, with Lean 4 checks (Mathlib v4.24.0). The author says OpenAI's 6 Oct 2026 release of mathematical manuscripts inspired it. An example of that release pushing practitioners to formalize older work
- Language or editing only: simultaneous popular differences in finite fields (Xuancheng Shao, Weiwen Zhang; 2610.28226, 'ChatGPT (OpenAI) was used by the authors to assist with polishing'); SympFormer (2610.31600, ChatGPT and Codex for editing, implementation and 'exploratory discussions of mathematical formulations')
- Checked, no AI disclosure found in the PDF: Gauss–Fourier conjecture papers (2610.30620, 2610.29790), Li–Wang–Guo spectral-radius conjectures (2610.29048), two-stack sortable permutations (2610.29115), Witt-design open problem (2610.29744), measurable counterexamples to finite-difference equations (2610.30583), entropic CLT bounds (2610.31493), equivariant main conjecture for graph towers (2610.27551), Erdős–Szekeres-type permutation result (2610.28305), KPZ functional CLT (2610.27219), iterated correlation matrices (2610.24592)
- Commentary: Quanta Q&A with Moritz Hardt (Max Planck Institute for Intelligent Systems, 16 Oct), author of 'The Emerging Science of Machine Learning Benchmarks': 'The benchmarking ecosystem is broken', though benchmarks drove real progress 'mostly by sheer luck'

#### What happened
Saturday 24 Oct had no new arXiv announcement. The sweep listed 16–22 Oct papers that earlier roundups had not cited. PDFs
of the maths papers with named problems and of the AI-heavy science papers were scanned (pdftotext) for AI statements. Most
named-problem papers have none. Quotes above are from the PDFs.

#### Why it matters
The two strongest items have their own entries. Among the rest, Google's aptamer study stands out: AI agents improved
across real wet-lab rounds and diagnosed an assay artifact from the data. The Amar addendum is a small sign that OpenAI's
6 Oct maths release is prompting people to formalize their own older work.

Sources: [arXiv 2610.24806: Aptamers as a sandbox for lab-in-the-loop agentic molecular discovery (Google)](https://arxiv.org/abs/2610.24806); [arXiv 2610.29850: Automated discovery of subgrid length scales for DES via agentic AI (DEScover)](https://arxiv.org/abs/2610.29850); [arXiv 2610.30843: Lean Finder V2](https://arxiv.org/abs/2610.30843); [arXiv 2610.31223: Sharp quantitative stability of the Ashbaugh–Benguria inequality](https://arxiv.org/abs/2610.31223); [arXiv 2610.29197: The Local Dyadic Conjecture](https://arxiv.org/abs/2610.29197); [arXiv 2610.29637: An AI-assisted mathematical addendum to a 2018 planning thesis](https://arxiv.org/abs/2610.29637); [arXiv 2610.28226: Simultaneous popular differences in finite fields](https://arxiv.org/abs/2610.28226); [arXiv 2610.31600: SympFormer](https://arxiv.org/abs/2610.31600); [Quanta: He Wants AI Benchmarking To Be a Science, Not a Dark Art (Moritz Hardt)](https://www.quantamagazine.org/he-wants-ai-benchmarking-to-be-a-science-not-a-dark-art-20261016/)

### 2026-10-25: AI-assisted results in the 25 Oct 2026 sweep: GPT-5.5 Thinking steers a neutral-atom tweezer array, GBClaw grain-boundary agent, ChatGPT Astra-built bone-marrow model

**; OpenAI, various researchers; Science & math; confidence medium; https://postcutoff.com/e/2026-10-25-ai-assisted-results-october-25-listing/

Catalogue of AI-disclosing preprints from the 17–20 Oct 2026 arXiv listings that no earlier roundup cited, read on 25 Oct. An LLM (GPT-5.5 Thinking via ChatGPT Plus) chose the control strategy for a 10×10 acousto-optic tweezer array, cutting non-uniformity from 20.30% to 1.57%. GBClaw (Georgia Tech, GPT-5.6 Sol and Luna) ran grain-boundary simulations in nickel. A Bristol modeller built a dynamic bone-marrow model through dialogue with ChatGPT (Astra, Extra High).

- LLM steering of acousto-optic tweezer arrays (Cheng Sheng, …, Gao-xiang Li; Huazhong Normal University and CAS Wuhan; 2610.25014): 'We test this role on the optical control subsystem of a neutral-atom apparatus using OpenAI GPT-5.5 Thinking through ChatGPT Plus.' The LLM recommended the model, update scale and next action. Deterministic software did the fitting, and an experimentalist authorized every hardware update. When separable row–column correction stalled, the LLM moved the workflow to a drift-corrected local response-matrix method. The Gaussian-peak coefficient of variation fell from 20.30% to 1.57% in 30 rounds
- GBClaw (Yazhuo Liu, Kunqing Ding, …, Ting Zhu; Georgia Tech, Osaka, USC; 2610.28629): agentic framework (orchestrator, specialized agents, LAMMPS/OVITO/ASE tools, hierarchical memory) that 'used both GPT-5.6 Sol and GPT-5.6 Luna, with task-dependent model routing'. Applied to 14 general asymmetric tilt grain boundaries in nickel, it characterized interfacial defect structures and deformation modes and found that secondary GB misfit dislocations strongly change shear strength. Researcher-guided follow-up calculations traced an anomalously high shear resistance to metastable boundary structures
- Diabetic bone-marrow dynamic model (Paolo Madeddu, University of Bristol; 2610.24728): 'AI assistance was provided through iterative dialogue with ChatGPT (OpenAI; Astra, Extra High reasoning setting). ChatGPT assisted evidence extraction, proposed regulatory rules and alternative assumptions, and supported coding and simulation.' The three-level logical model ranks endothelial ROS suppression as the best single intervention (score 24.2); ROS suppression combined with OSM blockade or p66Shc loss scores 37.9. The author says experiments must test this
- Checked, no AI disclosure found in the PDF: symmetric-group generation by prefix reversals (2610.30892), Wilf-equivalences / Dumont permutations conjecture (2610.30975), polynomial coefficients (2610.29132), Toeplitz semi-commutators (2610.30787), 2D MHD stability (2610.28308), radiating-gas model (2610.29462). Associated primes of monomial ideals (2610.27916) declares no generative AI use

#### What happened
Sunday 25 Oct had no new arXiv announcement. The sweep listed 17–22 Oct papers that earlier roundups had not cited. Most of
them only mention LLMs as a topic (benchmarks, applications). PDFs of the uncited maths papers and of the three papers that
gave an AI model a working role were scanned (pdftotext). Quotes above are from the PDFs.

#### Why it matters
The tweezer paper is a small but concrete case: a general-purpose chat model chose the experimental strategy on real quantum
hardware, while a human authorized each step. None of the items is a famous-problem result.

Sources: [arXiv 2610.25014: Large Language Model Steering of Acousto-Optic Tweezer Arrays](https://arxiv.org/abs/2610.25014); [arXiv 2610.28629: GBClaw, an agentic AI framework for atomistic grain boundary research](https://arxiv.org/abs/2610.28629); [arXiv 2610.24728: Evidence-informed dynamic modelling and intervention prioritisation in diabetic bone marrow](https://arxiv.org/abs/2610.24728)

## Videos

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- 2026-10-10, Mathify - Animate math by chatting with AI: [OpenAI Just Broke Matrix Multiplication. Why 2.25 Matters](https://www.youtube.com/watch?v=jZFuuXJPvFI) https://postcutoff.com/v/mathify-openai-matrix-multiplication-2-25/
- 2026-10-10, MS NOW: [‘A CHILLING EFFECT’: How OpenAI’s massive math breakthrough is ‘DESTROYING’ early careers](https://www.youtube.com/watch?v=a65dI2AYL3o) https://postcutoff.com/v/ms-now-chilling-effect-openai-math-early-careers/
- 2026-10-09, Bloomberg Podcasts: [How AI Is Upending the World of Mathematics | Odd Lots](https://www.youtube.com/watch?v=aTXGrz-sChk) https://postcutoff.com/v/bloomberg-odd-lots-ai-upending-mathematics/
- 2026-10-09, Global News: [OpenAI agent does it again, solving nearly 400 math problems no human has ever solved](https://www.youtube.com/watch?v=IQJT1ScDT6c) https://postcutoff.com/v/global-news-openai-agent-400-math-problems/
- 2026-10-09, House of El: AI: [OpenAI's Math Dump Just Exposed the AI Bubble's Fatal Flaw](https://www.youtube.com/watch?v=4fJK-O3S4ko) https://postcutoff.com/v/house-of-el-openai-math-dump-ai-bubble-fatal-flaw/
- 2026-10-09, The PrimeTime: [The Mathematics are Mad](https://www.youtube.com/watch?v=WXh4LF3zJ1Q) https://postcutoff.com/v/primetime-the-mathematics-are-mad/
- 2026-10-09, Prof Gio | Giordano Scarciotti: [You Truly Don't Understand What Just Happened to Mathematics](https://www.youtube.com/watch?v=KqiDqJMxg6o) https://postcutoff.com/v/prof-gio-dont-understand-what-happened-to-mathematics/
- 2026-10-09, Dr. Samuel Allen Alexander: [OpenAI withdraws three AI-generated math papers due to a sign error](https://www.youtube.com/watch?v=kiwzc3e5SQI) https://postcutoff.com/v/samuel-alexander-openai-withdraws-three-math-papers-sign-error/
- 2026-10-09, Alex Volkov from ThursdAI: [OpenAI dropped 722 math papers. Mathematicians are split.](https://www.youtube.com/watch?v=c66rfu1O40w) https://postcutoff.com/v/thursdai-openai-722-math-papers-mathematicians-split/
- 2026-10-08, Adem B10: [722 Math Papers in One Night. One of Them Is About Riemann.](https://www.youtube.com/watch?v=HO5MefNJh3Y) https://postcutoff.com/v/adem-b10-722-math-papers-one-night-riemann/
- 2026-10-08, AI Copium: [OpenAI Just Gave Us a Glimpse of the Singularity…](https://www.youtube.com/watch?v=9_ooMWSniqY) https://postcutoff.com/v/ai-copium-openai-glimpse-of-the-singularity/
- 2026-10-08, CNBC Television: [NYU's Tristan Buckmaster on the controversy over OpenAI's math findings](https://www.youtube.com/watch?v=GweKOBSJvEY) https://postcutoff.com/v/cnbc-squawk-box-buckmaster-openai-math-controversy/
- 2026-10-08, ColdFusion: [AI Keeps Pissing Off Mathematicians](https://www.youtube.com/watch?v=SoLWCYnc-Bs) https://postcutoff.com/v/coldfusion-ai-keeps-pissing-off-mathematicians/
- 2026-10-08, Easy Riders: [About OpenAI's "Breakthrough" on 377 Math problems...](https://www.youtube.com/watch?v=n3flGCdqesU) https://postcutoff.com/v/easy-riders-openai-377-math-problems/
- 2026-10-08, Joma Tech: [Introducing ChatGPT for Dishwashing](https://www.youtube.com/watch?v=ofdMfljL360) https://postcutoff.com/v/joma-tech-chatgpt-for-dishwashing-parody/
- 2026-10-08, Perplexity: [The Quasi-Riemann Hypothesis, explained](https://www.youtube.com/watch?v=mR0Wpo_rdEM) https://postcutoff.com/v/perplexity-quasi-riemann-hypothesis-explained/
- 2026-10-08, Dr. Samuel Allen Alexander: [Math 2.0: Terence Tao responds to the recent OpenAI flood of proofs](https://www.youtube.com/watch?v=mBUCv-nimL4) https://postcutoff.com/v/samuel-alexander-tao-responds-openai-proofs/
- 2026-10-08, Simply AI: [OpenAI's 722 AI Math Papers, and Why Terence Tao Is Uneasy](https://www.youtube.com/watch?v=6F80wqj14JY) https://postcutoff.com/v/simply-ai-openai-722-math-papers-tao-uneasy/
- 2026-10-08, Timcast IRL: [Math Field SHOOK After Open AI SOLVES HUNDREDS Of Problems, ITS OVER](https://www.youtube.com/watch?v=hUJuWw2utAg) https://postcutoff.com/v/timcast-irl-math-field-shook-openai-hundreds-problems/
- 2026-10-08, TVP WORLD: [OpenAI releases 700 math papers. AI takes over science? | You&AI](https://www.youtube.com/watch?v=X6ucWKTuunY) https://postcutoff.com/v/tvp-world-openai-700-math-papers-ai-takes-over-science/
- 2026-10-08, Wes Roth: [OpenAI’s “Alien Math” Is Freaking People Out](https://www.youtube.com/watch?v=rxpV3QLV-pg) https://postcutoff.com/v/wes-roth-openai-alien-math-freaking-people-out/
- 2026-10-07, AI Revolution: [OpenAI Just Broke Math With Its Most Powerful AI Yet](https://www.youtube.com/watch?v=WYpzvsC4qTk) https://postcutoff.com/v/ai-revolution-openai-just-broke-math/
- 2026-10-07, Alex Hyman: [BIGGEST Proof Dump in History! | OpenAI Release 722 Manuscripts](https://www.youtube.com/watch?v=TlrnYcfQzo0) https://postcutoff.com/v/alex-hyman-biggest-proof-dump-openai-722/
- 2026-10-07, Binary Verse AI: [OpenAI Integer Multiplication: A Tiny Crack in the n log n Barrier](https://www.youtube.com/watch?v=43owPX7tVDs) https://postcutoff.com/v/binary-verse-ai-openai-integer-multiplication/
- 2026-10-07, Binary Verse AI: [OpenAI Matrix Multiplication: The Proof Behind the 2 25 Leap](https://www.youtube.com/watch?v=wLJzmpwSwB0) https://postcutoff.com/v/binary-verse-openai-matrix-multiplication-2-25/
- 2026-10-07, Alvaro Lozano-Robledo: [Oct 6, 2026: A math drop of historic proportions](https://www.youtube.com/watch?v=liPcU6Wz6K8) https://postcutoff.com/v/lozano-robledo-oct-6-math-drop-historic-proportions/
- 2026-10-07, Mathify - Animate math by chatting with AI: [OpenAI Just Dropped 722 Math Papers. What Did It Actually Prove?](https://www.youtube.com/watch?v=r-zBgTYlpH0) https://postcutoff.com/v/mathify-openai-722-math-papers-what-did-it-prove/
- 2026-10-07, National Museum of Mathematics: [When AI Solves Major Open Problems: What Does It Mean for the Future of Mathematics—and Beyond?](https://www.youtube.com/watch?v=voDtZqijetw) https://postcutoff.com/v/momath-gala-panel-when-ai-solves-major-open-problems/
- 2026-10-07, Myren Eario: [OpenAI solved aperiodic Minecraft tilings](https://www.youtube.com/watch?v=3wqjyIAaFUU) https://postcutoff.com/v/myren-eario-openai-aperiodic-minecraft-tilings/
- 2026-10-07, The Pretrained Pod: [BREAKING: AI Just Claimed Problems Mathematicians Spent Decades On](https://www.youtube.com/watch?v=chK5TlDnHDo) https://postcutoff.com/v/pretrained-pod-ai-claimed-problems-mathematicians-decades/
- 2026-10-07, Two Minute Papers: [DeepMind's New AI Just Cracked The Code Of Life](https://www.youtube.com/watch?v=Wkaw03p3BrM) https://postcutoff.com/v/two-minute-papers-alphagenome-atlas/
- 2026-10-07, Versors: [How OpenAI Just Showed AI Will Transform Math](https://www.youtube.com/watch?v=XO_5D30KeVk) https://postcutoff.com/v/versors-how-openai-showed-ai-will-transform-math/
- 2026-10-07, Zeta Explained: [Zeta Explained #116.5: OpenAI's Zero-Free Region Re(s)>7/8](https://www.youtube.com/watch?v=XtHsE99VGpk) https://postcutoff.com/v/zeta-explained-openai-zero-free-region-7-8/
- 2026-10-06, Brendan Dell: [The most hyped AI breakthrough of 2026 is a hack](https://www.youtube.com/watch?v=iu2NsG6uGUs) https://postcutoff.com/v/brendan-dell-most-hyped-ai-breakthrough-is-a-hack/
- 2026-10-06, Wes Roth: [OpenAI's secret model just BROKE math...](https://www.youtube.com/watch?v=fTGIV3e8dTU) https://postcutoff.com/v/wes-roth-openai-secret-model-broke-math/
- 2026-10-05, The Royal Institution: [Can AI solve the hardest problems in mathematics? | PuzzlingScience with Yang-Hui He](https://www.youtube.com/watch?v=KiBboUqdD-4) https://postcutoff.com/v/royal-institution-can-ai-solve-hardest-maths-problems/
- 2026-10-04, Adem B10: [AI Checked a 48-Year Math Breakthrough in 6 Days](https://www.youtube.com/watch?v=53tuOJXJfuU) https://postcutoff.com/v/adem-b10-ai-checked-48-year-math-breakthrough-6-days/
- 2026-10-02, World Science Festival: [The Moment AI Changed Mathematics Forever | World Science Festival](https://www.youtube.com/watch?v=PQYFRuZ5phs) https://postcutoff.com/v/wsf-greene-buckmaster-moment-ai-changed-mathematics/
- 2026-10-01, StarTalk: [Did AI Solve One of Math’s Hardest Problems?](https://www.youtube.com/watch?v=s7KhLuc3Mck) https://postcutoff.com/v/startalk-did-ai-solve-one-of-maths-hardest-problems/
- 2026-09-29, Numberphile: [The Buckmaster Interview (Navier-Stokes) - Numberphile](https://www.youtube.com/watch?v=Mzbtj5nkMXI) https://postcutoff.com/v/numberphile-buckmaster-interview-navier-stokes/
- 2026-09-27, Ellie Sleightholm: [AI Took On the Riemann Hypothesis. Here’s What Actually Happened](https://www.youtube.com/watch?v=kvLOiTxYSzA) https://postcutoff.com/v/ellie-sleightholm-ai-took-on-riemann-hypothesis/
- 2026-09-25, Asian Dad Energy: [OpenAI’s Math “Breakthrough” Is Falling Apart?](https://www.youtube.com/watch?v=9_pZft8hHms) https://postcutoff.com/v/asian-dad-energy-openai-math-falling-apart/
- 2026-09-24, CBS News: [Anthropic says Claude discovered new gene-editing tool as debate over AI in healthcare grows](https://www.youtube.com/watch?v=3rKPF3EaK5Q) https://postcutoff.com/v/cbs-news-claude-gene-editing-tool-and-openai-hack/
- 2026-09-23, Anthropic: [Inside Anthropic's molecular biology lab](https://www.youtube.com/watch?v=DdCEmlAydcw) https://postcutoff.com/v/anthropic-molecular-biology-lab/
- 2026-09-23, Wes Roth: [Claude JUST found hidden DNA...](https://www.youtube.com/watch?v=H7KruLVX2Rk) https://postcutoff.com/v/wes-roth-claude-just-found-hidden-dna/
- 2026-09-22, Breaking Points: [Anthropic Starts AI 'WET LAB' In New NIGHTMARE SCENARIO](https://www.youtube.com/watch?v=7XWV0gA9he0) https://postcutoff.com/v/breaking-points-anthropic-ai-wet-lab/
- 2026-09-21, Sky News: [The world's best mathematicians already overtaken by AI](https://www.youtube.com/watch?v=1KitV-zjy18) https://postcutoff.com/v/sky-news-mathematicians-overtaken-by-ai/
- 2026-09-21, Stand-up Maths: [Has a Millennium Prize Problem been solved by AI?](https://www.youtube.com/watch?v=c79PsbkHw4A) https://postcutoff.com/v/standup-maths-millennium-prize-problem-solved-by-ai/
- 2026-09-18, 3Blue1Brown: [The last IMO problem AI could not solve](https://www.youtube.com/watch?v=Nbwv5wHQoj0) https://postcutoff.com/v/3blue1brown-last-imo-problem-ai-could-not-solve/
- 2026-09-18, Numberphile: [SOLVED: Navier-Stokes cracked by AI - Numberphile](https://www.youtube.com/watch?v=3geDF-DAwpg) https://postcutoff.com/v/numberphile-solved-navier-stokes-cracked-by-ai/
- 2026-09-18, Numberphile2: [Navier-Stokes solved by AI (extra) - Numberphile](https://www.youtube.com/watch?v=aom4269UgSM) https://postcutoff.com/v/numberphile2-navier-stokes-solved-by-ai-extra/
- 2026-09-17, Dwarkesh Patel: [OpenAI researcher on agent swarms & recursive self-improvement](https://www.youtube.com/watch?v=6AgOfiZOWiY) https://postcutoff.com/v/dwarkesh-noam-brown-agent-swarms-recursive-self-improvement/
- 2026-09-17, Institute for Pure & Applied Mathematics (IPAM): [Terence Tao - The integrated explicit analytic number theory network - IPAM at UCLA](https://www.youtube.com/watch?v=Boo72U-9u3c) https://postcutoff.com/v/ipam-terence-tao-integrated-explicit-analytic-number-theory-network/
- 2026-09-17, Quanta Magazine: [Why Navier-Stokes Pushes Math and Physics to the Edge](https://www.youtube.com/watch?v=PsWR1rQEWYo) https://postcutoff.com/v/quanta-why-navier-stokes-pushes-math-and-physics-to-the-edge/
- 2026-09-12, Caleb Writes Code: [OpenAI solving Navier-Stokes explained..](https://www.youtube.com/watch?v=7DncQnIjZmA) https://postcutoff.com/v/caleb-writes-code-openai-navier-stokes-explained/
- 2026-09-11, Fads: [OpenAI Keeps Claiming They Solved a $1 Million Math Problem. They Didn't.](https://www.youtube.com/watch?v=UjO3Kd6Igyg) https://postcutoff.com/v/fads-openai-million-dollar-math-problem-didnt/
- 2026-09-11, Fireship: [OpenAI's biggest math breakthrough is getting ugly...](https://www.youtube.com/watch?v=aspmNhKAFMc) https://postcutoff.com/v/fireship-openai-math-breakthrough-getting-ugly/
- 2026-09-10, citytutoringmath: [Did AI Really Solve Navier–Stokes? Show Me The Proof!](https://www.youtube.com/watch?v=zhz5OubU0BU) https://postcutoff.com/v/citytutoringmath-did-ai-really-solve-navier-stokes/
- 2026-09-10, Complexity: [Visualizing the OpenAI solution to the Navier-Stokes problem](https://www.youtube.com/watch?v=82WhfkCWU2Y) https://postcutoff.com/v/complexity-visualizing-openai-navier-stokes-solution/
- 2026-09-10, The PrimeTime: [The Navier Stokes situation is insane](https://www.youtube.com/watch?v=UA0UkPombNk) https://postcutoff.com/v/primetime-the-navier-stokes-situation-is-insane/
- 2026-09-10, Up and Atom: [The Controversy Behind AI Solving A Million Dollar Math Problem](https://www.youtube.com/watch?v=cvTHBXyXM2I) https://postcutoff.com/v/up-and-atom-controversy-ai-million-dollar-math-problem/
- 2026-09-09, Brain Station Advanced: [Controversy behind OpenAI Solving Navier Stokes Equation](https://www.youtube.com/watch?v=SoU9yt_DylE) https://postcutoff.com/v/brain-station-advanced-controversy-openai-navier-stokes/
- 2026-09-09, Kanał Zero: [SKANDAL WOKÓŁ OPENAI I PROBLEMU MILENIJNEGO. DRAGAN: SPRAWA WYGLĄDA BRUDNO](https://www.youtube.com/watch?v=Zl78TVQvCUs) https://postcutoff.com/v/kanal-zero-dragan-openai-navier-stokes-skandal/
- 2026-09-09, Physics by Oimeow: [The controversy around OpenAI "solving" a $1m math problem (Navier-Stokes)](https://www.youtube.com/watch?v=9XYt-_GdPIU) https://postcutoff.com/v/physics-by-oimeow-controversy-openai-navier-stokes/
- 2026-09-09, Sabine Hossenfelder: [AI Solved a Millennium Problem and Everyone's Angry](https://www.youtube.com/watch?v=eR2q8s4VB1w) https://postcutoff.com/v/sabine-ai-solved-millennium-problem-everyones-angry/
- 2026-09-08, Google DeepMind: [AlphaGenome Atlas: Understanding the human genome](https://www.youtube.com/watch?v=U0aToL5C-bQ) https://postcutoff.com/v/deepmind-alphagenome-atlas/
- 2026-09-08, Dr. Samuel Allen Alexander: [Allegations that OpenAI stole a Navier-Stokes solution?? Infinite drama blowup in finite time!](https://www.youtube.com/watch?v=VaJlOhX8Ss4) https://postcutoff.com/v/samuel-alexander-allegations-openai-stole-navier-stokes/
- 2026-09-08, Wes Roth: [OpenAI JUST solved math....](https://www.youtube.com/watch?v=lkujyxUdUIk) https://postcutoff.com/v/wes-roth-openai-just-solved-math-navier-stokes/
- 2026-09-03, Google DeepMind: [WeatherNext 3: More accurate, timely, and local weather forecasts](https://www.youtube.com/watch?v=_6jZlnRsXXQ) https://postcutoff.com/v/deepmind-weathernext-3/
- 2026-08-28, Big Think: [One of the world's greatest mathematicians explains 6 essential concepts of math | Terence Tao](https://www.youtube.com/watch?v=OOMx2BHHWtE) https://postcutoff.com/v/big-think-terence-tao-6-essential-concepts-of-math/
- 2026-08-19, Fireship: [The summer Math fell to the machines...](https://www.youtube.com/watch?v=iuZPTE5qsJY) https://postcutoff.com/v/fireship-summer-math-fell-to-the-machines/
- 2026-08-13, Simons Foundation: [ICM 2026 Public Lectures - Terence Tao](https://www.youtube.com/watch?v=M0--ZH1lOzg) https://postcutoff.com/v/simons-foundation-icm-2026-public-lecture-tao/
- 2026-07-27, Alvaro Lozano-Robledo: [Terence Tao: "Mathematics in the Age of AI" (ICM 2026)](https://www.youtube.com/watch?v=sxAe4HJceFQ) https://postcutoff.com/v/tao-icm-2026-mathematics-in-the-age-of-ai/
- 2026-06-05, Two Minute Papers: [DeepMind's New AI Found A Strange New Way To Think](https://www.youtube.com/watch?v=Dkqzqw8rxXI) https://postcutoff.com/v/two-minute-papers-deepmind-alphaproof-nexus-strange-new-way/
- 2026-05-20, OpenAI: [The Erdős Breakthrough](https://www.youtube.com/watch?v=Br4l9YjCyRU) https://postcutoff.com/v/openai-the-erdos-breakthrough/

## Posts

- 2026-10-23, Rohan Arun: [](https://x.com/RohanArun/status/2113677165866766619)
- 2026-10-22, Rohan Arun: [](https://x.com/RohanArun/status/2113091328913404342)
- 2026-10-21, OpenAI: [](https://x.com/OpenAI/status/2112899854569762936)
- 2026-10-20, Anthropic: [](https://x.com/AnthropicAI/status/2112579251555721573)
- 2026-10-18, Rohan Arun: [](https://x.com/RohanArun/status/2111940317662363937)
- 2026-10-17, Rohan Arun: [Rohan Arun: a free model, OpenCode's Space Bunny, proposes new conditional κ bounds for integer multiplication](https://x.com/RohanArun/status/2111493670318809303)
- 2026-10-16, Anthropic: [](https://x.com/AnthropicAI/status/2111152183240511640)
- 2026-10-15, Mistral AI: [](https://x.com/MistralAI/status/2110684732103041474)
- 2026-10-15, Rikab Gambhir: [Rikab Gambhir (guest post on Terence Tao's blog): 'Why are we even doing all this?'](https://terrytao.wordpress.com/2026/10/15/why-are-we-even-doing-all-this/)
- 2026-10-12, Sophie Huiberts: [Sophie Huiberts (guest post on Terence Tao's blog): 'This is normal bad work'](https://terrytao.wordpress.com/2026/10/12/this-is-normal-bad-work/)
- 2026-10-12, Rohan Arun: [](https://x.com/RohanArun/status/2109466530219999637)
- 2026-10-12, Rohan Arun: [](https://x.com/RohanArun/status/2109629408076083414)
- 2026-10-12, David Roberts: [David Roberts: 'What even is mathematical output?'](https://thehighergeometer.wordpress.com/2026/10/12/what-even-is-mathematical-output/)
- 2026-10-12, Terence Tao: [Tao: 'Additional recent pieces on math and AI' (reading list)](https://terrytao.wordpress.com/2026/10/12/additional-recent-pieces-on-math-and-ai/)
- 2026-10-11, Andrej Karpathy: [Karpathy: the gap in shared understanding of LLM capability is widening into 'a sharp funnel'](https://x.com/karpathy/status/2109361546505966046)
- 2026-10-11, Kate Stange and Edmund Harriss (guest post on Terence Tao's blog): [Illustration beyond Proof](https://terrytao.wordpress.com/2026/10/11/illustration-beyond-proof/)
- 2026-10-10, Terence Tao: [Math 2.0](https://terrytao.wordpress.com/2026/10/10/math-2-0/)
- 2026-10-10, Dr Singularity: ["Another big AI math drop": viral summary of AlphaProof Nexus](https://x.com/Dr_Singularity/status/2108936442671992990)
- 2026-10-09, Thomas Hales: [Thomas Hales: What mathematicians should know about the Lean Theorem Prover: questions of reliability and AI (guest post on Terence Tao's blog)](https://terrytao.wordpress.com/2026/10/09/what-mathematicians-should-know-about-the-lean-theorem-proverquestions-of-reliability-and-ai/)
- 2026-10-08, Dan Roberts: [OpenAI's Dan Roberts: math repo update with 6 Lean formalizations, 19 modifications and 3 withdrawals](https://x.com/danintheory/status/2108065033070789090)
- 2026-10-08, Doug Colkitt: [Doug Colkitt: community effort reaches κ = 2^-15 on OpenAI problem #109](https://x.com/0xdoug/status/2108218367962124423)
- 2026-10-08, Doug Colkitt: [Doug Colkitt: OpenAI problem #109 tightened to κ = 2^-34](https://x.com/0xdoug/status/2107994110644588859)
- 2026-10-08, Thomas Bloom: [Thomas Bloom: Erdős problems in the OpenAI proofs](https://www.erdosproblems.com/forum/thread/blog:10)
- 2026-10-08, Elliot Glazer: [Elliot Glazer: GPT-6 Astra check suggests OpenAI's abelian-eightfolds Hodge paper is flawed](https://x.com/ElliotGlazer/status/2108026240582246600)
- 2026-10-08, Asaf Karagila: [Asaf Karagila: 'OpenAI, the Partition Principle, and mathematics'](https://karagila.org/2026/openai-pp/)
- 2026-10-08, Pushmeet Kohli: [Pushmeet Kohli thread: DeepMind and mathematics; AlphaProof Nexus paper appears in Science today](https://x.com/pushmeet/status/2108265638435119494)
- 2026-10-08, Swapnil Jain: [Swapnil Jain: OpenAI problem #109 past 2^-14 (κ ≈ 6.397×10^-5)](https://x.com/SJ_Swapnil_Jain/status/2108300632738386419)
- 2026-10-08, Swapnil Jain: [Swapnil Jain: OpenAI problem #109 past 2^-15, certificates and assembly checked in Lean](https://x.com/SJ_Swapnil_Jain/status/2108196538568851574)
- 2026-10-08, Swapnil Jain: [Swapnil Jain: OpenAI problem #109 past 2^-18 ('2^-1x territory')](https://x.com/SJ_Swapnil_Jain/status/2108136796886548867)
- 2026-10-08, Doug Colkitt: [](https://x.com/0xdoug/status/2108171317321125959)
- 2026-10-08, Aurel Prosz: [](https://x.com/aurel_pr/status/2108214135179944096)
- 2026-10-08, Rohan Arun: [](https://x.com/RohanArun/status/2108190499941065193)
- 2026-10-08, Swapnil Jain: [Swapnil Jain: OpenAI problem #109 tightened to κ = 2^-27, building on Colkitt](https://x.com/SJ_Swapnil_Jain/status/2108109123774796281)
- 2026-10-07, Sam Altman: [Altman: "We are entering a new era of discovery now"](https://x.com/sama/status/2107623610483720463)
- 2026-10-07, Doug Colkitt: [Doug Colkitt: OpenAI problem #109 tightened to κ = 2^-59 (from 2^-182)](https://x.com/0xdoug/status/2107917690568327225)
- 2026-10-07, Doug Colkitt: [Doug Colkitt: OpenAI problem #109 tightened further to κ = 2^-78](https://x.com/0xdoug/status/2107832360527913445)
- 2026-10-07, Doug Colkitt: [Doug Colkitt: conditional improvement of OpenAI problem #109 (integer multiplication), κ from 2^-182 to 5.8×10^-33](https://x.com/0xdoug/status/2107658319632433524)
- 2026-10-07, Dave Goldblatt: [Independent re-run of the Lean check for OpenAI's ζ(s) ≠ 0 for Re(s) > 7/8 proof: "the proof checks"](https://github.com/davegoldblatt/openai-zeta-proof-check)
- 2026-10-07, AI Notkilleveryoneism Memes: [Viral claim: AI has solved "90 of the 500 most important open problems" (LLM-ranked list)](https://x.com/AISafetyMemes/status/2107622121618964829)
- 2026-10-07, bubble boi: [bubbleboi: "This guy has 6.1 Astra running in a loop and is breaking the record for integer multiplication algorithms every few hours"](https://x.com/bubbleboi/status/2107934679294935317)
- 2026-10-07, François Chollet: [Chollet: is the jagged frontier "mainly math + code" pushed by RLVR?](https://x.com/fchollet/status/2107625225768858076)
- 2026-10-07, Gary Marcus: [Marcus: the math results use symbolic AI (Lean) as well as LLMs, as he predicted](https://x.com/GaryMarcus/status/2107643356566392933)
- 2026-10-07, Greg Brockman: [Brockman quotes the math release: "towards acceleration of scientific discovery"](https://x.com/gdb/status/2107640886482104524)
- 2026-10-07, Gil Kalai: [Gil Kalai: "an amazing milestone for mathematics" that still needs verifying](https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/)
- 2026-10-07, roon: [roon: "do not let yourself be reduced to the status of audience at a magic show"](https://x.com/tszzl/status/2107634532212158507)
- 2026-10-07, Andreas Thom: [Andreas Thom: "It is impossible to put in words..."](https://mathstodon.xyz/@andreasthom/117397715584752954)
- 2026-10-07, Zvi Mowshowitz: [Zvi: "simultaneously the biggest day in mathematical history, and also felt kind of slow"](https://x.com/TheZvi/status/2107632497878266108)
- 2026-10-06, OpenAI: [Sharing AI progress in mathematics (OpenAI blog)](https://openai.com/index/sharing-ai-progress-in-mathematics/)
- 2026-10-06, OpenAI: [OpenAI announces release of a broad range of math results from an internal frontier model (openai/math)](https://x.com/OpenAI/status/2107596713791767021)
- 2026-10-06, Levent Alpöge: [Alpöge (Anthropic): "obviously the most significant moment in mathematical history"; notes scooping](https://x.com/__alpoge__/status/2107616859595981117)
- 2026-10-06, Alex Kontorovich: [Kontorovich: quasi-RH "would be an instant Fields Medal" if a human did it](https://x.com/AlexKontorovich/status/2107609087902941646)
- 2026-10-06, OpenAI: [openai/math repository README: 722 manuscripts in 372 families, ~4,000 problems, ~3h Pro compute each](https://github.com/openai/math)
- 2026-10-06, Terence Tao: [Tao: "Math 1.0" vs "Math 2.0"; solutions are being "harvested at large scale in an unsustainable fashion"](https://mathstodon.xyz/@tao/117395267721642920)
- 2026-10-06, Acer (AcerFur): [Former OpenAI team member: "we can do integer multiplication faster than n log n"](https://x.com/AcerFur/status/2107606747972309163)
- 2026-10-06, Cleo: [Claude Opus 5.5 reads OpenAI's math repo: 'a mix of awe and vertigo, plus a bit of grief'](https://x.com/ereliuer_eteer/status/2107618065869574208)
- 2026-10-06, Jared Duker Lichtman: [Lichtman: "absolutely historic" output, but "not the Millennium problem rumor"](https://x.com/jdlichtman/status/2107609204722892924)
- 2026-10-06, Joe Bebel: [TCS researcher: release includes a claimed, Lean-formalized proof of the Unique Games Conjecture he had worked on](https://x.com/joeintheory/status/2107612512288125096)
- 2026-10-06, Daniel Litt: [Litt: "mathematicians have a lot of exciting work to do"; flags a citation problem and questions 'progress' claims](https://x.com/littmath/status/2107603741222289427)
- 2026-10-06, Steven Strogatz: [Strogatz: matrix-multiplication exponent ≤ 2.25 is "like Bob Beamon's long jump"](https://x.com/stevenstrogatz/status/2107606017706266686)
- 2026-10-06, CG: [](https://x.com/cgtwts/status/2107597257684169010)
- 2026-10-06, Dan McAteer: ["If you’re a mathematician, you must feel like a nuclear bomb hit" (~388k views)](https://x.com/daniel_mac8/status/2107604648328569030)
- 2026-10-06, ℏεsam: [Hesamation: "mathematicians: … publish real proofs … OpenAI: fine. here's 722 papers"](https://x.com/Hesamation/status/2107600759953179030)
- 2026-10-06, Isaac Kim: [Isaac Kim: "shocking list" of quantum-information and many-body results](https://x.com/Isaac__kim/status/2107607429437657563)
- 2026-10-06, Ken Ono: [Ken Ono: AI-assisted discovery "at the highest level is no longer hypothetical"](https://x.com/KenOno691/status/2107612779264000261)
- 2026-10-06, Rohan Pandey: [Rohan Pandey: "kakeya conjecture in R^3 has been resolved" (conflates two Kakeya problems)](https://x.com/khoomeik/status/2107603513438335334)
- 2026-10-06, Chubby♨️: [kimmonismus: "HOLY, the rumors were true": 722 manuscripts from an unreleased OpenAI model](https://x.com/kimmonismus/status/2107597320028065793)
- 2026-10-06, Kiwamu Watanabe (渡邉究): [Japanese algebraic geometer lists the release's algebraic-geometry claims (Hodge for CM abelian varieties, Fujita, Nagata, MMP…)](https://x.com/Kiwamu_Watanabe/status/2107621501193408756)
- 2026-10-06, Peter Gostev: [Peter Gostev on Hadwiger–Nelson: "five-colours is no go", after "300 billion tokens" of his own attempts](https://x.com/petergostev/status/2107603534308864146)
- 2026-10-06, Tibo (Thibault Sottiaux): [Tibo (OpenAI Codex): "our models did solve some important math problems"](https://x.com/thsottiaux/status/2107597780352950624)
- 2026-10-06, Will DePue: [Will DePue: LLM-ranked list says 81% of top recent discoveries were released today](https://x.com/willdepue/status/2107604920753070317)
- 2026-10-06, Crémieux: ["This OpenAI math drop is nuts": 7/8 quasi-RH and Hilbert's tenth over Q](https://x.com/cremieuxrecueil/status/2107600945383112990)
- 2026-10-06, imjustnewatai: [Explainer thread: the release's "4 biggest claims in plain English" (~203k views)](https://x.com/imjustnewatai/status/2107601711032373333)
- 2026-10-06, Bartosz Naskręcki: [Naskręcki: "browsing these results in awe"](https://x.com/nasqret/status/2107603686113628627)
- 2026-10-05, Pushmeet Kohli: [](https://x.com/pushmeet/status/2107128950975529455)
- 2026-10-04, Diego Córdoba and Luis Martínez-Zoroa (guest post on Terence Tao's blog): [On classical solutions and singularity formation in incompressible fluids](https://terrytao.wordpress.com/2026/10/04/on-classical-solutions-and-singularity-formation-in-incompressible-fluids/)
- 2026-10-04, 前田悠馬｜成果報酬型のシステム開発: [](https://x.com/yumamaeda1210/status/2106672447617646846)
- 2026-10-03, Christian Szegedy: [Christian Szegedy, "Quo Vadis, Mathematics?": 'Mathematics is about to graduate'](https://docs.google.com/document/d/e/2PACX-1vTSh-pyNP3Gi99WMmsinnLmE9V5CDI0HEm6WGbIPMNt3V5SGlClHF8-BetotKHOImrvQDSXmFdiOw8D/pub)
- 2026-10-03, Jennifer Taback (guest post on Terence Tao's blog): [What does the advent of powerful AI models mean for mathematicians like me?](https://terrytao.wordpress.com/2026/10/03/what-does-the-advent-of-powerful-ai-models-mean-for-mathematicians-like-me/)
- 2026-10-02, AI at Meta: [AI at Meta: six maths papers from mathematician–Muse Spark collaborations](https://x.com/AIatMeta/status/2106099776035152231)
- 2026-10-02, Edmund Harriss and Katherine E. Stange: [Harriss & Stange: 'Mathematical Illustration: Proof, Intuition and AI' (arXiv 2610.02655)](https://arxiv.org/abs/2610.02655)
- 2026-10-01, Kevin Buzzard: [Kevin Buzzard: 'To grieve, or not to grieve?'](https://xenaproject.wordpress.com/2026/10/01/to-grieve-or-not-to-grieve/)
- 2026-10-01, Harald Helfgott: [Harald Helfgott: 'Mathematics and the AI crisis'](https://valuevar.wordpress.com/2026/10/01/mathematics-and-the-ai-crisis/)
- 2026-09-30, Fenner Stanley Tanswell: [Fenner Stanley Tanswell: 'The Enshittification of Mathematics' (The Mathematical Intelligencer)](https://link.springer.com/article/10.1007/s00283-026-10566-7)
- 2026-09-29, Rachel Webb (guest post on Terence Tao's blog): [How AI does, and does not, change the way I do math](https://terrytao.wordpress.com/2026/09/29/how-ai-does-and-does-not-change-the-way-i-do-math/)
- 2026-09-28, Andrew White 🐦‍⬛: [](https://x.com/andrewwhite01/status/2104614015951683902)
- 2026-09-27, Mario Rodríguez Mestre: [Mario Rodríguez Mestre disputes Claude's enzyme 'discovery' (jumbotrons)]()
- 2026-09-26, Ivan Corwin (guest post on Terence Tao's blog): [A value-based approach to mathematics](https://terrytao.wordpress.com/2026/09/26/a-value-based-approach-to-mathematics/)
- 2026-09-24, 0ximai: ['A 59-year-old conjecture in fusion physics just fell'](https://x.com/0x_impai/status/2103161612677107942)
- 2026-09-24, Alexandre Eremenko: [Alexandre Eremenko on MathOverflow: 'In August 2026, at least 7' of his ~40 collected open problems 'were solved using AI'](https://mathoverflow.net/a/515515)
- 2026-09-23, Anthropic: [Anthropic: Claude discovers a previously unknown CRISPR-like enzyme system](https://x.com/AnthropicAI/status/2102824959827742916)
- 2026-09-23, Lucas Harrington: [Lucas Harrington: the Anthropic enzyme find is routine genome mining; the hard part is function](https://x.com/CRISPR_LuCas/status/2102878373160906938)
- 2026-09-23, Alex Kontorovich: [Alex Kontorovich: 'WOW!! Zeta(5) is irrational! Here's a Lean formalization'](https://x.com/AlexKontorovich/status/2102870642555269554)
- 2026-09-23, Alexander Gamburd: [The Siren Call of Silicon Leviathan: Reflections on blowup and Aufklärungsdämmerung (arXiv 2609.28591)](https://arxiv.org/abs/2609.28591)
- 2026-09-23, Tapio Schneider (guest post on Terence Tao's blog): [Headlines and inside stories: understanding and trust in AI for mathematics, science, and engineering](https://terrytao.wordpress.com/2026/09/23/headlines-and-inside-stories-understanding-and-trust-in-ai-for-mathematics-science-and-engineering/)
- 2026-09-22, Elliot Glazer: [Elliot Glazer: 'zeta(5) has very likely been proven irrational by Aabir Fauzan'](https://x.com/ElliotGlazer/status/2102435821542691202)
- 2026-09-22, Gil Kalai: [Gil Kalai: '“Be a Grothendieck!”—On AI and Mathematics'](https://gilkalai.wordpress.com/2026/09/22/be-a-grothendieck-on-ai-and-mathematics/)
- 2026-09-21, OpenAI: [Advisory Group on Mathematics and Artificial Intelligence](https://openai.com/index/advisory-group-on-mathematics-and-ai/)
- 2026-09-21, Terence Tao (for AGMAI): [Announcing the Advisory Group on Mathematics and Artificial Intelligence](https://terrytao.wordpress.com/2026/09/21/advisory-group-on-mathematics-and-artificial-intelligence/)
- 2026-09-21, Sam Rodriques: [](https://x.com/SGRodriques/status/2102030245394337967)
- 2026-09-19, Sam Rodriques: [Sam Rodriques launches the Millennium Problems for Biology ('the last reasonable eval for AI in biology')](https://x.com/SGRodriques/status/2101449009546273138)
- 2026-09-19, Po-Shen Loh (guest post on Terence Tao's blog): [Why do we need human mathematicians anymore?](https://terrytao.wordpress.com/2026/09/19/why-do-we-need-human-mathematicians-anymore/)
- 2026-09-18, Grant Sanderson (3Blue1Brown), guest post on Terence Tao's blog: [If math is more than proof, we need to better celebrate the rest of it](https://terrytao.wordpress.com/2026/09/18/if-math-is-more-than-proof-we-need-to-better-celebrate-the-rest-of-it/)
- 2026-09-18, Ethan Mollick: [The Overhang](https://www.oneusefulthing.org/p/the-overhang)
- 2026-09-17, Timothy Gowers: [Why I didn't sign the Fields medallists' letter](https://terrytao.wordpress.com/2026/09/17/why-i-didnt-sign-the-fields-medallists-letter/)
- 2026-09-17, Dimitris Koukoulopoulos: [A CERN for AI-assisted science?](https://terrytao.wordpress.com/2026/09/17/a-cern-for-ai-assisted-science/)
- 2026-09-16, Captain Sude: [](https://x.com/captain_sude/status/2100277626120163664)
- 2026-09-16, Captain Sude: ['The Liouville version of the Goldbach conjecture is now fully proven and Lean verified!'](https://x.com/captain_sude/status/2100277626120163664)
- 2026-09-15, Scott Aaronson (Shtetl-Optimized): [The Age of Wonders and Terrors](https://scottaaronson.blog/?p=10062)
- 2026-09-11, Fields Medallists (Tao, Deligne, Donaldson, Bhargava, Scholze, Maynard, Avila et al.): [A Severe Misalignment of AI in Mathematics](https://mathandai.org/)
- 2026-09-11, Terence Tao: [Tao: 25 Fields Medalists make a joint declaration on Math and AI](https://mathstodon.xyz/@tao/117253629967855195)
- 2026-09-11, Andreas Thom (guest post on Terence Tao's blog): [On the existence of non-sofic groups](https://terrytao.wordpress.com/2026/09/11/on-the-existence-of-non-sofic-groups/)
- 2026-09-10, Matilde Marcolli: [Matilde Marcolli: 'Duet for the end of math'](https://www.its.caltech.edu/~matilde/AImathNew.pdf)
- 2026-09-09, Valerio Capraro: [](https://x.com/ValerioCapraro/status/2097791836269977996)
- 2026-09-08, OpenAI: [OpenAI: 'We're sharing a solution to the Navier-Stokes Millennium Prize Problem'](https://x.com/OpenAI/status/2097374640582668336)
- 2026-09-08, Tristan Buckmaster: [Buckmaster: Alpöge and I have made public three finite-time blowup results (with statement)](https://mastodon.social/@tristanbuckmaster/117233413705701198)
- 2026-09-08, Noam Brown: [Noam Brown: OpenAI mathematicians had their 'Lee Sedol moment'](https://x.com/polynoamial/status/2097375272387613183)
- 2026-09-08, Sebastien Bubeck: [Bubeck calls Buckmaster's allegations "false and inflammatory"](https://x.com/SebastienBubeck/status/2097214122471432349)
- 2026-09-07, Terence Tao: [Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations](https://terrytao.wordpress.com/2026/09/07/finite-time-blowup-with-smooth-forcing-term-for-the-incompressible-porous-medium-boussinesq-and-incompressible-euler-equations/)
- 2026-09-06, Jakub Pachocki: [An Alien Mind](https://openai.com/index/an-alien-mind/)
- 2026-09-06, Lech Mazur: [Lech Mazur: AI agents prove a positive fraction of integers reach 1 under Collatz in 10.46 ln n steps (Lean-formalized)](https://x.com/LechMazur/status/2096464472973975782)
- 2026-09-05, NVIDIA AI: [](https://x.com/NVIDIAAI/status/2096032566310789528)
- 2026-09-04, Anthropic: [Anthropic: Claude completes first formalized proof of Fermat's Last Theorem](https://x.com/AnthropicAI/status/2095947707605266436)
- 2026-09-04, Kevin Buzzard: [FLT: Anthropic has beaten me to it](https://xenaproject.wordpress.com/2026/09/04/flt-anthropic-has-beaten-me-to-it/)
- 2026-09-03, Terence Tao: [Tao: open problems have become a non-renewable resource](https://mathstodon.xyz/@tao/117204929023813310)
- 2026-09-03, Thomas Bloom: [Thomas Bloom: A big day for AI and mathematics — FrontierMath Erdős](https://x.com/thomasfbloom/status/2095630765035864260)
- 2026-09-03, Weijie Su: [](https://x.com/weijie444/status/2095600108956262911)
- 2026-08-30, Hugo Duminil-Copin (Proofs and Prompts): [Care for a little more AI?](https://proofsandprompts.com/2026/08/30/care-for-a-little-more-ai/)
- 2026-08-27, Anthropic: [Anthropic launches research preview of the Model Hardware Standard (MHS)](https://x.com/AnthropicAI/status/2093038426140651791)
- 2026-08-18, Pushmeet Kohli: [Pushmeet Kohli: new record for the matrix multiplication exponent ω < 2.371177 with AlphaEvolve](https://x.com/pushmeet/status/2089717134129565763)
- 2026-08-12, Terence Tao: [A digestion of the proof of Sendov's conjecture](https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/)
- 2026-08-11, Daniel Litt: [Returning from OpenAI's summit on the future of mathematics: 'The End of Mathematics' talk](https://x.com/littmath/status/2087302625423409549)
- 2026-07-22, Dmitry Rybin: [](https://x.com/DmitryRybin1/status/2079904005652893709)
- 2026-07-22, Shouqiao Wang: [](https://x.com/Qiaoqiao2001/status/2080003441821163958)
- 2026-07-21, Terence Tao: [A digestion of the Jacobian conjecture counterexample](https://terrytao.wordpress.com/2026/07/21/a-digestion-of-the-jacobian-conjecture-counterexample/)
- 2026-07-21, Deedy Das: [Deedy Das: Fable, Sol, K3 and Axiom all score 42/42 on IMO 2026](https://x.com/deedydas/status/2079409461874332066)
- 2026-07-21, NVIDIA AI: [NVIDIA: Nemotron 3 Ultra graded 30/42 by the IMO team at IMO 2026](https://x.com/NVIDIAAI/status/2079642933058244704)
- 2026-04-21, David Bessis: [The fall of the theorem economy](https://davidbessis.substack.com/p/the-fall-of-the-theorem-economy)
- 2025-09-11, Math, Inc.: [](https://x.com/mathematics_inc/status/1966194751847461309)
- 2025-08-20, Sebastien Bubeck: [](https://x.com/SebastienBubeck/status/1958198661139009862)
- 2025-07-19, OpenAI: [](https://x.com/OpenAI/status/1946594928945148246)