{"schema":"postcutoff/scienceboard@1","as_of":"2026-11-13T13:49:00+01:00","url":"https://postcutoff.com/science/","md":"https://postcutoff.com/science/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"rule":"A claim counts when an entry dated after 30 June 2026 reports an AI result on a named open problem, has importance 4 or 5, and credits an AI system. Formal proofs of theorems already proved, batches of many results and other science (forecasts, drug and protein design, competitions) are listed under ‘All AI science results’ instead. The status says where the field stands on the claim: ‘confirmed’, ‘awaiting review’ (no verdict yet from peer review or the experts in the area) or ‘disputed’. The ‘How it was checked’ column says what checking exists so far: peer review, a Lean proof, named experts, or a preprint only. A Lean proof whose formal statement nobody has reviewed is labelled ‘Lean proof, not yet reviewed’. A Lean proof can check a theorem while experts still dispute what it settles, so a row can be both ‘Lean-verified’ and ‘disputed’.","counts":{"claims":194,"confirmed":29,"pending":164,"disputed":1,"retracted":0},"median_years_open":52,"problem_years":8338,"claims_with_start_year":151,"since":"2026-07-17","checks":[{"key":"peer-review","label":"Peer-reviewed","claims":20},{"key":"lean","label":"Lean-verified","claims":76},{"key":"expert","label":"Expert-checked","claims":11},{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed","claims":12},{"key":"preprint","label":"Preprint only","claims":67},{"key":"none","label":"Unverified claim","claims":8}],"claims":[{"entry":{"id":"2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean","url":"https://postcutoff.com/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/","date":"2026-10-15","date_precision":"day","short_title":"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","deck":null,"takeaway":"The first downstream theorem built on OpenAI's claimed exclusion of Landau–Siegel zeros: the unconditional class-number table jumps from h ≤ 100 (Watkins, 2004) to h ≤ 2000, with the key bounds checked in Lean.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-16","updated":"2026-10-16","orgs":["OpenAI","Anthropic"]},"problem":"Gauss class number problem: list all imaginary quadratic fields of a given class number (effectively, unconditionally)","short":"Gauss class number problem","year_posed":1801,"years_open":225,"ai_system":["OpenAI internal model (Siegel-zero proof and its Lean formalization)","Claude Opus 5.5 (Claude Code)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Human-directed: the author chose the problem, checks and trusted base; Claude did constants, Lean, code and most text; the underlying theorem is OpenAI's","result":"Unconditional determination of all imaginary quadratic fields with class number ≤ 2000 (16,453,493 fields), via an explicit, Lean-verified version of OpenAI's Landau–Siegel zero exclusion.","field":"mathematics"},{"entry":{"id":"2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover","url":"https://postcutoff.com/e/2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover/","date":"2026-10-12","date_precision":"day","short_title":"Cureton and Thorner, with AxiomProver, extend the quasi-Riemann zero-free half-plane to every Dedekind zeta function (Re s > 31/32, Lean-formalized)","deck":null,"takeaway":"It is one of the first results to build directly on OpenAI's claimed quasi-Riemann breakthrough.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Axiom Math"]},"problem":"Quasi-Riemann hypothesis for Dedekind zeta functions of arbitrary number fields","short":"Quasi-Riemann hypothesis for Dedekind zeta functions of arbitrary number fields","year_posed":1859,"years_open":167,"ai_system":["AxiomProver"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-assisted: Cureton and Thorner directed the research and made the mathematical judgments; AxiomProver generated the proofs and the Lean formalization","result":"For every number field K, ζ_K has no zeros with Re s > 31/32; likewise finite-order Hecke L-functions over K.","field":"mathematics"},{"entry":{"id":"2026-10-06-quasi-riemann-hypothesis-openai","url":"https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the quasi-Riemann hypothesis","deck":"No zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized)","takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":19,"official":14,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Quasi-Riemann hypothesis (a zero-free half-plane Re s > θ, θ < 1, for ζ and Dirichlet L-functions); Landau–Siegel zeros","short":"Quasi-Riemann hypothesis","year_posed":1859,"years_open":167,"ai_system":["OpenAI internal model (unreleased)"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"Listed by OpenAI as an exception to its fixed automated procedure (details not given); the 11/12 write-up was human-edited","result":"Claimed: ζ, all Dirichlet L-functions and finite-order Hecke L-functions over ℚ(√−3) have no zeros with Re s > 7/8; real zeros of real primitive Dirichlet L-functions satisfy 1 − β ≥ c/log q.","field":"mathematics"},{"entry":{"id":"2026-09-14-sylvester-conjecture-proved-burungale-tian","url":"https://postcutoff.com/e/2026-09-14-sylvester-conjecture-proved-burungale-tian/","date":"2026-09-14","date_precision":"day","short_title":"Sylvester's 1879 conjecture on sums of two cubes is fully proved","deck":"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","takeaway":"This is one of the oldest named problems in Diophantine number theory, and the main proofs are human-led.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-10","updated":"2026-10-19","orgs":["University of Texas at Austin","Chinese Academy of Sciences","Shandong University"]},"problem":"Sylvester's conjecture (1879): every prime p ≡ 4, 7, 8 (mod 9) is a sum of two rational cubes","short":"Sylvester's conjecture","year_posed":1879,"years_open":147,"ai_system":["ChatGPT","Claude","unnamed AI (Yin, Oct 2026)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led with AI tools. Burungale–Tian used ChatGPT and Claude for exploratory computations, editing and reorganization. In Yin's Oct 2026 short proof, the ideas are his and the detailed working-out was done by AI and then rewritten by him.","result":"Proved in full: the 4, 7 cases by Yin (arXiv 2605.25917, May 2026), the 8 case by Burungale–Tian (arXiv 2609.14893, Sept 2026), who show E_p: y² = x³ + p²/4 has analytic rank one; a shorter proof of the 8 case followed from Yin (arXiv 2610.08015, Oct 2026).","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-hilbert-sixteenth-uniform-limit-cycle-bound-claim","url":"https://postcutoff.com/e/2026-10-06-openai-hilbert-sixteenth-uniform-limit-cycle-bound-claim/","date":"2026-10-06","date_precision":"day","short_title":"Hilbert's 16th problem: uniform limit-cycle bound claimed (OpenAI)","deck":null,"takeaway":"A 160-page manuscript credited to 'OpenAI' (dated 24 Sept, public on 6 Oct) claims that for every degree d there is a finite B(d) bounding the number of limit cycles of every real planar polynomial vector field of degree at most d. That is the uniform-boundedness assertion in the second part of Hilbert's sixteenth problem (1900), open for every degree above one. The bound is not effective. None of the release's Lean statements covers it: only a companion result, at most two limit cycles for quintic Liénard systems, is formalized. No dynamical-systems expert has published an assessment.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Hilbert's sixteenth problem, second part: is the number of limit cycles of a real planar polynomial vector field bounded by a function of its degree alone (finiteness of the Hilbert number H(d))?","short":"Hilbert's sixteenth problem, second part","year_posed":1900,"years_open":126,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; no human exposition or check yet","result":"Claimed existence of a finite, non-effective bound B(d) for every degree d; companion theorem (in Lean): quintic classical Liénard systems have at most two limit cycles, and two are attained","field":"mathematics"},{"entry":{"id":"2026-08-19-caratheodory-loewner-smooth-counterexamples-alpoge-claude","url":"https://postcutoff.com/e/2026-08-19-caratheodory-loewner-smooth-counterexamples-alpoge-claude/","date":"2026-08-19","date_precision":"day","short_title":"Smooth Carathéodory and Loewner conjectures disproved with Claude","deck":null,"takeaway":"On 19 Aug 2026 Levent Alpöge (Anthropic) posted an explicit formula on X: a smooth strictly convex closed surface in ℝ³ with exactly one umbilic point, against Carathéodory's conjecture that every such surface has at least two, and umbilics of index 1 + k/2, against Loewner's index ≤ 1. He credited John-Paul Smith 'and of course claude'. On 25 Oct the umbilic specialists Naoya Ando and Masaaki Umehara posted a simpler, self-contained construction on arXiv, found with GPT-6.1's help. The analytic case, the classical form of both conjectures, is not affected.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-27","updated":"2026-10-27","orgs":["Anthropic","Kumamoto University","Institute of Science Tokyo","OpenAI"]},"problem":"Carathéodory's conjecture (every closed convex surface in ℝ³ has ≥ 2 umbilics) and Loewner's conjecture (umbilic index ≤ 1), in the C^∞ category","short":"Carathéodory's conjecture","year_posed":1924,"years_open":102,"ai_system":["Claude (Anthropic)","ChatGPT (GPT-6.1)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: Alpöge and John-Paul Smith worked on the problem with Claude, which also checked it; Ando and Umehara used GPT-6.1 to search for candidate seed functions","result":"Explicit C^∞ counterexamples: umbilics of index 1 + k/2 for every positive integer k, and a strictly convex smooth closed surface with exactly one umbilic (index 2); simplified independently by Ando and Umehara","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-artin-primitive-root-conjecture-infinitude","url":"https://postcutoff.com/e/2026-10-06-openai-artin-primitive-root-conjecture-infinitude/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Artin's primitive root conjecture (infinitude)","deck":null,"takeaway":"Family 029 of OpenAI's 6 Oct 2026 release claims that every integer other than −1 and the squares is a primitive root modulo infinitely many primes, at least c·x/(log x)² of them in each interval (x, 2x). Before, this was known only under the generalized Riemann hypothesis (Hooley, 1967) or for all but at most two prime bases (Heath-Brown, so one of 2, 3, 5 works), not for any single named base such as 2. The argument rests on the release's own zero-free half-planes. Frank Calegari calls the argument interesting but 'not at all surprising' once quasi-RH is known. Manjul Bhargava has posted a positive-density strengthening.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Artin's primitive root conjecture (1927): every integer other than −1 and the squares is a primitive root modulo infinitely many primes","short":"Artin's primitive root conjecture","year_posed":1927,"years_open":99,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI; follow-ups by human mathematicians with AI assistance","result":"Claimed: for every admissible a, at least c_a·x/(log x)² primes in (x, 2x) have primitive root a (infinitude, not the conjectured density); a positive lower density was added by a follow-up","field":"mathematics"},{"entry":{"id":"2026-08-03-schiffer-pompeiu-conjectures-disproved","url":"https://postcutoff.com/e/2026-08-03-schiffer-pompeiu-conjectures-disproved/","date":"2026-08-03","date_precision":"day","short_title":"Planar Schiffer and Pompeiu conjectures disproved by two independent groups","deck":"One proof has a Lean certificate written by GPT-5.6","takeaway":"Schiffer's conjecture is a classic rigidity question and appears on Yau's list of open problems.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":7,"official":5,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Matthew Colbrook","George Stepaniants","Gonzalo Cao-Labora","Jaume de Dios Pont"]},"problem":"Schiffer's conjecture (Yau Problem 80) and the planar Pompeiu problem","short":"Schiffer's conjecture","year_posed":1929,"years_open":97,"ai_system":["GPT-5.6","GPT-5.5","Claude Opus 4.8","Claude Fable 5","ChatGPT/Codex"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Human-led. Colbrook–Stepaniants: AI only helped debug code. Cao-Labora–de Dios Pont: humans devised the bifurcation strategy; LLMs checked estimates numerically and drafted some lemma proofs, and GPT-5.6 wrote the Lean formalisation","result":"Non-circular bounded planar domains carrying a nonconstant Neumann eigenfunction that is constant on the boundary, which disproves both Schiffer's conjecture and the Pompeiu conjecture in the plane (one explicit domain, and an infinite family).","field":"mathematics"},{"entry":{"id":"2026-09-30-mt0-svg-ai-generated-tammes-15-robbins-frog-model-lean","url":"https://postcutoff.com/e/2026-09-30-mt0-svg-ai-generated-tammes-15-robbins-frog-model-lean/","date":"2026-09-30","date_precision":"day","short_title":"mt0-svg: AI-made Tammes N=15, Robbins V>2, frog model (Lean)","deck":null,"takeaway":"Between Sept 28 and Oct 18, 2026 one pseudonymous GitHub account released six papers whose 'results, proofs and code were found and written by AI', each with a Lean 4 proof and comparator check: the first new case of the Tammes problem since N=14 (2015), a 1996 question on Robbins' problem, and both open cases of a 2017 frog-model conjecture. No model is named and no expert review is known.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":10,"filed":"2026-10-18","updated":"2026-10-26","orgs":["mt0-svg (pseudonymous)"]},"problem":"Tammes problem for 15 points (open since Tammes 1930; solved for N ≤ 14 and N = 24); Robbins' problem (is V > 2?, Bruss–Ferguson 1996); Hoffman–Johnson–Junge frog-model conjecture for d = 3, 4 (2017)","short":"Tammes problem for 15 points","year_posed":1930,"years_open":96,"ai_system":["unnamed AI system(s)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Claimed fully AI-generated ('results, proofs and code were found and written by AI'); the human operator is anonymous. One Tammes theorem came from a third party using Claude","result":"Tammes optimum for N = 15 determined, with exactly two optimal configurations; V > 2 for Robbins' problem; frog model recurrent on the 3-ary and transient on the 4-ary tree; plus three smaller conjectures settled.","field":"mathematics"},{"entry":{"id":"2026-09-07-koethe-conjecture-disproved","url":"https://postcutoff.com/e/2026-09-07-koethe-conjecture-disproved/","date":"2026-09-07","date_precision":"day","short_title":"Pre-release GPT-6 Astra disproves the Köthe conjecture with a Lean-verified counterexample","deck":null,"takeaway":"If it survives review, it resolves one of the most famous open problems in ring theory, found autonomously and verified formally.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI","Epoch AI"]},"problem":"Köthe conjecture","short":"Köthe conjecture","year_posed":1930,"years_open":96,"ai_system":["GPT-6 Astra (pre-release)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous discovery; humans checked and wrote up","result":"Explicit counterexample disproving the Köthe conjecture, formally verified in Lean.","field":"mathematics"},{"entry":{"id":"2026-10-15-hopf-conjecture-t3-symmetry-chatgpt-astra","url":"https://postcutoff.com/e/2026-10-15-hopf-conjecture-t3-symmetry-chatgpt-astra/","date":"2026-10-15","date_precision":"day","short_title":"Hopf's Euler characteristic conjecture proved under T³ symmetry","deck":"The first proof came from ChatGPT-6-Astra 'with minimal human input'","takeaway":"The symmetry needed for Hopf's 1930s conjecture drops from a 4-torus to a 3-torus; the paper says the starting proof of its main theorem was obtained with GPT-6 Astra.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-16","updated":"2026-10-16","orgs":["OpenAI","University of Münster"]},"problem":"Hopf's conjecture: even-dimensional manifolds of positive sectional curvature have positive Euler characteristic (symmetric case)","short":"Hopf's conjecture","year_posed":1931,"years_open":95,"ai_system":["GPT-6 Astra"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: GPT-6 Astra produced the first proof of Theorem A with minimal human input; the stronger structure theorem, orbifold case and the write-up are human","result":"Proved for manifolds and orbifolds admitting an effective isometric T³-action (previously T⁴), with the rational cohomology of fixed-point components classified.","field":"mathematics"},{"entry":{"id":"2026-09-06-odin-ai-agent-positive-curvature-s3xs3-hopf-sign-conjecture","url":"https://postcutoff.com/e/2026-09-06-odin-ai-agent-positive-curvature-s3xs3-hopf-sign-conjecture/","date":"2026-09-06","date_precision":"day","short_title":"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)","deck":null,"takeaway":"If correct, a 1930s conjecture of Hopf falls in dimension 6, with the construction and proof credited to an AI agent; no geometer has confirmed it yet.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":6,"filed":"2026-10-16","updated":"2026-10-16","orgs":["Odin Automatic AI Research Agent","Harvard T.H. Chan School of Public Health"]},"problem":"Hopf's sign conjecture: closed even-dimensional manifolds with positive sectional curvature have positive Euler characteristic","short":"Hopf's sign conjecture","year_posed":1931,"years_open":95,"ai_system":["Odin Automatic AI Research Agent"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous per the authors: the agent 'discovered' the metric and the proof; human authors posted and presumably edited the paper","result":"Claimed smooth metric of strictly positive sectional curvature on S³×S³ (χ = 0), a counterexample in dimension 6; preceded by a claimed metric on S²×S³","field":"mathematics"},{"entry":{"id":"2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1","url":"https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/","date":"2026-09-03","date_precision":"day","short_title":"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","deck":null,"takeaway":"Problem #1 is probably the longest-standing open Erdős problem.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":10,"official":7,"filed":"2026-09-30","updated":"2026-10-16","orgs":["OpenAI","Epoch AI"]},"problem":"Erdős problems #1 (distinct subset sums), #74, #126 and #571 (rational exponents conjecture)","short":"Erdős problems #1, #74, #126 and #571","year_posed":1931,"years_open":95,"ai_system":["GPT-6 Astra (pre-release)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous: the model worked in an agent harness without human steering; humans chose the problems, wrote the Lean statements and wrote informal expositions afterwards","result":"#1 disproved (N ≤ ε·2^n possible for arbitrarily large n); #571 proved (every rational α in [1,2) is a Turán exponent of a bipartite graph); #74 answered in the negative direction; #126 proved with |S(A)| ≫ n^(1/2).","field":"mathematics"},{"entry":{"id":"2026-08-13-banach-isometric-conjecture-solved","url":"https://postcutoff.com/e/2026-08-13-banach-isometric-conjecture-solved/","date":"2026-08-13","date_precision":"day","short_title":"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","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Xinbao Lu","Kaiwen Yang","Antonio Acuaviva","Tomasz Kania","OpenAI"]},"problem":"Banach's isometric subspace conjecture","short":"Banach's isometric subspace conjecture","year_posed":1932,"years_open":94,"ai_system":["ChatGPT 5.5 Pro","ChatGPT 5.6 Pro","GPT-5.6 Sol"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: the humans made the reduction; the key approach came out of dialogue with ChatGPT Pro, and GPT-5.6 Sol drafted proofs that the authors checked and rewrote","result":"Real case completed (all odd n, with Gromov's even n); complex and quaternionic analogues also proved.","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-borsuk-conjecture-dimension-nine-counterexample","url":"https://postcutoff.com/e/2026-10-06-openai-borsuk-conjecture-dimension-nine-counterexample/","date":"2026-10-06","date_precision":"day","short_title":"Borsuk's conjecture fails in dimension 9 (OpenAI claim, Lean-checked)","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 maths release (family 156, dated 23 Sept) claims that the compact set of rank-one projectors onto lines in ℝ⁴, a copy of real projective 3-space in a 9-dimensional space, cannot be cut into ten pieces of smaller diameter. That would make 9 the smallest known dimension in which Borsuk's 1933 conjecture fails. The record had been 63, set in May 2026 with a finite 321-point set. The main theorem has a Lean proof, which a community Comparator run accepted. Gil Kalai called it 'a major progress' and said the topological proof 'looks very beautiful', but added that it still needs careful checking. The example is infinite, so for finite sets the record is still 63. Gábor Damásdi showed that every finite subset of it can be split into eight smaller pieces.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":18,"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Borsuk's conjecture (1933): every bounded set in ℝᵈ can be partitioned into d + 1 sets of smaller diameter; smallest dimension in which it fails","short":"Borsuk's conjecture","year_posed":1933,"years_open":93,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Autonomous output released without named authors; the construction was known (Kahn–Kalai, Kalai 2015), the covering obstruction is the model's","result":"Claimed: the compact projector image of ℝP³ in the 9-dimensional trace-one space needs more than ten pieces, so the conjecture fails in every dimension d ≥ 9 (previous smallest known dimension 63, for finite sets). Every finite subset needs at most eight pieces (Damásdi), so the finite version below 63 stays open","field":"mathematics"},{"entry":{"id":"2026-11-06-navier-stokes-unforced-leray-hopf-nonuniqueness-buckmaster-coiculescu-li","url":"https://postcutoff.com/e/2026-11-06-navier-stokes-unforced-leray-hopf-nonuniqueness-buckmaster-coiculescu-li/","date":"2026-11-06","date_precision":"day","short_title":"Navier–Stokes: unforced Leray–Hopf non-uniqueness, AI-written","deck":null,"takeaway":"On 6 Nov 2026 three NYU mathematicians deposited on Hexagon an 80-page proof, written mainly by Claude Opus and Codex, that the unforced 3D Navier–Stokes equations have a continuum of Leray–Hopf solutions with the same initial data, following the Jia–Šverák programme. The authors say they understand the broad steps; a Lean formalization is announced but not yet public.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-11-07","updated":"2026-11-08","orgs":["New York University","Anthropic","OpenAI"]},"problem":"Uniqueness of Leray–Hopf weak solutions of the unforced 3D incompressible Navier–Stokes equations (Leray 1934; Jia–Šverák programme 2014–15)","short":"Uniqueness of Leray–Hopf weak solutions of the unforced 3D incompressible Navier–Stokes equations","year_posed":1934,"years_open":92,"ai_system":["Claude Opus","Codex"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"AI-generated text and proofs; authors understood the broad steps of the construction","result":"Claimed continuum of distinct suitable Leray–Hopf solutions from one compactly supported axisymmetric datum in L^{3,∞}, without force","field":"mathematics"},{"entry":{"id":"2026-10-11-landau-siegel-zeros-banks-explicit-account","url":"https://postcutoff.com/e/2026-10-11-landau-siegel-zeros-banks-explicit-account/","date":"2026-10-11","date_precision":"day","short_title":"William Banks writes out OpenAI's Landau–Siegel proof with explicit constants","deck":"(1 − β) log q ≥ 0.003 for every real character, and says he verified it in full","takeaway":"The first detailed human account of OpenAI's claim that Landau–Siegel zeros do not exist gives an explicit, effective constant. Banks used Claude to draft it but says he worked through every line.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":3,"filed":"2026-10-14","updated":"2026-10-23","orgs":["OpenAI","Anthropic","University of Missouri"]},"problem":"Landau–Siegel zeros (exceptional real zeros of real Dirichlet L-functions)","short":"Landau–Siegel zeros","year_posed":1935,"years_open":91,"ai_system":["OpenAI internal model (original proof)","Claude"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Expert reconstruction: the proof is OpenAI's; Claude drafted the report, expansion and constants; Banks checked every line","result":"Explicit version of OpenAI's claimed exclusion: (1 − β) log q ≥ 3.0 × 10⁻³ for every primitive real character of conductor q ≥ 3 (with Lu–Zaman–Zhao for small q).","field":"mathematics"},{"entry":{"id":"2026-10-12-bloch-constant-conjecture-disproved","url":"https://postcutoff.com/e/2026-10-12-bloch-constant-conjecture-disproved/","date":"2026-10-12","date_precision":"day","short_title":"Bloch constant conjecture (Ahlfors–Grunsky, 1937) disproved","deck":"Teng Zhang shows B < 0.47186… − 1.3×10⁻⁹, with a Lean proof generated by Codex","takeaway":"The value Ahlfors and Grunsky conjectured for Bloch's constant in 1937 is not the true value. The human proof uses a genus-two translation surface; ChatGPT helped explore approaches and Codex generated the Lean formalization.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-14","updated":"2026-10-14","orgs":["Xi'an Jiaotong University","OpenAI"]},"problem":"Bloch constant conjecture: the Bloch constant equals the Ahlfors–Grunsky value 0.4718616…","short":"Bloch constant conjecture","year_posed":1937,"years_open":89,"ai_system":["ChatGPT","OpenAI Codex"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"Human-led: proofs developed and written by the author; ChatGPT for exploring approaches and editing; Codex generated the Lean formalization","result":"Disproved: B < A_AG − 1.3 × 10⁻⁹.","field":"mathematics"},{"entry":{"id":"2026-09-06-collatz-positive-density-log-time-mazur-codex","url":"https://postcutoff.com/e/2026-09-06-collatz-positive-density-log-time-mazur-codex/","date":"2026-09-06","date_precision":"day","short_title":"Collatz positive-density convergence, AI-developed, Lean-checked","deck":null,"takeaway":"The first linear lower bound for Collatz-convergent integers (earlier bounds were X^0.84, then X^0.90): an explicit positive proportion of n < X reach 1 within 10.46 ln n steps. Developed mainly by AI agents, Lean-checked twice, not yet peer reviewed. The Collatz conjecture itself stays open.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":3,"filed":"2026-10-18","updated":"2026-10-19","orgs":["ProofAtlas","OpenAI","Axiom Math"]},"problem":"Collatz conjecture: lower bounds on the number of integers below X whose orbit reaches 1","short":"Collatz conjecture","year_posed":1937,"years_open":89,"ai_system":["OpenAI GPT models via Codex","ProofAtlas.ai harness","Claude (manuscript review)","AxiomProver (formalization of Kenny Lau's refinement)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-led: 'developed primarily by AI agents' with high-level guidance from Lech Mazur, who directed the work and is responsible for the claims","result":"A positive lower natural density of integers reach 1 within 10.46 ln n ordinary Collatz steps (linear count cX, with explicit but astronomically small c), improving earlier sublinear bounds X^0.84 and X^0.90.","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-mahler-conjecture-lean","url":"https://postcutoff.com/e/2026-10-06-openai-mahler-conjecture-lean/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI proves the Mahler conjectures (Lean)","deck":null,"takeaway":"Family 087 of OpenAI's 6 Oct 2026 release claims Mahler's conjecture for symmetric convex bodies (volume product at least 4ⁿ/n!, equality exactly for Hanner polytopes) and for general convex bodies (equality exactly for simplices) in every dimension, plus a symplectic statement that gives a second proof of the symmetric case. The symmetric case was known only up to dimension 3. Both main theorems have Lean proofs whose statements read as the standard conjectures, and the community comparator run accepted them. No expert has yet published a full check.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":8,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Mahler's conjecture (1939) on the minimal volume product |K||K°| of convex bodies, symmetric and general, with equality cases","short":"Mahler's conjecture on the minimal volume product |K||K°| of convex bodies, symmetric and general, with equality cases","year_posed":1939,"years_open":87,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","result":"Claimed: sharp lower bounds 4ⁿ/n! (symmetric; equality for Hanner polytopes) and (n+1)ⁿ⁺¹/(n!)² (general; equality for simplices) in every dimension; Gromov width of K × K° equals 4","field":"mathematics"},{"entry":{"id":"2026-07-20-jacobian-conjecture-counterexample","url":"https://postcutoff.com/e/2026-07-20-jacobian-conjecture-counterexample/","date":"2026-07-20","date_precision":"day","short_title":"Claude Fable 5 finds a counterexample to the Jacobian conjecture in dimension 3","deck":null,"takeaway":"The Jacobian conjecture is one of the most famous open problems in algebra.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":6,"filed":"2026-09-29","updated":"2026-11-10","orgs":["Anthropic"]},"problem":"Jacobian conjecture (Keller 1939): a polynomial map with non-zero constant Jacobian determinant is invertible","short":"Jacobian conjecture","year_posed":1939,"years_open":87,"ai_system":["Claude Fable 5"],"status":"confirmed","verification":{"key":"none","label":"Unverified claim"},"human_role":"AI-assisted: human chose the problem and verified; Claude found the counterexample","result":"Explicit non-injective polynomial map C³→C³ with constant Jacobian determinant, disproving the conjecture for all n≥3.","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-kaplansky-zero-divisor-direct-finiteness-counterexamples-claim","url":"https://postcutoff.com/e/2026-10-06-openai-kaplansky-zero-divisor-direct-finiteness-counterexamples-claim/","date":"2026-10-06","date_precision":"day","short_title":"Kaplansky's zero-divisor and direct-finiteness conjectures refuted? (OpenAI, Lean)","deck":null,"takeaway":"Two families of OpenAI's 6 Oct release claim to refute group-ring conjectures from Kaplansky's 1950s list. One claims a finitely presented torsion-free group G whose group algebra F₂[G] has zero divisors, where the conjecture predicted none. The other claims group algebras in characteristic 2, in odd characteristic and over a torsion-free group in which ab = 1 but ba ≠ 1. Such examples need non-sofic groups, so they would also refute Gottschalk's 1973 surjunctivity conjecture. A further paper refutes the unrestricted Determinant Conjecture. Lean covers the zero-divisor, the torsion examples and the determinant claim, not the torsion-free direct-finiteness example. Specialists have commented on the presentation, not on correctness.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Kaplansky's zero-divisor conjecture (K[G] has no zero divisors for torsion-free G) and direct-finiteness conjecture (ab = 1 implies ba = 1 in K[G]); Gottschalk's surjunctivity conjecture; the group-ring Determinant Conjecture","short":"Kaplansky's zero-divisor conjecture","year_posed":1940,"years_open":86,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; specialists have commented on presentation and consequences, not on correctness","result":"Claimed counterexamples: zero divisors in F₂[G] for a finitely presented torsion-free G with a 2-dimensional classifying space; one-sided inverses in K[G] in characteristic 2 (finitely presented G), odd characteristic and over F₂ for torsion-free G; injective non-surjective cellular automata; a Z[G]-matrix with Fuglede–Kadison determinant in (0, 1)","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-hilbert-smith-conjecture-claim","url":"https://postcutoff.com/e/2026-10-06-openai-hilbert-smith-conjecture-claim/","date":"2026-10-06","date_precision":"day","short_title":"Hilbert–Smith conjecture claimed by OpenAI, Lean-checked by a separate AI proof","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release (family 304, dated 23 Sept) claims the Hilbert–Smith conjecture, the manifold-action form of Hilbert's fifth problem, in every dimension. Only dimensions up to three were known (Pardon, 2013). OpenAI's paper has no Lean proof. On 8 Oct a GitHub author posted a separate 17-page proof written by 'A. Dabrowski's AI agents', with a Lean 4 formalization that the Palomar registry lists since 16 Oct. We found no public expert assessment of either proof as of 29 Oct.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Hilbert–Smith conjecture: a locally compact group acting faithfully and continuously on a connected manifold is a Lie group (equivalently, ℤ_p has no faithful action on a manifold)","short":"Hilbert–Smith conjecture","year_posed":1941,"years_open":85,"ai_system":["OpenAI internal model (math release)","Claude Opus 5.5","GPT 6.1"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"OpenAI: autonomous output released without named authors. Separate proof: AI agents run 'under human guidance' by the repository owner (adbrw)","result":"Claimed proof in every finite dimension (OpenAI manuscript); separate AI-agent proof with a registered Lean 4 formalization under second-countability assumptions","field":"mathematics"},{"entry":{"id":"2026-10-16-max-hamiltonian-cycles-tournaments-gpt-5-6-wormald","url":"https://postcutoff.com/e/2026-10-16-max-hamiltonian-cycles-tournaments-gpt-5-6-wormald/","date":"2026-10-16","date_precision":"day","short_title":"Tournament Hamiltonian cycles: GPT-5.6 settles Alon's question","deck":null,"takeaway":"Princeton's Divoux and Wang asked GPT-5.6 for a lower bound; a second instance turned the method into the matching upper bound, closing a line of work that runs from Szele's 1943 paper through Alon (1990) and Friedgut–Kahn.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Princeton University"]},"problem":"Maximum number of Hamiltonian cycles (and paths) in an n-vertex tournament: Alon's question (1990) and Wormald's conjecture on the extremal tournament","short":"Maximum number of Hamiltonian cycles in an n-vertex tournament","year_posed":1943,"years_open":83,"ai_system":["GPT-5.6"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Authors chose the problem and the strategy hint (permanent estimates); GPT-5.6 found the lower-bound proof and a second instance the upper bound; authors rewrote, simplified and exposited","result":"C(n) = (cosh 1/cos 1 ± o(1))(n−1)!/2ⁿ, attained asymptotically by the cyclic regular tournament; asymptotic maximum number of Hamiltonian paths follows","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-hadwiger-conjecture-counterexample-claim","url":"https://postcutoff.com/e/2026-10-06-openai-hadwiger-conjecture-counterexample-claim/","date":"2026-10-06","date_precision":"day","short_title":"Hadwiger's conjecture disproved? (OpenAI, Lean)","deck":null,"takeaway":"A 104-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims arbitrarily large graphs whose chromatic number exceeds their largest clique minor. That would refute Hadwiger's 1943 conjecture, a far-reaching generalization of the four colour theorem known to hold only up to K6. The examples have no three pairwise non-adjacent vertices and also beat the fractional weakening. A Lean formalization of the ordinary counterexample, added in OpenAI's 7 Oct update, was accepted in a community comparator run, and an independent Lean project derived the fractional version. The graphs come from a random construction; no explicit example has been published, and no expert has published an assessment.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Hadwiger's conjecture (1943): every graph with no K_t minor is (t−1)-colourable, i.e. χ(G) ≤ h(G); also its fractional weakening χ_f(G) ≤ h(G)","short":"Hadwiger's conjecture","year_posed":1943,"years_open":83,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; independent Lean re-packaging by a volunteer; no human exposition yet","result":"Claimed arbitrarily large counterexamples with independence number at most two, beating even the fractional form (Lean covers the ordinary form; an independent Lean project derives the fractional one); companion claims: Colin de Verdière chromatic conjecture false, linear list-Hadwiger bound true","field":"mathematics"},{"entry":{"id":"2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra/","date":"2026-10-04","date_precision":"day","short_title":"Linear Hadwiger conjecture proved","deck":"K_t-minor-free graphs are O(t)-colourable; proof found by GPT-6 Astra (Norin & Steiner), Lean-verified by Codex","takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-06","updated":"2026-10-19","orgs":["OpenAI","McGill University","ETH Zurich"]},"problem":"Linear Hadwiger conjecture (K_t-minor-free graphs have chromatic number O(t))","short":"Linear Hadwiger conjecture","year_posed":1943,"years_open":83,"ai_system":["GPT-6 Astra","OpenAI Codex (6 Sol, Astra)"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"AI-generated proof under human direction: authors set the strategy and sub-goals; GPT-6 Astra wrote essentially all of the proof; Codex produced the Lean formalization","result":"Proved: a universal C with χ(G) ≤ Ct for all K_t-minor-free G","field":"mathematics"},{"entry":{"id":"2026-10-02-linnik-constant-3-99-naslund-claude-gpt-astra","url":"https://postcutoff.com/e/2026-10-02-linnik-constant-3-99-naslund-claude-gpt-astra/","date":"2026-10-02","date_precision":"day","short_title":"Linnik's constant ≤ 3.99, written by Claude Opus 5.5 and GPT Astra","deck":null,"takeaway":"Linnik proved in 1944 that the least prime congruent to a mod q is at most q^L for an absolute constant L. Heath-Brown brought L down to 5.5 in 1992 and Xylouris to 5. On 2 Oct Eric Naslund posted on Hexagon a computer-assisted proof that L ≤ 3.99. It rests on a weighted form of a Heath-Brown lemma that Heath-Brown said no neat way of achieving 'seems available'. The paper says it was written, and the research carried out, by Claude Opus 5.5 and GPT Astra 6 under Naslund's prompting and supervision, with some formalization by Grok. Two components are Lean-verified on Palomar; the final bound itself is not. Hexagon lists human understanding as 'some parts', and we found no expert assessment as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","OpenAI","xAI"]},"problem":"Linnik's constant: the least L with P(a, q) ≪ q^L for the least prime in every reduced residue class mod q","short":"Linnik's constant","year_posed":1944,"years_open":82,"ai_system":["Claude Opus 5.5","GPT-6 Astra","Grok 4.7"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Naslund prompted and supervised; the models did the research and wrote the paper; Grok helped with formalization","result":"Claimed L ≤ 3.99 (P(a, q) < q^3.99 for all large q), improving Xylouris's L ≤ 5","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-unit-distance-upper-bound-power-saving-claim","url":"https://postcutoff.com/e/2026-10-06-openai-unit-distance-upper-bound-power-saving-claim/","date":"2026-10-06","date_precision":"day","short_title":"Unit-distance bound below n^(4/3) claimed (OpenAI, Lean)","deck":null,"takeaway":"A 53-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims that n points in the plane determine at most O(n^(4/3−δ)) pairs at distance one, for some unspecified δ > 0. The bound n^(4/3) of Spencer, Szemerédi and Trotter had not been improved since 1984. A companion proves the weak pinned Erdős distance conjecture. Both are formalized in Lean and passed a community comparator run, and Terence Tao's optimization-constants repository now lists the new exponent. On 25 Oct a commenter posting as WillSawin explained the argument on erdosproblems.com and said he sees how to make its ineffective step effective. Since OpenAI's May counterexample gave a lower bound of n^(1+c), the exponent now lies strictly between 1 and 4/3.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Erdős unit distance problem: the maximum number u(n) of unit distances among n points in the plane; the upper bound O(n^(4/3)) (Spencer–Szemerédi–Trotter, 1984) had not been improved. Also the weak pinned distinct-distance conjecture","short":"Erdős unit distance problem","year_posed":1946,"years_open":80,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; detailed public reading by an outside mathematician (forum comment), no written check","result":"Claimed u(n) = O(n^(4/3−δ)) for an absolute, unspecified δ > 0, and the weak pinned distance theorem; both formalized in Lean","field":"mathematics"},{"entry":{"id":"2026-08-23-hopf-problem-s6-complex-structure","url":"https://postcutoff.com/e/2026-08-23-hopf-problem-s6-complex-structure/","date":"2026-08-23","date_precision":"day","short_title":"Claude-assisted construction claims a complex structure on the 6-sphere, answering Hopf's 1947 problem (pending verification)","deck":null,"takeaway":"The existence of a complex structure on S⁶ is one of the best-known open problems in geometry.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":3,"filed":"2026-09-29","updated":"2026-10-30","orgs":["Anthropic"]},"problem":"Hopf problem: does S⁶ admit an integrable complex structure?","short":"Hopf problem","year_posed":1947,"years_open":79,"ai_system":["Claude (internal research model)"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"AI-assisted: human mathematician (Alpöge) directed and wrote up","result":"Claimed explicit construction of integrable complex structures on S⁶ (answer: yes).","field":"mathematics"},{"entry":{"id":"2026-11-06-axiom-weil-conjectures-lean-formalization","url":"https://postcutoff.com/e/2026-11-06-axiom-weil-conjectures-lean-formalization/","date":"2026-11-06","date_precision":"day","short_title":"Axiom: the Weil conjectures proved in Lean","deck":null,"takeaway":"On 6 Nov 2026 Axiom Math published AxiomMath/Weil, a kernel-checked Lean 4 proof of rationality, the functional equation, the Riemann hypothesis (Deligne) and the Betti-number statement, following Freitag–Kiehl. About 980 modules were written by AI agents; the code has not been reviewed by humans.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-11-07","updated":"2026-11-08","orgs":["Axiom Math"]},"problem":"Weil conjectures (Weil 1949; Dwork 1960, Grothendieck 1960s, Deligne 1974): machine-checked proof","short":"Weil conjectures","year_posed":1949,"years_open":77,"ai_system":["Axiom agents"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI agents wrote the library on top of Axiom's Stacks formalization; no human code review yet","result":"Claimed sorry-free Lean 4 proof of rationality, functional equation, Riemann hypothesis and Betti-number statement for smooth projective varieties over finite fields","field":"mathematics"},{"entry":{"id":"2026-10-06-hodge-conjecture-cm-abelian-varieties-openai","url":"https://postcutoff.com/e/2026-10-06-hodge-conjecture-cm-abelian-varieties-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI release proves the rational Hodge conjecture for all CM abelian varieties, giving the Tate conjecture for abelian varieties over finite fields; Lean proof added 26 Oct, human-written proof by van Hoften and Sempliner, and Milne calls it proved","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":13,"official":8,"filed":"2026-10-07","updated":"2026-11-10","orgs":["OpenAI"]},"problem":"Hodge conjecture for CM abelian varieties; Tate conjecture for abelian varieties over finite fields","short":"Hodge conjecture (CM abelian varieties)","year_posed":1950,"years_open":76,"ai_system":["OpenAI internal model (unreleased)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Listed by OpenAI as an exception to its fixed automated procedure (details not given)","result":"Claimed: every rational Hodge class on a complex CM abelian variety is a ℚ-linear combination of algebraic cycle classes; consequences for Tate and the Hodge standard conjecture.","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-plane-not-five-colourable-hadwiger-nelson-claim","url":"https://postcutoff.com/e/2026-10-06-openai-plane-not-five-colourable-hadwiger-nelson-claim/","date":"2026-10-06","date_precision":"day","short_title":"Chromatic number of the plane is 6 or 7, OpenAI claims (Lean)","deck":null,"takeaway":"A 62-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims that every 5-colouring of the plane has two points at distance one with the same colour, with no regularity assumed on the colour classes. That raises de Grey's 2018 lower bound of five to six, so the chromatic number of the plane is 6 or 7. The proof first transfers arbitrary colourings to a weak measurability condition, then rules out five colours under it. It is formalized in Lean and passed a community comparator run and a stricter independent check. Mathematicians called the method unlike anything in the prior literature; by compactness a finite non-5-colourable unit-distance graph must exist, but none has been displayed.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":10,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Hadwiger–Nelson problem: the least number of colours needed so that no two points of the plane at distance one share a colour (bounds 5 ≤ χ ≤ 7 before the release)","short":"Hadwiger–Nelson problem","year_posed":1950,"years_open":76,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; no human exposition yet; already used as a tool by human authors (with AI help)","result":"Claimed proof that 5 colours never suffice, for arbitrary colour classes, so χ(R²) ∈ {6, 7}; Lean formalization of both bounds","field":"mathematics"},{"entry":{"id":"2026-10-19-quasitrace-monotone-completeness-counterexamples-openai","url":"https://postcutoff.com/e/2026-10-19-quasitrace-monotone-completeness-counterexamples-openai/","date":"2026-10-19","date_precision":"day","short_title":"Kaplansky's quasitrace conjecture reported disproved by an unreleased OpenAI model","deck":"Experts reproduce it and refute the monotone completeness conjecture","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-21","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Kaplansky's quasitrace conjecture (2-quasitraces / AW*-factors) and the monotone completeness conjecture for AW*-algebras","short":"Kaplansky's quasitrace conjecture","year_posed":1951,"years_open":75,"ai_system":["Unreleased OpenAI model","GPT-6 Astra"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"OpenAI model produced the original quasitrace counterexample; Arulseelan and Hanson re-derived it, devised the monotone completeness argument, and reworked GPT 6 Astra's preliminary write-up","result":"Counterexample to the quasitrace conjecture (from an unreleased OpenAI model, reproduced with a modified construction) and a finite AW*-factor that is not monotone complete","field":"mathematics"},{"entry":{"id":"2026-10-06-irrationality-exponent-pi-openai","url":"https://postcutoff.com/e/2026-10-06-irrationality-exponent-pi-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the irrationality exponent of π is exactly 2 (Lean-formalized), which implies the Flint–Hills series converges","deck":null,"takeaway":"Since Mahler's 1953 bound of 42, mathematicians had lowered the irrationality exponent of π only to about 7.1. OpenAI's release claims the conjectured value 2 in a 23-page paper with a Lean proof. Six days on, no expert has publicly assessed it; an outsider has already adapted the Lean proof to log 2.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":8,"filed":"2026-10-12","updated":"2026-10-13","orgs":["OpenAI"]},"problem":"Irrationality exponent (measure) of π, conjectured to be 2; convergence of the Flint–Hills series","short":"Irrationality exponent of π","year_posed":1953,"years_open":73,"ai_system":["OpenAI internal model (unreleased)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous per OpenAI's README: 'The vast majority of results were obtained with the same procedure using an unreleased internal OpenAI model' (about three hours of ChatGPT Pro thinking compute per result on average); family 017 is not among the listed exceptions; no human author is named","result":"Claimed: μ(π) = 2 (ineffective); hence Σ 1/(n³ sin² n) converges, and Σ 1/(n^a |sin n|^b) converges iff a > max{1, b}.","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-borel-conjecture-dimension-four-free-group-not-good","url":"https://postcutoff.com/e/2026-10-06-openai-borel-conjecture-dimension-four-free-group-not-good/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims 4D Borel counterexample and that F₂ is not good","deck":null,"takeaway":"Four manuscripts in OpenAI's 6 Oct 2026 math release (families 305 and 320) make three claims in 4-manifold topology. Freedman's disk embedding theorem fails without a fundamental-group hypothesis, so the free group F₂ is not 'good', the central open question of topological 4-manifold surgery. A four-dimensional Poincaré duality group has no closed aspherical manifold model, against Wall's question. And two closed aspherical 4-manifolds with a hyperbolic fundamental group are homotopy equivalent but not homeomorphic, against the topological Borel conjecture. None has a Lean proof or a published expert check as of 29 Oct.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Is every group good (Freedman's disk embedding for all fundamental groups)? Wall's realization of PD4 groups; topological Borel conjecture in dimension 4","short":"Is every group good","year_posed":1953,"years_open":73,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI","result":"Claimed: disk embedding fails in general and F₂ is not good; a PD4 group with no closed aspherical manifold model; homotopy-equivalent but non-homeomorphic closed aspherical 4-manifolds with hyperbolic fundamental group","field":"mathematics"},{"entry":{"id":"2026-10-26-openai-yang-mills-qcd-existence-claim","url":"https://postcutoff.com/e/2026-10-26-openai-yang-mills-qcd-existence-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims 4D Yang–Mills and QCD existence","deck":null,"takeaway":"The physics third of OpenAI's 26 Oct announcement claims that certain interacting 4D quantum field theories exist mathematically, among them pure Yang–Mills and quantum chromodynamics. That is the existence half of the Yang–Mills Millennium problem, not the mass gap. A group led by David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, is reviewing it. At the group's request the AI-produced material is not public while they write their own account. Nothing can be checked yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":18,"official":3,"filed":"2026-10-27","updated":"2026-11-13","orgs":["OpenAI"]},"problem":"Existence of four-dimensional interacting quantum field theories (pure Yang–Mills, QCD); existence half of the Yang–Mills Millennium problem","short":"Existence of four-dimensional interacting quantum field theories","year_posed":1954,"years_open":72,"ai_system":["OpenAI internal model"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Unknown; a physics panel led by David Gross reviews it and will write its own account","result":"Claimed rigorous existence proof; details, method and paper not public","field":"physics"},{"entry":{"id":"2026-10-06-harary-hill-zarankiewicz-crossing-numbers-openai","url":"https://postcutoff.com/e/2026-10-06-harary-hill-zarankiewicz-crossing-numbers-openai/","date":"2026-10-06","date_precision":"day","short_title":"Harary–Hill and Zarankiewicz crossing-number conjectures (OpenAI, Lean)","deck":null,"takeaway":"Two short manuscripts credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claim the exact crossing numbers of complete graphs and complete bipartite graphs. Those are the Harary–Hill conjecture (late 1950s) and Zarankiewicz's 1954 formula for Turán's 1944 brick-factory problem. Lean proves both formulas, and a community comparator run accepted both Lean challenges. On 25 Oct the graph-drawing specialist Martin Balko posted a simplified exposition of the Harary–Hill proof, which he says is based on the polynomial method. Gelasio Salazar is preparing another.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":6,"filed":"2026-10-27","updated":"2026-10-28","orgs":["OpenAI","Charles University"]},"problem":"Harary–Hill (Guy) conjecture on cr(K_n); Zarankiewicz's crossing-number conjecture for K_m,n (Turán's brick-factory problem, 1944)","short":"Harary–Hill conjecture on cr(K_n); Zarankiewicz's crossing-number conjecture for K_m,n","year_posed":1954,"years_open":72,"ai_system":["OpenAI internal model (openai/math release)","ChatGPT (Astra) for Balko's exposition"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Manuscripts credited to 'OpenAI' as model output with no human author; human reading, simplification and exposition by Balko (with ChatGPT's help in understanding the proof), another exposition in preparation by Salazar","result":"Claimed exact formulas for all n and all m, n, with Lean proofs of both equalities; independent simplified human proof of the K_n case (Balko)","field":"mathematics"},{"entry":{"id":"2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean","url":"https://postcutoff.com/e/2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean/","date":"2026-10-20","date_precision":"day","short_title":"MDS conjecture settled in even characteristic","deck":"Two independent proofs, one checked in Lean with Claude's help, the other written with ChatGPT","takeaway":"If it holds, this is the biggest advance on the conjecture since Ball's prime-field theorem.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-21","updated":"2026-10-21","orgs":["South China University of Technology","Guangzhou University","OpenAI","Anthropic"]},"problem":"Linear MDS conjecture (Segre 1955; maximum length of MDS codes / arcs), even characteristic","short":"Linear MDS conjecture","year_posed":1955,"years_open":71,"ai_system":["ChatGPT","Claude","Claude Opus 5.5"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Human-led with AI tools: Kumar devised the argument and Claude helped formalize it in Lean; Chen and Huang used ChatGPT to explore proof strategies and check calculations","result":"For q even and 2 ≤ k ≤ q, the maximum length of a k-dimensional linear MDS code over F_q is q + 1, or q + 2 when k ∈ {3, q − 1}","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-kadison-similarity-problem-lean","url":"https://postcutoff.com/e/2026-10-06-openai-kadison-similarity-problem-lean/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Kadison's similarity problem (Lean)","deck":null,"takeaway":"Family 288 of OpenAI's 6 Oct 2026 release (paper dated 23 Sept) claims a positive answer to Kadison's similarity problem of 1955. The claim: every bounded unital homomorphism from a unital C*-algebra into the operators on a Hilbert space becomes a *-homomorphism after a bounded change of coordinates. The paper also claims one hyperreflexivity constant valid for every von Neumann algebra. Before, only special cases were known: nuclear algebras, cyclic representations, algebras without tracial states, and II₁ factors with property Γ. Lean statements of the similarity theorem, the hyperreflexivity bound and the underlying commutator estimate were accepted in two independent comparator runs. As of 29 Oct we found no assessment by an operator algebraist.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Kadison's similarity problem (1955): is every bounded representation of a C*-algebra on a Hilbert space similar to a *-representation?","short":"Kadison's similarity problem","year_posed":1955,"years_open":71,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","result":"Claimed: yes, for every unital C*-algebra and every Hilbert space, together with a uniform amplified commutator estimate and one hyperreflexivity constant for all von Neumann algebras","field":"mathematics"},{"entry":{"id":"2026-10-29-berry-esseen-constant-esseen-sharp-certificate-claim","url":"https://postcutoff.com/e/2026-10-29-berry-esseen-constant-esseen-sharp-certificate-claim/","date":"2026-10-29","date_precision":"day","short_title":"Sharp Berry–Esseen constant claimed: C = Esseen's 0.4097","deck":null,"takeaway":"The Berry–Esseen inequality bounds how far a normalised sum of i.i.d. variables is from the normal law. Its best constant has been open since Esseen showed in 1956 that it cannot be below (3+√10)/(6√(2π)) ≈ 0.4097; the best published upper bound is Shevtsova's 0.4690. On 29 Oct Bartosz Kołodziejek and Kacper Rodziewicz (Warsaw) posted on Hexagon a claimed proof that Esseen's value is the exact constant for every sample size n. The proof combines analytic reductions with exact finite certificates in 13 index ranges, checked end to end by their own programs, with a one-command Docker replay. They used ChatGPT, Codex and Gemini/Antigravity 'extensively'. It is not a Lean proof, and no specialist had checked it by 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Warsaw University of Technology","University of Warsaw","OpenAI","Google"]},"problem":"Optimal constant in the i.i.d. Berry–Esseen inequality (Esseen 1956 lower bound; sharpness conjecture attributed to Zolotarev, 1966–67)","short":"Optimal constant in the i.i.d. Berry–Esseen inequality","year_posed":1956,"years_open":70,"ai_system":["ChatGPT","Codex","Gemini","Antigravity"],"status":"pending","verification":{"key":"expert","label":"Expert-checked"},"human_role":"AI-assisted: the two authors directed the reductions, certificate schemas and verification runs; AI systems helped with exploration, checking code, distributed verification and drafting; Hexagon text label 'primarily AI-generated', understanding 'some parts'","result":"Claimed: C_n ≤ C_E = (3+√10)/(6√(2π)) ≈ 0.40973 for every n ≥ 1, so the optimal constant equals Esseen's lower bound","field":"mathematics"},{"entry":{"id":"2026-09-09-pierce-birkhoff-conjecture-disproved","url":"https://postcutoff.com/e/2026-09-09-pierce-birkhoff-conjecture-disproved/","date":"2026-09-09","date_precision":"day","short_title":"Pierce–Birkhoff conjecture disproved by a multi-agent GPT + Claude harness","deck":"Counterexamples Lean-verified","takeaway":"Pierce–Birkhoff is a classic problem in real algebraic geometry and ordered rings, open for 70 years.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-10-19","orgs":["University of Chicago","Lek-Heng Lim","Zehua Lai","Junyu Ren","OpenAI","Anthropic"]},"problem":"Pierce–Birkhoff conjecture","short":"Pierce–Birkhoff conjecture","year_posed":1956,"years_open":70,"ai_system":["GPT-5.6","GPT-6","Claude Opus 5","Claude Fable 5.1"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"Humans gave high-level direction on which targets mattered and wrote the final exposition; a multi-model agent system found and reviewed the construction","result":"Explicit continuous piecewise-quadratic function on R^30 that is not a finite lattice combination of polynomials; further counterexamples in dimensions 5, 6, 7, 22, 24 and 72.","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-anderson-localization-2d-delocalization-3d","url":"https://postcutoff.com/e/2026-10-06-openai-anderson-localization-2d-delocalization-3d/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Anderson localization in 2D and extended states in 3D","deck":null,"takeaway":"Two manuscripts in OpenAI's 6 Oct 2026 math release (family 261, dated 23 Sept) claim the two central conjectures about electrons in a random lattice. In two dimensions, the Anderson model has pure-point spectrum, so all states are localized, at every disorder strength. In three or more dimensions, weak disorder leaves purely absolutely continuous spectrum, meaning extended states, on an energy interval. These are Barry Simon's Problems 1 and 2 of 2000, for uniform disorder. Specialists Jakšić and Last shortened the last step of the 3D proof in an appendix, and a third party has posted a Lean formalization of the 3D result. OpenAI's own Lean covers only the 2D spectrum's location.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Simon's Problem 2 (localization at all disorder in 2D) and Problem 1 (absolutely continuous spectrum / extended states at weak disorder in d ≥ 3) for the Anderson model","short":"Simon's Problem 2","year_posed":1958,"years_open":68,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"Autonomous AI output released by OpenAI; an expert appendix shortens one step; a third-party AI-assisted Lean formalization of the 3D result","result":"Claimed for uniform site potentials: pure-point spectrum throughout the spectrum in 2D at every disorder; purely absolutely continuous spectrum on an open energy interval in every d ≥ 3 at small disorder","field":"physics"},{"entry":{"id":"2026-08-05-sendov-conjecture-proved","url":"https://postcutoff.com/e/2026-08-05-sendov-conjecture-proved/","date":"2026-08-05","date_precision":"day","short_title":"Sendov's 1958 conjecture on polynomial roots proved with GPT-5.6 Pro","deck":"Tao simplifies and formalises it","takeaway":"It is a classic, well-known conjecture closed by AI, with the leading expert on the problem verifying and formalising the result.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":1,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},"problem":"Sendov's conjecture","short":"Sendov's conjecture","year_posed":1958,"years_open":68,"ai_system":["GPT-5.6 Pro"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Human orchestrated (Mazur); Tao simplified and formalised with AI agents","result":"Proof of Sendov's conjecture for all degrees, formally verified in Lean.","field":"mathematics"},{"entry":{"id":"2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition","url":"https://postcutoff.com/e/2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition/","date":"2026-10-21","date_precision":"day","short_title":"Nagata's 1959 conjecture","deck":"Algebraic geometer Jenia Tevelev writes out OpenAI's 'remarkably simple' proof and calls the release 'a stunning achievement'","takeaway":"Nagata's conjecture on plane curves through very general points, open since 1959, was one of the claims in OpenAI's 6 Oct maths release. A UMass Amherst specialist has now reorganized, simplified and published the full proof himself, while criticizing how the manuscripts were dumped on the community.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","UMass Amherst"]},"problem":"Nagata's conjecture on plane curves (1959)","short":"Nagata's conjecture on plane curves","year_posed":1959,"years_open":67,"ai_system":["OpenAI internal model (unreleased)"],"status":"confirmed","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI proof (OpenAI release, no human author); Tevelev re-derived, simplified and presented it, using ChatGPT Astra only for editing and figures","result":"For r ≥ 10 very general points in P², every nonzero curve of degree d with multiplicities m_i satisfies Σ m_i < d√r (claimed by OpenAI; reorganized and re-proved by Tevelev).","field":"mathematics"},{"entry":{"id":"2026-10-28-kisin-serre-artin-character-conjecture-gpt-astra","url":"https://postcutoff.com/e/2026-10-28-kisin-serre-artin-character-conjecture-gpt-astra/","date":"2026-10-28","date_precision":"day","short_title":"Kisin proves Serre's 1960 Artin-character conjecture with GPT Astra","deck":null,"takeaway":"In 1960 Serre conjectured that a function built from fixed-point multiplicities of a finite group acting on a regular local ring is the character of an ℓ-adic representation. It is the higher-dimensional analogue of the Artin character of local fields. Tomoyuki Abe proved the positive-characteristic case (Inventiones, 2025). On 28 Oct Harvard's Mark Kisin posted the mixed-characteristic case on Hexagon, which finishes the conjecture. He writes that 'extensive use of GPT Astra was made both to arrive at the proofs … and to write a first draft', and that 'significant human effort' went into checking them. Part of the work was done at 'The Overhang', a San Francisco meeting on 9–10 Sept 2026.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Harvard University","OpenAI"]},"problem":"Serre's 1960 conjecture: the Artin function of a finite group acting on a regular local ring with isolated fixed points is the character of an ℓ-adic representation","short":"Serre's 1960 conjecture","year_posed":1960,"years_open":66,"ai_system":["GPT-6 Astra"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Kisin directed the work; GPT Astra was used extensively to find the proofs and write the first draft; Kisin checked, refined and rewrote","result":"Proof in mixed characteristic, completing the conjecture (positive characteristic: Abe 2025; residue characteristic zero: classical)","field":"mathematics"},{"entry":{"id":"2026-10-13-ulam-problem-lie-groups-countably-representable-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-13-ulam-problem-lie-groups-countably-representable-gpt-6-astra/","date":"2026-10-13","date_precision":"day","short_title":"Ulam's question answered","deck":"Every Lie group is a group of permutations of a countable set (Monod & Thom; key construction from GPT-6 Astra)","takeaway":"Nicolas Monod and Andreas Thom prove that every Lie group embeds in the permutation group of a countable set, answering a classical question of Ulam. They say the key deformation was first built by GPT-6 Astra over three weeks of discussions; Thom earlier criticised how OpenAI credited human work.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":1,"filed":"2026-10-14","updated":"2026-10-14","orgs":["EPFL","TU Dresden","OpenAI"]},"problem":"Ulam's problem: is every Lie group countably representable (isomorphic to a group of permutations of a countable set)?","short":"Ulam's problem","year_posed":1960,"years_open":66,"ai_system":["GPT-6 Astra"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: GPT-6 Astra produced the key countable-index deformation for SL3(R) after three weeks of discussion; the authors extended it to all simple Lie groups, checked it and rewrote the proof","result":"Yes: every Lie group, and hence every locally compact second countable group, embeds in Sym(N).","field":"mathematics"},{"entry":{"id":"2026-11-04-huffman-unequal-letter-costs-np-complete-codex","url":"https://postcutoff.com/e/2026-11-04-huffman-unequal-letter-costs-np-complete-codex/","date":"2026-11-04","date_precision":"day","short_title":"Huffman coding with unequal letter costs is NP-complete (Codex)","deck":null,"takeaway":"A classical problem in information theory, listed as open in surveys of prefix coding for decades, appears settled with the key reduction attributed to a coding agent.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-04","updated":"2026-11-04","orgs":["OpenAI","Indian Institute of Science"]},"problem":"Complexity of optimal prefix-free coding with unequal letter costs","short":"Complexity of optimal prefix-free coding with unequal letter costs","year_posed":1961,"years_open":65,"ai_system":["Codex"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: authors posed the question and guided; Codex produced the reduction strategy and much of the proof","result":"NP-complete even for letter costs (M, M+1) given in the input","field":"computer-science"},{"entry":{"id":"2026-10-10-vaught-conjecture-infinitary-counterexample-lean","url":"https://postcutoff.com/e/2026-10-10-vaught-conjecture-infinitary-counterexample-lean/","date":"2026-10-10","date_precision":"day","short_title":"Vaught's conjecture fails for infinitary logic","deck":"Ackerman, Freer and Knight post a Lean-verified counterexample, built over 8 months with Claude Code and Codex","takeaway":"A sentence of L_ω1,ω with exactly ℵ1 countable models and no perfect set of them refutes the infinitary Vaught conjecture (1961). It follows Robin Knight's 2002 approach, which was never accepted, and the whole argument is now certified in Lean.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-14","updated":"2026-10-29","orgs":["Harvard University","MIT","University of Oxford","Anthropic","OpenAI"]},"problem":"Vaught's conjecture (1961) for L_ω1,ω: a countable theory/sentence has either countably many or perfectly many (2^ℵ0) countable models","short":"Vaught's conjecture for L_ω1,ω","year_posed":1961,"years_open":65,"ai_system":["Claude Code","Codex","Claude Opus 4.5–5.5","Claude Fable 5/5.1","GPT-5.2–GPT-6","GPT Pro"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Human-led: Knight's framework and draft; Ackerman and Freer directed the work; coding agents did the Lean formalization and helped search for the combinatorial template, with SMT/Horn-clause solvers","result":"Disproved for L_ω1,ω: an explicit sentence with exactly ℵ1 countable models and no perfect set of non-isomorphic models.","field":"mathematics"},{"entry":{"id":"2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"GPT-6 Astra proves the Erdős–Sós conjecture with a short counting argument","deck":"Mathematicians race to simplify and extend it","takeaway":"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\".","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":10,"official":9,"filed":"2026-09-30","updated":"2026-09-30","orgs":["OpenAI","Epoch AI"]},"problem":"Erdős–Sós conjecture (Erdős problem #548)","short":"Erdős–Sós conjecture","year_posed":1962,"years_open":64,"ai_system":["GPT-6 Astra (pre-release)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous discovery and Lean proof in Epoch's benchmark harness; humans checked, simplified and generalised it afterwards","result":"Full proof: every n-vertex graph with more than (k−2)n/2 edges contains every tree on k vertices.","field":"mathematics"},{"entry":{"id":"2026-10-17-2d-euler-unforced-nonuniqueness-lp-vorticity-ai","url":"https://postcutoff.com/e/2026-10-17-2d-euler-unforced-nonuniqueness-lp-vorticity-ai/","date":"2026-10-17","date_precision":"day","short_title":"2D Euler non-uniqueness without force claimed three times, all with AI","deck":null,"takeaway":"Uniqueness of 2D Euler solutions is known for bounded vorticity (Yudovich, 1963). With a force, Vishik showed in 2018 that it fails for vorticity in L^p, p > 2. Without a force this stayed open. Within ten days, three groups posted proofs that it fails there as well. On 17 Oct Umberto Pappalettera and Massimo Sorella posted on Hexagon a manuscript 'developed autonomously by ChatGPT 6.1 Sol'. They claim no authorship and run a reading group to check it. On 22 Oct Matei Coiculescu (NYU) posted Part I of a two-part proof whose details, code and text were 'largely produced with Claude Opus 5.5'. His repository has a Lean 4 formalization of the whole proof, with no sorry. On 27 Oct Hyungjun Choi (Princeton) posted an 86-page construction, with ChatGPT completing technical steps. As of 29 Oct we found no expert assessment of any of the three.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":10,"official":7,"filed":"2026-10-29","updated":"2026-11-10","orgs":["OpenAI","Anthropic","Hexagon Mathematics Foundation"]},"problem":"Non-uniqueness of weak solutions of the unforced 2D Euler equations with vorticity in L^p, 2 < p < ∞ (beyond Yudovich's uniqueness class; the unforced form of Vishik's theorem)","short":"Non-uniqueness of weak solutions of the unforced 2D Euler equations with vorticity in L^p, 2 < p < ∞","year_posed":1963,"years_open":63,"ai_system":["GPT-6.1 Sol","Claude Opus 5.5","ChatGPT 5.6","ChatGPT 6.0","ChatGPT 6.1","GPT-6 Astra"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Ranges from none (Pappalettera–Sorella: prompted only, no edits, no claim of authorship) to human strategy with AI-written details (Coiculescu: idea and strategy his, proofs, code and text largely by Claude; Choi: main ideas his, ChatGPT completed technical details)","result":"Three independent claimed constructions of two distinct solutions with the same initial vorticity: vanishing-viscosity solutions for every p > 2 (Pappalettera–Sorella, fully AI-generated), renormalized weak solutions in L¹ ∩ L^q for every q < ∞ (Coiculescu, Lean-formalized), and finite-energy solutions in every finite L^p (Choi)","field":"mathematics"},{"entry":{"id":"2026-10-06-hironaka-resolution-of-singularities-lean-fable-agents","url":"https://postcutoff.com/e/2026-10-06-hironaka-resolution-of-singularities-lean-fable-agents/","date":"2026-10-06","date_precision":"day","short_title":"Hironaka's resolution of singularities formalized in Lean by agents","deck":null,"takeaway":"Resolution (resolution.org) reports that an orchestrated team of agents, mostly Claude Fable 5.1 with occasional checks by GPT-6 Astra, autoformalized Hironaka's 1964 theorem in characteristic zero between 4 and 10 Sept 2026, together with refinements by Kollár, Włodarczyk and Bierstone–Milman. That took about 100B input tokens, 300M output tokens and roughly $135k. A human mathematician intervened about 30 times. The repository says neither library contains a sorry.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":4,"filed":"2026-10-26","updated":"2026-10-29","orgs":["Resolution","Anthropic"]},"problem":"Formalization of Hironaka's resolution of singularities in characteristic zero (1964)","short":"Formalization of Hironaka's resolution of singularities in characteristic zero","year_posed":1964,"years_open":62,"ai_system":["Claude Fable 5.1","GPT-6 Astra"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Humans wrote the 112-page plan, chose sources and intervened ~30 times; agents wrote the proofs and Lean code","result":"Sorry-free Lean 4 + Mathlib library (~500–600k lines) of Hironaka's main theorems and modern refinements, for schemes and analytic spaces","field":"mathematics"},{"entry":{"id":"2026-10-12-erdos-matching-conjecture-large-matchings-ai","url":"https://postcutoff.com/e/2026-10-12-erdos-matching-conjecture-large-matchings-ai/","date":"2026-10-12","date_precision":"day","short_title":"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","deck":null,"takeaway":"Three Shanghai Jiao Tong mathematicians prove Erdős's 1965 matching conjecture for all n once the matching size is large, by first settling a probability conjecture of Łuczak, Mieczkowska and Šileikis. Their disclosure credits AI with the unified argument for k ≥ 6 and with closing a gap.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-14","updated":"2026-10-29","orgs":["Shanghai Jiao Tong University"]},"problem":"Erdős Matching Conjecture (1965); Łuczak–Mieczkowska–Šileikis tail-probability conjecture","short":"Erdős Matching Conjecture","year_posed":1965,"years_open":61,"ai_system":["unnamed generative AI"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: humans built the framework and the k = 2–5 and k ≥ 10^100 cases; AI produced the unified k ≥ 6 argument and the idea that closed a gap","result":"Tail conjecture proved for all k; Erdős Matching Conjecture proved for fixed k ≥ 3, s sufficiently large, all n ≥ k(s+1).","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim","url":"https://postcutoff.com/e/2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims BSD in Selmer corank ≤ 1 and Goldfeld's conjecture","deck":null,"takeaway":"Families 002 and 006 of OpenAI's 6 Oct 2026 maths release claim the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose p-power Selmer group has corank 0 or 1 at some prime, and Goldfeld's conjecture that half of all quadratic twists have analytic rank 0 and half rank 1. Together they would give full BSD for 100% of the quadratic twists of every curve. None of it is formalized, and the experts who have commented read it as a strong technical result in the rank ≤ 1 range that leaves the general conjecture untouched.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Birch and Swinnerton-Dyer conjecture in rank ≤ 1 (full leading-term formula at every prime); Goldfeld's conjecture (1979) on the ranks of quadratic twists","short":"Birch and Swinnerton-Dyer conjecture in rank ≤ 1","year_posed":1965,"years_open":61,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","result":"Claimed: full BSD formula whenever some p-power Selmer group has corank 0 or 1; Goldfeld's 50/50 density and mean analytic rank 1/2 for the quadratic twists of every E/ℚ; hence full BSD for a density-one set of twists of every curve","field":"mathematics"},{"entry":{"id":"2026-10-01-lehmer-permutation-conjecture-opus-5-5","url":"https://postcutoff.com/e/2026-10-01-lehmer-permutation-conjecture-opus-5-5/","date":"2026-10-01","date_precision":"day","short_title":"Lehmer's 1965 permutation conjecture proved","deck":"Claude Opus 5.5 found the short hypercube proof","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":1,"filed":"2026-10-02","updated":"2026-10-09","orgs":["Eindhoven University of Technology","Anthropic","Harmonic"]},"problem":"Lehmer's permutation conjecture (1965): imperfect Hamiltonian traversals of multiset permutations by adjacent swaps; an unsolved research problem in Knuth's TAOCP","short":"Lehmer's permutation conjecture","year_posed":1965,"years_open":61,"ai_system":["Claude Opus 5.5","Aristotle"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Human-led long project; the final short proof was found and developed by Claude Opus 5.5 in a multi-agent setup, then re-derived and checked by the author","result":"Proof of Verhoeff's reformulation (Hamiltonicity of the non-stutter-word graph N(S), with two exceptional families), and hence of Lehmer's conjecture.","field":"mathematics"},{"entry":{"id":"2026-10-09-kinetic-krylov-safonov-pucci-conjecture-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-09-kinetic-krylov-safonov-pucci-conjecture-gpt-6-astra/","date":"2026-10-09","date_precision":"day","short_title":"\"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)","deck":null,"takeaway":"Six analysts, among them Clément Mouhot and Luis Silvestre, say the missing estimate behind a long-open regularity theorem for kinetic equations was found by GPT-6 Astra Ultra on 6 Sept 2026 with no human contribution. They wrote a human exposition and published the prompts, chats and AI manuscripts, and state that they claim no credit.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-12","updated":"2026-10-29","orgs":["OpenAI","University of Cambridge","University of Chicago","UC Irvine"]},"problem":"A Krylov–Safonov Harnack inequality for kinetic equations in nondivergence form with measurable coefficients (missing: an Aleksandrov–Bakelman–Pucci estimate), and Pucci's 1966 conjecture on the optimal integrability exponent in the ABP estimate","short":"Kinetic Krylov–Safonov theorem and Pucci's conjecture","year_posed":1966,"years_open":60,"ai_system":["GPT-6 Astra Ultra","GPT-6 Astra","GPT-6 Pro"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous per the authors: each AI source is marked 'Human contribution: None' (the authors wrote the prompt and research instructions). The six mathematicians checked the kinetic proof, wrote a human exposition and published the chats","result":"Aleksandrov estimate sup u ≤ C‖f‖_{L^p} for Kolmogorov equations with rough coefficients, giving the kinetic Krylov–Safonov theorem; AI manuscripts also claim Pucci's conjecture in every dimension and its parabolic version.","field":"mathematics"},{"entry":{"id":"2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt","url":"https://postcutoff.com/e/2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt/","date":"2026-10-16","date_precision":"day","short_title":"Lonely runner: (1+c)/2n bound with ChatGPT 6","deck":null,"takeaway":"A human-led result on the 1967 lonely runner problem in which ChatGPT 6 Pro proposed the GCD-sum route, pointed to Green–Walker and fixed gaps, but 'the models were unable to improve the lonely runner bound on their own'.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","University of Chinese Academy of Sciences","Ohio State University"]},"problem":"Lonely Runner Conjecture (Wills 1967, Cusick 1973): every runner is at some time at distance ≥ 1/(n+1) from all others","short":"Lonely Runner Conjecture","year_posed":1967,"years_open":59,"ai_system":["ChatGPT 6 Pro","GPT-6 Astra"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led: authors developed the strategy; ChatGPT 6 Pro/Astra proposed the GCD-sum ingredient, pointed to Green–Walker, fixed gaps and turned informal proofs into rigorous ones","result":"ML(V) > (1+c)/(2n) for an absolute constant c > 0, the first constant-factor improvement over the union bound 1/(2n)","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-release-3d-restriction-bochner-riesz-claims","url":"https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims 3D restriction and Bochner–Riesz conjectures","deck":null,"takeaway":"Beside quasi-RH and Unique Games, OpenAI's 6 Oct catalogue claims a whole cluster of central harmonic-analysis conjectures in three dimensions: Stein's restriction conjecture for the sphere, the diagonal extension conjecture, Bochner–Riesz, Sogge's local smoothing and the planar endpoint of Carleson's Schrödinger problem. None has a Lean formalization and no expert review has been published; the first arXiv paper to cite one appeared on 15 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":7,"filed":"2026-10-19","updated":"2026-10-26","orgs":["OpenAI"]},"problem":"Stein's Fourier restriction conjecture in ℝ³ (sphere, bounded data; diagonal form for positively curved surfaces); 3D Bochner–Riesz conjecture; Sogge's local smoothing conjecture in 3D; Sobolev endpoint of Carleson's Schrödinger convergence problem","short":"Stein's Fourier restriction conjecture in ℝ³","year_posed":1967,"years_open":59,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Credited to 'OpenAI' with no named human authors; OpenAI marks the release's review status as unchecked","result":"Claimed full resolution of each in three dimensions (planar endpoint for Carleson) in OpenAI manuscripts of 24 Sept 2026, released 6 Oct; not formally verified and not yet reviewed","field":"mathematics"},{"entry":{"id":"2026-09-21-grad-conjecture-counterexamples","url":"https://postcutoff.com/e/2026-09-21-grad-conjecture-counterexamples/","date":"2026-09-21","date_precision":"day","short_title":"Grad's 1967 conjecture on 3D plasma equilibria falls","deck":"Two AI-assisted papers give three families of counterexamples","takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":7,"official":5,"filed":"2026-09-30","updated":"2026-11-13","orgs":["University of Maryland","OpenAI","Anthropic"]},"problem":"Grad's conjecture on the non-existence of smooth non-symmetric 3D MHD (magnetohydrostatic) equilibria with nested flux surfaces","short":"Grad's conjecture","year_posed":1967,"years_open":59,"ai_system":["GPT-6 Astra Pro","GPT-5.6 Sol","Claude Fable 5","Claude Opus 5","GPT-6 Astra","Claude Fable 5.1"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Landreman: solutions discovered by GPT-6 Astra Pro, equations checked by hand. Gómez-Serrano–Liehr–Taylor: human roadmap, LLMs filled in technical details and wrote the Lean proof under guidance.","result":"Three families of smooth, non-symmetric toroidal equilibria with nested pressure surfaces and non-constant pressure: one C_N-symmetric family proved with Nash–Moser methods and certified in Lean (Gómez-Serrano–Liehr–Taylor), and two explicit analytic families (Landreman).","field":"physics"},{"entry":{"id":"2026-10-21-barnette-conjecture-openai-proof-zhentao-li-exposition","url":"https://postcutoff.com/e/2026-10-21-barnette-conjecture-openai-proof-zhentao-li-exposition/","date":"2026-10-21","date_precision":"day","short_title":"Barnette's 1969 Hamiltonicity conjecture","deck":"Graph theorist Zhentao Li publishes a digested, partly re-proved version of OpenAI's proof","takeaway":"OpenAI's 6 Oct release claimed Barnette's conjecture (every cubic 3-connected bipartite planar graph is Hamiltonian), with a Lean proof. A combinatorialist has now written the argument out by hand, replacing its exponential-sum step with explicit sign-reversing involutions.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-10-23","updated":"2026-10-23","orgs":["OpenAI"]},"problem":"Barnette's conjecture (Hamiltonicity of cubic 3-connected bipartite planar graphs)","short":"Barnette's conjecture","year_posed":1969,"years_open":57,"ai_system":["OpenAI internal model (unreleased)"],"status":"confirmed","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"Autonomous AI proof (OpenAI release); Zhentao Li re-derived it by hand with combinatorial replacements, using LLMs to read the preprint and format the note","result":"Every cubic 3-connected bipartite planar graph has a Hamiltonian cycle; indeed G − e is Hamiltonian for every edge e (claimed by OpenAI; written out and partly re-proved by Li).","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-matrix-multiplication-omega-9-4","url":"https://postcutoff.com/e/2026-10-06-openai-matrix-multiplication-omega-9-4/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI proves ω ≤ 9/4 for matrix multiplication","deck":null,"takeaway":"Family 107 of OpenAI's 6 Oct 2026 release proves that two n × n matrices can be multiplied with O(n^(9/4+ε)) arithmetic operations, the largest drop in the exponent in decades; the best bound before was ω < 2.371177, set in August 2026 with AlphaEvolve. The 13-page proof is non-constructive and comes with a Lean formalization. By 21 Oct, Amanov, van den Berg and Zuiddam had re-proved it over every field and Rafael Pass had given an explicit construction. The resulting algorithms are galactic and give no practical speedup.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":13,"official":9,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Exponent ω of matrix multiplication (Strassen 1969: ω < 2.81; laser-method records near 2.371)","short":"Exponent ω of matrix multiplication","year_posed":1969,"years_open":57,"ai_system":["OpenAI internal model (openai/math release)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous AI output released by OpenAI; independent human-led re-proofs (Amanov–van den Berg–Zuiddam, Pass), both AI-assisted","result":"ω ≤ 9/4 over ℂ (OpenAI, non-constructive, Lean-formalized); re-proved over every field and made constructive by independent authors within three weeks","field":"computer-science"},{"entry":{"id":"2026-09-15-erdos-108-high-girth-problem-disproved-conjectures-io","url":"https://postcutoff.com/e/2026-09-15-erdos-108-high-girth-problem-disproved-conjectures-io/","date":"2026-09-15","date_precision":"day","short_title":"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","deck":null,"takeaway":"This is a well-known problem from Erdős's list, and the counterexample comes with a machine-checked proof.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":7,"official":7,"filed":"2026-10-05","updated":"2026-10-07","orgs":["Conjectures.io","Purdue University","OpenAI"]},"problem":"Erdős problem #108: Erdős–Hajnal high-girth, high-chromatic subgraph problem","short":"Erdős problem #108","year_posed":1969,"years_open":57,"ai_system":["ChatGPT","Codex","GPT-6 Astra"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-assisted: two Purdue authors developed the construction and the Lean proof with substantial ChatGPT/Codex help. In the follow-up, GPT-6 Astra found the proof of the 3-colourable lemma and the experts streamlined it.","result":"Negative answer: for every M there is a triangle-free graph with χ > M whose C4-free subgraphs are all 6-colourable (Kohlmeyer–Kruer, Lean-verified); improved to 3-colourable by Nguyen–Walczak.","field":"mathematics"},{"entry":{"id":"2026-11-04-matrix-mortality-2x2-undecidable-codex-claude-lean","url":"https://postcutoff.com/e/2026-11-04-matrix-mortality-2x2-undecidable-codex-claude-lean/","date":"2026-11-04","date_precision":"day","short_title":"2×2 matrix mortality claimed undecidable (AI-assisted, Lean)","deck":null,"takeaway":"Paterson showed in 1970 that mortality is undecidable for 3×3 integer matrices; the 2×2 case was the one dimension left open. A Waterloo preprint claims it is undecidable for families of 5,799 2×2 integer matrices, with AI used for research, proofs and a Lean formalization of the reduction.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-11-08","updated":"2026-11-08","orgs":["University of Waterloo","OpenAI","Anthropic"]},"problem":"Matrix mortality problem in dimension 2 (open since Paterson 1970)","short":"Matrix mortality problem in dimension 2","year_posed":1970,"years_open":56,"ai_system":["OpenAI Codex","GPT-5.6 Sol","Claude Code"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-assisted: authors chose the final proof architecture and reworked the exposition; AI used for research, proofs and Lean code","result":"Claimed undecidable for finite families (5,799 matrices) of 2×2 integer matrices","field":"computer-science"},{"entry":{"id":"2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt","url":"https://postcutoff.com/e/2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt/","date":"2026-10-25","date_precision":"day","short_title":"Golomb–Welch conjecture (perfect Lee codes) claimed, with ChatGPT","deck":null,"takeaway":"Bocong Chen, Jing Huang and Gaojun Luo posted two arXiv papers on 25 Oct 2026. They claim that Lee balls of radius r ≥ 2 never tile ℤⁿ for n ≥ 3, first for lattice tilings and then for arbitrary centre sets. That is the full Golomb–Welch conjecture from around 1970, the central open problem on perfect Lee codes. Both papers say ChatGPT helped explore proof strategies and check calculations. They are unrefereed, and no expert has commented yet.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-27","updated":"2026-10-27","orgs":["South China University of Technology","Guangzhou University","Nanjing University of Aeronautics and Astronautics","OpenAI"]},"problem":"Golomb–Welch conjecture (c. 1968–1970): no perfect r-error-correcting Lee code in ℤⁿ for n ≥ 3, r ≥ 2","short":"Golomb–Welch conjecture","year_posed":1970,"years_open":56,"ai_system":["ChatGPT"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: authors used ChatGPT to explore proof strategies and check calculations; the authors say they wrote all results and proofs","result":"Claimed full proof, in both the lattice and the arbitrary-centre forms","field":"mathematics"},{"entry":{"id":"2026-10-19-erdos-szemeredi-sunflower-conjecture-gpt-6-astra-pro","url":"https://postcutoff.com/e/2026-10-19-erdos-szemeredi-sunflower-conjecture-gpt-6-astra-pro/","date":"2026-10-19","date_precision":"day","short_title":"Erdős–Szemerédi sunflower conjecture proved for every k ≥ 3","deck":"The key vector bound (Theorem 2) was found by ChatGPT 6 Astra Pro (Bucić, He, Lee)","takeaway":"A 1970s Erdős–Szemerédi conjecture is claimed proved for every k: families of subsets of [n] with no k-sunflower have size at most c_k^n with c_k < 2. Before this only k = 3 was known, through the 2016 cap-set breakthrough. The authors say ChatGPT 6 Astra Pro found the proof of the central theorem. The paper also says a Lean check exists, but it has not been released yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["University of Vienna","KAIST","IBS","OpenAI"]},"problem":"Erdős–Szemerédi sunflower conjecture: k-sunflower-free families in 2^[n] have size at most c_k^n with c_k < 2","short":"Erdős–Szemerédi sunflower conjecture, all k","year_posed":1970,"years_open":56,"ai_system":["ChatGPT 6 Astra Pro"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-assisted: the authors posed the generalization of Naslund–Sawin and rewrote the argument; ChatGPT 6 Astra Pro found the proof of the key Theorem 2","result":"Claimed proof for every k ≥ 3, with the bound 2^((1 − 1/(16k³))n), via a new polynomial-method bound for vector sunflowers.","field":"mathematics"},{"entry":{"id":"2026-10-08-alphaproof-nexus-science-paper","url":"https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/","date":"2026-10-08","date_precision":"day","short_title":"Google DeepMind's AlphaProof Nexus paper published in Science","deck":"LLM + Lean agent resolved 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures","takeaway":"This is a peer-reviewed, fully formal account of LLM agents solving open research problems at scale and at low cost.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":16,"official":4,"filed":"2026-10-10","updated":"2026-10-10","orgs":["Google DeepMind"]},"problem":"Open Erdős problems (erdosproblems.com), open OEIS conjectures, Zanello's log-concavity conjecture, last-iterate rate of anchored GDA","short":"Open Erdős problems","year_posed":1970,"years_open":56,"ai_system":["AlphaProof Nexus","Gemini 3.1 Pro","AlphaProof"],"status":"confirmed","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Autonomous on the Erdős and OEIS evaluation (humans chose and formalized problems, reviewed formalizations); AI-assisted collaboration in the deployment results","result":"Lean-verified resolutions of 9 of 353 attempted open Erdős (sub-)problems, 44 of 492 open OEIS conjectures, 2 of 4 open Hilbert-function problems, and an exact O(1/t) rate for anchored GDA.","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-hilbert-tenth-problem-rationals-claim","url":"https://postcutoff.com/e/2026-10-06-openai-hilbert-tenth-problem-rationals-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Hilbert's tenth problem over ℚ is undecidable","deck":null,"takeaway":"Family 004 of OpenAI's 6 Oct 2026 release claims that no algorithm decides whether an integer polynomial has a rational zero, the rational-number version of Hilbert's tenth problem, open since Matiyasevich settled the integer case in 1970. The 62-page proof avoids defining the integers inside the rationals; it reduces integer solvability to infinitely many finite rational tests. It is not formalized and leans on three companion manuscripts. Frank Calegari said it 'threw me for a loop', and Hector Pasten called the strategy 'original'.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Hilbert's tenth problem over ℚ: is there an algorithm deciding whether a polynomial with integer coefficients has a rational zero?","short":"Hilbert's tenth problem over ℚ","year_posed":1970,"years_open":56,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","result":"Claimed negative answer: rational solvability is undecidable (number of variables part of the input), via a Turing reduction from integer solvability to finitely many rational tests","field":"mathematics"},{"entry":{"id":"2026-11-05-singmaster-bound-claude-opus-5-5-whitmore","url":"https://postcutoff.com/e/2026-11-05-singmaster-bound-claude-opus-5-5-whitmore/","date":"2026-11-05","date_precision":"day","short_title":"Singmaster's problem: new bound by Claude Opus 5.5","deck":null,"takeaway":"Singmaster's problem is a well-known open problem in elementary number theory, and the claimed exponent 1/2 improves even on what was previously known under Cramér's conjecture.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-06","updated":"2026-11-06","orgs":["Anthropic"]},"problem":"Singmaster's conjecture on multiplicities of binomial coefficients","short":"Singmaster's conjecture on multiplicities of binomial coefficients","year_posed":1971,"years_open":55,"ai_system":["Claude Opus 5.5"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-generated proofs, formalization and most text; human contributor checked some parts","result":"Claimed unconditional bound N(t) = O((log t)^(1/2+o(1)))","field":"mathematics"},{"entry":{"id":"2026-09-16-chvatal-conjecture-proved","url":"https://postcutoff.com/e/2026-09-16-chvatal-conjecture-proved/","date":"2026-09-16","date_precision":"day","short_title":"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","deck":null,"takeaway":"It is a clear example of the late-2026 pattern in mathematics.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":4,"filed":"2026-10-01","updated":"2026-10-07","orgs":["Institute for Basic Science","OpenAI"]},"problem":"Chvátal's conjecture on intersecting subfamilies of downsets","short":"Chvátal's conjecture","year_posed":1972,"years_open":54,"ai_system":["ChatGPT","GPT-6 Astra","Codex"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-assisted, human-led original proof (ChatGPT tested inequalities and proved a guiding special case); the short spectral proof was found by GPT-6 Astra under human direction","result":"Proof of Chvátal's conjecture via a sharp correlation inequality, followed by an AI-found short spectral proof and a Lean formalization.","field":"mathematics"},{"entry":{"id":"2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey","url":"https://postcutoff.com/e/2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey/","date":"2026-10-21","date_precision":"day","short_title":"Hochster's 1973 small Cohen–Macaulay conjecture","deck":"An OpenAI manuscript's counterexample is surveyed, re-proved and extended to every dimension by Cristian Anghel","takeaway":"OpenAI's 6 Oct release included a model-written counterexample to the domain form of Hochster's small Cohen–Macaulay conjecture, a central question of commutative algebra since 1973. A survey by Cristian Anghel, whose September preprints started this line of work, gives a complete proof of it and generalizes it to all dimensions.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","Simion Stoilow Institute of Mathematics of the Romanian Academy"]},"problem":"Hochster's small Cohen–Macaulay module conjecture (1973), domain form in equal characteristic zero; Eisenbud–Schreyer–Weyman Ulrich question (2003)","short":"Hochster's small Cohen–Macaulay module conjecture","year_posed":1973,"years_open":53,"ai_system":["OpenAI internal model (unreleased)","ChatGPT","Claude"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Counterexample autonomous (OpenAI release), building on human preprints by Anghel, Lopez and Chen; Anghel's survey re-proves and generalizes it, with proofs drafted with ChatGPT and Claude","result":"A 3-dimensional complete Noetherian normal local domain over C with no nonzero finitely generated maximal Cohen–Macaulay module (OpenAI); Anghel extends the obstruction to all dimensions ≥ 3.","field":"mathematics"},{"entry":{"id":"2026-10-14-penrose-conjecture-bray-khuri-claude-openai-provenance","url":"https://postcutoff.com/e/2026-10-14-penrose-conjecture-bray-khuri-claude-openai-provenance/","date":"2026-10-14","date_precision":"day","short_title":"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","deck":null,"takeaway":"Leading experts on the Penrose conjecture posted a special-case proof built with Claude's help. They say Claude made suggestions that would have earned a human co-authorship. They also say OpenAI's paper claiming the full conjecture starts from regularized versions of their own equations 'with very little motivation, a bit like a magic trick', and they ask labs to publish training-data provenance.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-15","updated":"2026-10-15","orgs":["Duke University","Stony Brook University","Anthropic","OpenAI"]},"problem":"Penrose conjecture (spacetime Penrose inequality): ADM mass bounded below by the area of the black holes, for initial data satisfying the dominant energy condition","short":"Penrose conjecture","year_posed":1973,"years_open":53,"ai_system":["Claude"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led: the key t = 1 identity was derived by hand; Claude did and checked the algebra for general t and made 'useful mathematical suggestions' the authors call co-author level but 'not decisive'","result":"Spacetime Penrose inequality in all dimensions n ≥ 3 for 'maximally parameterized' initial data (with a horizon nondegeneracy condition), via new generalized Schoen–Yau identities; the general case is reduced to solving a degenerate PDE system","field":"physics"},{"entry":{"id":"2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition","url":"https://postcutoff.com/e/2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition/","date":"2026-10-13","date_precision":"day","short_title":"Thompson's group F is nonamenable","deck":"Specialists Sang-hyun Kim and Yash Lodha publish a human-written version of OpenAI's proof","takeaway":"The amenability of Thompson's group F, open since the 1970s, was one of the headline claims of OpenAI's 6 Oct math release. Two experts on groups of piecewise linear maps have now written the proof out themselves and generalized it, without AI.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-10-14","updated":"2026-10-19","orgs":["OpenAI","KIAS","Purdue University"]},"problem":"Amenability of Thompson's group F (Thompson 1973; Geoghegan's conjecture)","short":"Amenability of Thompson's group F","year_posed":1973,"years_open":53,"ai_system":["OpenAI internal model (unreleased)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous AI proof (OpenAI release); human experts simplified, generalized and wrote it out without AI","result":"Claimed by OpenAI and re-proved in streamlined form by Kim and Lodha: F is nonamenable.","field":"mathematics"},{"entry":{"id":"2026-07-17-cycle-double-cover-conjecture-proved","url":"https://postcutoff.com/e/2026-07-17-cycle-double-cover-conjecture-proved/","date":"2026-07-17","date_precision":"day","short_title":"GPT-5.6 Sol Ultra proves the 50-year-old cycle double cover conjecture","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":3,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI"]},"problem":"Cycle double cover conjecture","short":"Cycle double cover conjecture","year_posed":1973,"years_open":53,"ai_system":["GPT-5.6 Sol Ultra"],"status":"confirmed","verification":{"key":"expert","label":"Expert-checked"},"human_role":"Autonomous proof per OpenAI; human experts wrote independent expositions","result":"Proof that every bridgeless graph admits a cycle double cover.","field":"mathematics"},{"entry":{"id":"2026-10-20-double-cap-conjecture-witsenhausen-lean-ai","url":"https://postcutoff.com/e/2026-10-20-double-cap-conjecture-witsenhausen-lean-ai/","date":"2026-10-20","date_precision":"day","short_title":"Double-cap conjecture claimed in all dimensions, with a separate AI-built Lean proof","deck":null,"takeaway":"If confirmed, a 52-year-old geometry problem falls in every dimension, and a machine-checked proof of the statement already exists.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-21","updated":"2026-10-29","orgs":["Universidade Federal do Ceará","Universidade de São Paulo","OpenAI"]},"problem":"Double-cap conjecture / Witsenhausen's problem on spherical sets without orthogonal pairs","short":"Double-cap conjecture / Witsenhausen's problem on spherical sets without orthogonal pairs","year_posed":1974,"years_open":52,"ai_system":["Codex","AI-assisted tools (unspecified)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Human-led paper with an assistive AI role; the separate Lean proof was produced almost entirely with AI-assisted tools","result":"For n ≥ 3, sets in S^(n−1) with no orthogonal pair have measure at most that of two opposite caps of radius π/4, with equality only for double caps (claimed)","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-hot-spots-conjecture-simply-connected-lean","url":"https://postcutoff.com/e/2026-10-06-openai-hot-spots-conjecture-simply-connected-lean/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims the planar hot spots conjecture (Lean)","deck":null,"takeaway":"Family 369 of OpenAI's 6 Oct 2026 release (paper dated 24 Sept) concerns smooth bounded simply connected planar domains. It claims that on every such domain each first nonconstant Neumann eigenfunction has no interior critical point, so its maximum and minimum occur only on the boundary. This is the strict form of the simply connected case of Rauch's hot spots problem (1974), stated as a conjecture by Krzysztof Burdzy in 2005. It includes smooth convex domains, long a main open case. The Lean statement passed the community comparator runs. Burdzy (8 Oct) and Rodrigo Bañuelos (23 Oct), whose 1999 paper set out the strict and non-strict forms of the problem, each used ChatGPT to extend the result to nonsmooth domains. Both took OpenAI's theorem as an input they had not checked. No specialist has published a check of the proof as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Rauch's hot spots conjecture, simply connected planar case (Burdzy 2005, Conjecture 1.2(ii)), in its strict form","short":"Rauch's hot spots conjecture, simply connected planar case","year_posed":1974,"years_open":52,"ai_system":["OpenAI internal model (openai/math release)","ChatGPT (follow-ups by Burdzy and Bañuelos)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README); the follow-ups were generated with ChatGPT from prompts by Burdzy and by Bañuelos","result":"Claimed: on every smooth bounded simply connected planar domain, every first nonconstant Neumann eigenfunction has no interior critical point, so its extrema lie only on the boundary; eigenvalue multiplicity allowed","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-strong-cosmic-censorship-near-kerr","url":"https://postcutoff.com/e/2026-10-06-openai-strong-cosmic-censorship-near-kerr/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims strong cosmic censorship near Kerr","deck":null,"takeaway":"Three manuscripts in OpenAI's 6 Oct 2026 math release (family 264, dated 23 Sept) claim local strong cosmic censorship for rotating Kerr black holes, with no symmetry assumption. For generic small perturbations of two-ended Kerr data, the spacetime cannot be extended past its maximal development with a continuous metric and square-integrable connection. That is the weak-solution formulation Christodoulou proposed in 2009. Berkeley's Sung-Jin Oh, who has worked toward this problem for thirteen years, wrote that the proof takes a route different from the field's and organised a reading group. No Lean proof; no expert verdict yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Strong cosmic censorship (Penrose 1974) near Kerr, in Christodoulou's formulation: generic data admit no extension with continuous metric and locally square-integrable Christoffel symbols","short":"Strong cosmic censorship near Kerr, in Christodoulou's formulation","year_posed":1974,"years_open":52,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI; reading group of specialists organised by Sung-Jin Oh","result":"Claimed local strong cosmic censorship near every fixed subextremal rotating Kerr spacetime (C⁰ ∩ W^{1,2}_loc, plus C¹ and C² versions), without symmetry","field":"physics"},{"entry":{"id":"2026-09-20-kung-traub-conjecture-proved-colbrook-lean-chatgpt","url":"https://postcutoff.com/e/2026-09-20-kung-traub-conjecture-proved-colbrook-lean-chatgpt/","date":"2026-09-20","date_precision":"day","short_title":"Kung–Traub conjecture proved, Lean check with ChatGPT 6","deck":null,"takeaway":"The Kung–Traub conjecture is a standard open question in numerical analysis textbooks for half a century.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-02","updated":"2026-11-02","orgs":["University of Cambridge","OpenAI"]},"problem":"Kung–Traub conjecture on the maximal order of multipoint iterations without memory","short":"Kung–Traub conjecture on the maximal order of multipoint iterations without memory","year_posed":1974,"years_open":52,"ai_system":["ChatGPT 6","ChatGPT 5.6"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Human-led: proofs by the author; ChatGPT 6 assisted the Lean formalization and ChatGPT 5.6/6 were used to discuss and check arguments","result":"Local order ≤ 2^(n−1) for any memoryless method with n evaluations per step, for real and complex scalar evaluations; sharper bound for grouped evaluations","field":"mathematics"},{"entry":{"id":"2026-10-28-murty-simon-conjecture-proof-gpt-5-6-sol-das-fox","url":"https://postcutoff.com/e/2026-10-28-murty-simon-conjecture-proof-gpt-5-6-sol-das-fox/","date":"2026-10-28","date_precision":"day","short_title":"Murty–Simon conjecture: proof for all n, bound found by GPT-5.6 Sol","deck":null,"takeaway":"The Murty–Simon conjecture (Plesník 1975, Murty and Simon) says a diameter-2-critical graph on n vertices has at most ⌊n²/4⌋ edges, with equality only for the balanced complete bipartite graph. Füredi proved it in 1992 only for n above a tower of 2s of height about 10¹⁴. On 28 Oct Sanjana Das and Jacob Fox (Stanford) posted a proof for every n. Their AI statement says the proof of the n²/4 bound 'was mostly found by AI, specifically GPT-5.6 Sol on August 11, 2026'. The authors simplified it and extended it to the equality case for odd n by hand. It is a preprint and has not been refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Stanford University"]},"problem":"Murty–Simon conjecture (Plesník 1975): diameter-2-critical graphs have at most ⌊n²/4⌋ edges, equality only for the balanced complete bipartite graph","short":"Murty–Simon conjecture","year_posed":1975,"years_open":51,"ai_system":["GPT-5.6 Sol","GPT-6 Astra"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: GPT-5.6 Sol found the n²/4 bound and the even-n equality case; the authors simplified it, proved the odd-n equality case and wrote the paper","result":"Claimed proof for every n (previously known only for n beyond a tower-type bound, Füredi 1992)","field":"mathematics"},{"entry":{"id":"2026-10-27-erdos-lovasz-intersecting-covering-number-frankl-ai-construction","url":"https://postcutoff.com/e/2026-10-27-erdos-lovasz-intersecting-covering-number-frankl-ai-construction/","date":"2026-10-27","date_precision":"day","short_title":"Erdős–Lovász covering-number problem: AI-found construction disproves Frankl's 1996 conjecture","deck":null,"takeaway":"Erdős and Lovász showed in 1975 that an intersecting family of k-sets that needs k points to cover it has at most k^k members. The best constructions stayed near (k/2)^(k−1) for 30 years. In 1996 Frankl, Ota and Tokushige conjectured that m(k) < (ck)^k for some c < 1. A 27 Oct preprint by Peter Frankl, Jian Wang and two Sichuan University students builds families of size k^k·exp(−50√(k log k)). With Zakharov's 2024 upper bound this pins m(k) down to k^k·exp(−k^(1/2+o(1))) and refutes the 1996 conjecture. The authors write that 'the construction was found using generative AI tools'. They do not name the models.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Rényi Institute","Sichuan University"]},"problem":"Erdős–Lovász problem: the maximum size m(k) of an intersecting k-uniform family with covering number k; Frankl–Ota–Tokushige conjecture m(k) < (ck)^k","short":"Erdős–Lovász problem","year_posed":1975,"years_open":51,"ai_system":["unnamed generative AI tools"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI found the construction; the authors verified the proofs, reworked the arguments and wrote the exposition","result":"m(k) ≥ k^k·exp(−50√(k log k)) for log k ≥ 32, so m(k) = k^k·exp(−k^(1/2+o(1))) with Zakharov's bound; the Frankl–Ota–Tokushige conjecture is false for large k","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-polyakov-o3-sigma-model-mass-gap-claim","url":"https://postcutoff.com/e/2026-10-06-openai-polyakov-o3-sigma-model-mass-gap-claim/","date":"2026-10-06","date_precision":"day","short_title":"2D O(n) mass gap (Polyakov) and O(3) continuum limit claimed (OpenAI)","deck":null,"takeaway":"Five manuscripts credited to 'OpenAI' (23 Sept to 5 Oct, public on 6 Oct) claim the lattice and continuum versions of a problem that physicists call the warm-up for Yang–Mills. The 2D classical O(n) model with n ≥ 3 is claimed to have exponentially decaying correlations at every positive temperature, as Polyakov predicted in 1975. A massive continuum O(3) field theory satisfying the Osterwalder–Schrader axioms, with a unique vacuum and a positive gap, is also claimed, along with the exact O(4) mass asymptotics. Only the 20-page decay theorem is in Lean. Roland Bauerschmidt called the first paper groundbreaking and comprehensible and the second 'horrible'. Zohar Komargodski noted the decay was known from numerics for decades.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Polyakov's conjecture (1975): the 2D O(n) model with n ≥ 3 has a mass gap (exponential decay) at every positive temperature; construction of the massive continuum O(3) nonlinear sigma model","short":"Polyakov's conjecture","year_posed":1975,"years_open":51,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; first reactions by mathematical physicists quoted in the press","result":"Claimed: exponential decay for all n ≥ 3 and all β (Lean-formalized in finite volume); an Osterwalder–Schrader continuum O(3) theory with unique vacuum, mass gap and an isolated particle pole; exact O(4) lattice mass asymptotics","field":"physics"},{"entry":{"id":"2026-10-06-openai-erdos-arithmetic-progressions-conjecture-claim","url":"https://postcutoff.com/e/2026-10-06-openai-erdos-arithmetic-progressions-conjecture-claim/","date":"2026-10-06","date_precision":"day","short_title":"Erdős's arithmetic-progression conjecture claimed (OpenAI, Lean)","deck":null,"takeaway":"A 198-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arbitrarily long arithmetic progressions, the problem for which Erdős offered $5000. It does so by claiming quasipolynomial bounds in Szemerédi's theorem for every progression length. A Lean proof of the qualitative conjecture was accepted in a community comparator run. Thomas Bloom of erdosproblems.com reads the Lean code as proving a weaker bound than the paper claims, and describes the proof as a combination of existing techniques. No expert has published a check of the 198 pages.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":10,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Erdős's conjecture on arithmetic progressions (often called the Erdős–Turán conjecture; erdosproblems.com #3, $5000): every set of positive integers with divergent reciprocal sum contains arbitrarily long arithmetic progressions","short":"Erdős's conjecture on arithmetic progressions","year_posed":1976,"years_open":50,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; no human exposition or check yet","result":"Claimed proof via quasipolynomial bounds r_k(N) ≤ C_k N exp(−c_k (log N)^ε_k) for every k ≥ 3; the qualitative conjecture is proved in Lean (with, per Bloom, a weaker quantitative bound inside the formal proof)","field":"mathematics"},{"entry":{"id":"2026-10-31-bhargava-cyclic-reduction-elliptic-curves-positive-proportion-gpt-6","url":"https://postcutoff.com/e/2026-10-31-bhargava-cyclic-reduction-elliptic-curves-positive-proportion-gpt-6/","date":"2026-10-31","date_precision":"day","short_title":"Bhargava: elliptic curves have cyclic reduction at a positive proportion of primes (claim)","deck":null,"takeaway":"Serre showed under the generalized Riemann hypothesis that an elliptic curve over ℚ without full rational 2-torsion has cyclic reduction E(F_p) at a positive proportion of primes; Murty removed GRH for CM curves, and Gupta–Murty proved infinitely many such primes in general. On 31 Oct Manjul Bhargava (Princeton) posted on Hexagon an unconditional positive lower proportion for every such curve. The argument uses Cureton–Thorner's zero-free half-plane and 'the marked Type II estimates of OpenAI'. Per the provenance statement, handwritten notes were turned into LaTeX 'with additional writing and details added by both the author and GPT-6'. GPT-6 also helped formalise the deduction in Lean, with the analytic inputs left as hypotheses. Only some parts have been checked by a human, and it is not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Princeton University"]},"problem":"Serre's cyclicity problem: unconditional positive density of primes p with E(F_p) cyclic","short":"Serre's cyclicity problem","year_posed":1977,"years_open":49,"ai_system":["GPT-6","ChatGPT"],"status":"pending","verification":{"key":"none","label":"Unverified claim"},"human_role":"Human-led with AI writing: Bhargava's handwritten argument; GPT-6 added writing and details and helped with the Lean formalisation; builds on OpenAI's AI-generated analytic estimates","result":"Claimed positive lower density c_E/(32 log 32) for every E/ℚ without full rational 2-torsion, CM and non-CM alike (known before under GRH, or for CM curves)","field":"mathematics"},{"entry":{"id":"2026-10-17-erdos-hajnal-all-paths-nguyen-astra-deepmind","url":"https://postcutoff.com/e/2026-10-17-erdos-hajnal-all-paths-nguyen-astra-deepmind/","date":"2026-10-17","date_precision":"day","short_title":"Erdős–Hajnal conjecture proved for every path P_t","deck":"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","takeaway":"Until now the Erdős–Hajnal property was known for paths only up to five vertices (P5, 2023). Oxford's Tung H. Nguyen claims it for every path, with GPT-6 Astra proving parts of the key lemmas. His paper says a Google DeepMind internal model got the P6 case and then all paths on its own, Lean-checked, and that two Claude Opus 5.5 manuscripts on GitHub claim P6 and all paths.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-20","updated":"2026-11-03","orgs":["University of Oxford","OpenAI","Google DeepMind","Google Research","Anthropic"]},"problem":"Erdős–Hajnal conjecture for H = P_t (induced paths), open beyond P5","short":"Erdős–Hajnal for all paths","year_posed":1977,"years_open":49,"ai_system":["ChatGPT-6 Astra","Google DeepMind internal model","Claude Opus 5.5"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Human-led (Nguyen): the author's framework, with Astra proving parts of key lemmas; DeepMind's and the Claude manuscripts are described as AI-generated","result":"Claimed proof that every path P_t (and the subdivided claws S_{1,2,t}) has the Erdős–Hajnal property; DeepMind's model and a Claude-assisted manuscript claim the P_t case independently.","field":"mathematics"},{"entry":{"id":"2026-10-15-delsarte-lp-bound-exact-rate-salmon-opus","url":"https://postcutoff.com/e/2026-10-15-delsarte-lp-bound-exact-rate-salmon-opus/","date":"2026-10-15","date_precision":"day","short_title":"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","deck":null,"takeaway":"A coding-theory question open since the 1970s, what Delsarte's LP bound converges to, now has a closed answer in elliptic integrals; the author credits Opus 5.5 with the step that made the lower bound work.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-16","updated":"2026-10-16","orgs":["Anthropic","OpenAI"]},"problem":"Asymptotic value of Delsarte's linear programming bound for binary codes and spherical codes (Kalai's question)","short":"Asymptotic value of Delsarte's linear programming bound for binary codes and spherical codes","year_posed":1977,"years_open":49,"ai_system":["Claude Opus 5.5","GPT-5.6 Sol","GPT-6 Astra","Codex"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: human-led program; GPT-5.6 Sol co-formulated the upper bounds; Opus 5.5 found the key compatibility and the obstacle-problem reformulation 'with light direction'; Codex drafted","result":"Exact asymptotic rates of the binary and spherical Delsarte LPs for all distances, in closed form via elliptic integrals (matching upper and lower bounds).","field":"mathematics"},{"entry":{"id":"2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov","url":"https://postcutoff.com/e/2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov/","date":"2026-10-16","date_precision":"day","short_title":"Brennan's conjecture: experts confirm and shorten OpenAI's proof","deck":null,"takeaway":"The first famous result from OpenAI's October maths release to be checked by leading experts in its field: four complex analysts, including Fields medallist Stanislav Smirnov, found no gap. They say the AI combined known techniques in a way 'missed by the human experts'.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":4,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","University of Toronto","Tel Aviv University","Åbo Akademi University","University of Geneva"]},"problem":"Brennan's conjecture (1978) on integrability of |g′|^p for conformal maps; Kraetzer's t²/4 conjecture for the universal integral-means spectrum","short":"Brennan's conjecture on integrability of |g′|^p for conformal maps; Kraetzer's t²/4 conjecture for the universal integral-means spectrum","year_posed":1978,"years_open":48,"ai_system":["OpenAI internal model (openai/math release)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-generated original proof; Binder, Ivrii, Prause and Smirnov validated it and wrote a shorter, stronger proof without AI help (some used AI agents to understand the AI proof)","result":"OpenAI's internal model proved Brennan's conjecture and disproved Kraetzer's conjecture at t = −1 (preprints dated 24 Sept, released 6 Oct). Four experts checked the work and gave a short independent proof that B_b(t) ≤ t²/4 on [−2, 2], strict except at 0 and ±2","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-de-giorgi-conjecture-dimension-eight","url":"https://postcutoff.com/e/2026-10-06-openai-de-giorgi-conjecture-dimension-eight/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims De Giorgi's conjecture in dimension eight","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release (family 375, dated 26 Sept) claims to complete De Giorgi's conjecture. Every bounded entire solution of the Allen–Cahn equation Δu = u³ − u in ℝ⁸ that is strictly increasing in one direction is a one-dimensional tanh profile. Dimensions 2 and 3 were known, as was Savin's result up to dimension 8 under an extra limit assumption, and counterexamples exist from dimension 9. The paper also classifies stable solutions in ℝ⁷ without energy bounds. A Chinese group has already built on it. No expert assessment and no Lean proof are known as of 28 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-28","updated":"2026-10-28","orgs":["OpenAI"]},"problem":"De Giorgi's conjecture (1978): monotone bounded entire solutions of Δu = u³ − u in ℝⁿ, n ≤ 8, are one-dimensional; also stable rigidity in ℝ⁷","short":"De Giorgi's conjecture","year_posed":1978,"years_open":48,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI","result":"Claimed proof in the sharp dimension 8 without Savin's limit assumption; classification of stable solutions in ℝ⁷ without energy growth","field":"mathematics"},{"entry":{"id":"2026-09-17-zeta-5-irrational-fauzan-lean-verified","url":"https://postcutoff.com/e/2026-09-17-zeta-5-irrational-fauzan-lean-verified/","date":"2026-09-17","date_precision":"day","short_title":"ζ(5) proved irrational","deck":"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","takeaway":"It is the most famous number-theory result of the AI-assisted 2026 wave: a problem experts had worked on for about 48 years.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":10,"official":5,"filed":"2026-10-05","updated":"2026-10-05","orgs":["Aalto University","Google DeepMind","Anthropic"]},"problem":"Irrationality of ζ(5) = Σ 1/n⁵ (open since Apéry's 1978 proof for ζ(3); Euler-era question)","short":"Irrationality of ζ(5)","year_posed":1978,"years_open":48,"ai_system":["Claude Opus 5","Claude Opus 5.5","Claude Code","unnamed generative AI tool (paper)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Paper by a single human author who discloses AI only for editing and consistency checks (disputed by Calegari). One Lean formalization was written by Claude agents directed by Dan Romik; the other, by Moritz Firsching, does not say how it was produced.","result":"ζ(5) is irrational, with integer polynomials Q_n of degree 37n satisfying 0 < Q_n(ζ(5)) < exp(−139n²/5), and irrationality measure at most 260.","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-log-abundance-nonvanishing-claim","url":"https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims the abundance conjecture; Jihao Liu raises priority","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release, 'Log abundance in characteristic zero' (dated 24 Sept), claims the log abundance conjecture in every dimension in characteristic zero, and with it canonical nonvanishing for smooth projective varieties. Both are central open problems of the minimal model program, proved only up to dimension three. Companion papers claim Iitaka's subadditivity conjecture and related results. On 7 Oct Jihao Liu (Peking University) submitted a human-led proof that nonvanishing implies abundance. arXiv announced it only on 28 Oct. His comment line says he was 'forced to submit here due to an announcement of the proof of the abundance conjecture by OpenAI'. In a remark he suggests that an earlier draft, sent to 'codex chat' before July 2026, may have fed OpenAI's work. No birational geometer has assessed the claims publicly, there is no Lean proof, and OpenAI has not responded.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Peking University"]},"problem":"Abundance conjecture (κ = ν; nef lc adjoints are semiample) and the nonvanishing conjecture, open in dimension ≥ 4; also Iitaka's subadditivity conjecture C_{n,m}","short":"Abundance conjecture","year_posed":1980,"years_open":46,"ai_system":["OpenAI internal model (math release)","Liu: DeepSeek, Fable, GLM, ChatGPT, Danus, Rethlas"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"OpenAI: autonomous AI output released by OpenAI. Liu: human-led paper written with several AI models and agent systems","result":"OpenAI claims log abundance for rational-boundary lc pairs and canonical nonvanishing in every dimension in characteristic zero. Liu (human-led, AI-assisted) proves nonvanishing ⇒ abundance for lc pairs","field":"mathematics"},{"entry":{"id":"2026-11-11-smale-mean-value-constant-equals-4-goel-lean","url":"https://postcutoff.com/e/2026-11-11-smale-mean-value-constant-equals-4-goel-lean/","date":"2026-11-11","date_precision":"day","short_title":"Smale's mean value constant = 4 (claimed, Lean-checked)","deck":null,"takeaway":"Two months after GPT-6 Astra refuted the constant-1 form, a Hexagon deposit says the best constant in Smale's mean value theorem is exactly 4, so Smale's 1981 bound cannot be improved at all; a Lean proof accompanies it, experts have not yet commented.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":5,"filed":"2026-11-12","updated":"2026-11-12","orgs":["Anthropic","OpenAI"]},"problem":"Smale's mean value problem (1981): the best constant K in |p(z) − p(c)|/|z − c| ≤ K|p'(z)| at some critical point c","short":"Smale's mean value problem","year_posed":1981,"years_open":45,"ai_system":["Claude","ChatGPT"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Author steered AI models in dialogue; work 'done overwhelmingly by AI models'","result":"Claimed: the best constant over all degrees is exactly 4 (Smale's original bound), approached but not attained as the degree grows","field":"mathematics"},{"entry":{"id":"2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"GPT-6 Astra autonomously disproves Smale's mean value conjecture in Lean","deck":"HKU mathematicians build on it to show Smale's constant exceeds 1.1018","takeaway":"A Smale problem from 1981 fell to an unattended model run in an Epoch AI benchmark, with a Lean proof; six weeks later two complex analysts turned the construction into a quantitative bound M > 1.1018.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":9,"filed":"2026-10-16","updated":"2026-11-12","orgs":["OpenAI","Epoch AI","University of Hong Kong"]},"problem":"Smale's mean value conjecture (1981), constant-1 form","short":"Smale's mean value conjecture","year_posed":1981,"years_open":45,"ai_system":["GPT-6 Astra (pre-release)","Grok 4.6"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous (benchmark run, no human steering); follow-up bound by Jatar & Ng with AI drafts rewritten by the authors","result":"Disproved: a polynomial with p(0) = 0, p'(0) = 1 and |p(c)/c| > 1 at every critical point exists (Lean). Follow-up: Smale's constant M > 1.1018.","field":"mathematics"},{"entry":{"id":"2026-10-30-number-of-c4-free-graphs-erdos-question-tianchi-yang-gpt-6-pro","url":"https://postcutoff.com/e/2026-10-30-number-of-c4-free-graphs-erdos-question-tianchi-yang-gpt-6-pro/","date":"2026-10-30","date_precision":"day","short_title":"Number of C₄-free graphs is 2^{(1+o(1))ex(n,C₄)} (claim, GPT-6 Pro)","deck":null,"takeaway":"Tianchi Yang's preprint (arXiv 2610.39595, 29 Oct 2026) claims that the number of graphs on n vertices with no 4-cycle is 2^{(1+o(1))ex(n,C₄)}. That answers a question of Erdős. The best earlier upper bound was Kleitman and Winston's 2^{(1.0819…+o(1))n^{3/2}} from 1982, and Morris and Saxton showed the same statement fails for C₆. The AI statement says the author supplied the main ideas, a modified hypergraph container method, and 'GPT-6 Pro assisted in completing the proofs.' Single author, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Anthropic"]},"problem":"Erdős's question: is the number of C₄-free graphs on n vertices 2^{(1+o(1))ex(n,C₄)}?","short":"Erdős's question","year_posed":1982,"years_open":44,"ai_system":["GPT-6 Pro","GPT-6.1","Claude Opus 5.5"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led: main ideas by the author; GPT-6 Pro helped complete the proofs; GPT-6.1 and Opus 5.5 revised the text","result":"Claimed affirmative answer (previous best upper bound 2^{1.0819…n^{3/2}}, Kleitman–Winston 1982)","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-baum-connes-kadison-kaplansky-counterexamples","url":"https://postcutoff.com/e/2026-10-06-openai-baum-connes-kadison-kaplansky-counterexamples/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Baum–Connes and Kadison–Kaplansky counterexamples","deck":null,"takeaway":"Three manuscripts in OpenAI's 6 Oct 2026 math release (family 285, dated 23 Sept) claim the first counterexamples to the Baum–Connes conjecture without coefficients, one of them for a torsion-free group, and a torsion-free group whose reduced C*-algebra has a nontrivial projection, which would refute the Kadison–Kaplansky conjecture. Until now, counterexamples were known only for the version with coefficients. On 26 Oct Guoliang Yu, a leading specialist, posted his own construction of projections of every trace in (0,1), saying it develops arguments from OpenAI's irrational-trace paper. None of the three papers has a Lean proof.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Baum–Connes conjecture without coefficients (reduced assembly map an isomorphism) and the Kadison–Kaplansky conjecture (no nontrivial projections in C*_r(G) for torsion-free G)","short":"Baum–Connes conjecture without coefficients","year_posed":1982,"years_open":44,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI; follow-up construction by Guoliang Yu, who states all proofs in it are his own","result":"Claimed counterexamples: failure of rational injectivity for a torsion-free group, failure of surjectivity via an irrational-trace projection, and a torsion-free group with a nontrivial projection; the surjectivity failure re-proved in a stronger form by Guoliang Yu","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-yau-nodal-conjecture-surfaces-counterexamples","url":"https://postcutoff.com/e/2026-10-06-openai-yau-nodal-conjecture-surfaces-counterexamples/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI: Yau's nodal bound holds on surfaces, fails in higher dimensions","deck":null,"takeaway":"Three manuscripts in OpenAI's 6 Oct 2026 math release (family 350, dated 23 Sept) claim to settle Yau's 1982 conjecture on the size of nodal sets of Laplace eigenfunctions for smooth metrics. On every smooth closed surface the nodal length is at most C√λ, which completes the conjecture there. In dimensions 3 and 4, explicit smooth metrics, including some arbitrarily close to the round 3-sphere, violate the upper bound, and on S⁴×S¹ it fails by a power of λ. Lean proofs of all three statements passed an independent Comparator run. Whether the formal statements match the papers has not been checked by specialists.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Yau's conjecture (1982): on a closed smooth Riemannian manifold, c√λ ≤ H^{n−1}(nodal set of an eigenfunction) ≤ C√λ","short":"Yau's conjecture","year_posed":1982,"years_open":44,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Autonomous AI output released by OpenAI, with Lean formalizations","result":"Claimed: upper bound true on all smooth surfaces (completing the conjecture in dimension 2); upper bound false for smooth metrics in dimensions 3 and 4, and by a power of λ on S⁴×S¹","field":"mathematics"},{"entry":{"id":"2026-10-27-axiom-cfsg-classification-finite-simple-groups-lean","url":"https://postcutoff.com/e/2026-10-27-axiom-cfsg-classification-finite-simple-groups-lean/","date":"2026-10-27","date_precision":"day","short_title":"Axiom: Classification of Finite Simple Groups formalized in Lean","deck":null,"takeaway":"On 27 Oct 2026 Axiom Math published a Lean 4 formalization of the Classification of Finite Simple Groups (CFSG). The proof was originally spread over hundreds of papers and an estimated 10,000–15,000 pages. Axiom says it is sorry-free, follows the first-generation proof, and was built by AI agents working with about 30 mathematicians and engineers. Press reports put the build time at 24 days. Along the way the team found a false step in one published route to the Unbalanced Group Theorem and wrote a replacement argument. The project is self-reviewed: outside experts have not yet re-run the build or checked that the Lean statement matches CFSG.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":8,"filed":"2026-10-28","updated":"2026-11-03","orgs":["Axiom Math"]},"problem":"Classification of Finite Simple Groups (announced c. 1983, quasithin gap closed by Aschbacher–Smith 2004): formal machine-checked proof","short":"Classification of Finite Simple Groups","year_posed":1983,"years_open":43,"ai_system":["AxiomProver","Axiom agents (models not disclosed)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI agents formalized the literature from a human-written blueprint; about 34 named mathematicians and engineers; experts Lyons and Solomon consulted","result":"Claimed complete, sorry-free Lean 4 formalization of the first-generation CFSG proof against TauCeti's statement, plus a repaired step in one published route to the Unbalanced Group Theorem","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-spin-one-haldane-gap-proof","url":"https://postcutoff.com/e/2026-10-06-openai-spin-one-haldane-gap-proof/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI proves the spin-one Haldane gap (periodic chains)","deck":null,"takeaway":"Two manuscripts in OpenAI's 6 Oct 2026 math release (dated 24 Sept) prove that the spin-one antiferromagnetic Heisenberg chain has a spectral gap that stays positive as even periodic rings grow. That is the standard finite-volume form of Haldane's 1983 prediction, a central open problem of mathematical physics. Hal Tasaki, a leading expert on the problem, studied the analytic part. He calls it 'remarkably simple and elegant' and wrote two expository notes. He has not checked the computer-assisted estimates. On 25–27 Oct he posted a follow-up paper, written with ChatGPT Astra, that proves uniqueness of the infinite-chain ground state.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":4,"filed":"2026-10-28","updated":"2026-10-28","orgs":["OpenAI"]},"problem":"Haldane conjecture / Haldane gap for the spin-one antiferromagnetic Heisenberg chain (Haldane 1981–1983)","short":"Haldane conjecture / Haldane gap for the spin-one antiferromagnetic Heisenberg chain","year_posed":1983,"years_open":43,"ai_system":["OpenAI internal model (math release)","ChatGPT Astra (Tasaki follow-up)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI; Tasaki's follow-up is AI-assisted (he set problems and strategy, ChatGPT Astra developed proofs)","result":"Uniform positive spectral gap above a unique ground state on even periodic rings (Δ₁ ≥ log 20/784), plus a boundary-field gap for odd open chains with a nontrivial topological index","field":"physics"},{"entry":{"id":"2026-11-05-hopfield-critical-capacity-claude-opus-5-5","url":"https://postcutoff.com/e/2026-11-05-hopfield-critical-capacity-claude-opus-5-5/","date":"2026-11-05","date_precision":"day","short_title":"Hopfield critical capacity bracketed, by Claude Opus 5.5","deck":null,"takeaway":"Turning the replica prediction for the Hopfield model into a theorem has been a long-standing goal in the mathematics of spin glasses.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-06","updated":"2026-11-06","orgs":["Anthropic","OpenAI"]},"problem":"Rigorous critical storage capacity of the Hopfield associative-memory model at zero temperature","short":"Rigorous critical storage capacity of the Hopfield associative-memory model at zero temperature","year_posed":1985,"years_open":41,"ai_system":["Claude Opus 5.5","GPT-6.1 Sol"],"status":"pending","verification":{"key":"none","label":"Unverified claim"},"human_role":"Mainly AI-generated; human contributor checked parts","result":"Claimed bracket αc ∈ [0.138198129, 0.13820106]","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-falconer-distance-conjecture-lean","url":"https://postcutoff.com/e/2026-10-06-openai-falconer-distance-conjecture-lean/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI proves the Falconer distance conjecture (Lean)","deck":null,"takeaway":"Family 073 of OpenAI's 6 Oct 2026 release proves that every compact set in ℝᵈ (d ≥ 2) of Hausdorff dimension greater than d/2 has a distance set of positive length. This is Falconer's 1985 conjecture, the continuous analogue of Erdős's distinct-distances problem; the best planar threshold before was 5/4. The 64-page proof comes with a Lean formalization whose statement uses Mathlib's own Hausdorff dimension and Lebesgue measure, and the community comparator run accepted it. A first follow-up already builds on its method.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":6,"filed":"2026-10-29","updated":"2026-11-12","orgs":["OpenAI"]},"problem":"Falconer's distance conjecture (1985): compact sets in ℝᵈ of Hausdorff dimension > d/2 have distance sets of positive Lebesgue measure","short":"Falconer's distance conjecture","year_posed":1985,"years_open":41,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","result":"Claimed in every dimension d ≥ 2, with a Lean proof of the exact statement (comparator accepted)","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-relativistic-vlasov-maxwell-global-solutions-lean","url":"https://postcutoff.com/e/2026-10-06-openai-relativistic-vlasov-maxwell-global-solutions-lean/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims global smoothness for 3D relativistic Vlasov–Maxwell (Lean)","deck":null,"takeaway":"Family 362 of OpenAI's 6 Oct 2026 release (paper dated 23 Sept) concerns the three-dimensional one-species relativistic Vlasov–Maxwell system, the basic model of a collisionless plasma. The claim: every smooth, compactly supported particle density with finite-energy fields generates a unique global smooth solution, with no smallness or symmetry assumption. Glassey and Strauss showed in 1986 that solutions stay smooth as long as particle momenta stay bounded. Since then the large-data problem had been settled only under size, symmetry or dimension restrictions. A Lean statement of the full theorem was accepted in two independent comparator runs. OpenAI's published reasoning summary says the final assertion rests on new estimates that the cited continuation theorem does not supply. We found no assessment by a kinetic theorist as of 29 Oct.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Large-data global classical existence for the three-dimensional relativistic Vlasov–Maxwell system (open since Glassey and Strauss's 1986 continuation criterion)","short":"Large-data global classical existence for the three-dimensional relativistic Vlasov–Maxwell system","year_posed":1986,"years_open":40,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README); a summary of the model's reasoning is published","result":"Claimed: every smooth compactly supported nonnegative particle density with smooth finite-energy fields satisfying the Gauss laws has a unique global classical solution, smooth on every finite time interval","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-sidorenko-conjecture-counterexample-claim","url":"https://postcutoff.com/e/2026-10-06-openai-sidorenko-conjecture-counterexample-claim/","date":"2026-10-06","date_precision":"day","short_title":"Sidorenko's conjecture disproved? (OpenAI, Lean)","deck":null,"takeaway":"A 51-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims that Sidorenko's 1986 conjecture is false. Sidorenko predicted that every bipartite graph H appears in any host graph at least as often as in a random graph of the same edge density. The counterexample H is explicit: the incidence graph of 22 triples on 13 points, with 35 vertices and 66 edges. The host graph is shown to exist by sampling from a finite weighted kernel but is not written down. Lean proves the existence statement and it passed a community comparator run. The same H also refutes the Skokan–Thoma forcing conjecture.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Sidorenko's conjecture (1986): every bipartite graph H satisfies t(H,G) ≥ p(G)^|E(H)| in every host graph G; also the Skokan–Thoma forcing conjecture","short":"Sidorenko's conjecture","year_posed":1986,"years_open":40,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; no human exposition or check yet","result":"Claimed counterexample: an explicit 35-vertex, 66-edge bipartite incidence graph H and a (non-explicit) finite host G with t(H,G) < p(G)^66; the same H is not forcing","field":"mathematics"},{"entry":{"id":"2026-09-27-erdos-993-forest-independence-polynomials-unimodal","url":"https://postcutoff.com/e/2026-09-27-erdos-993-forest-independence-polynomials-unimodal/","date":"2026-09-27","date_precision":"day","short_title":"Erdős Problem #993 (1987)","deck":"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)","takeaway":"A 1987 question of Alavi, Malde, Schwenk and Erdős asks whether the numbers of independent sets of each size in a tree or forest always rise and then fall. Tong Zhang and Wei Li claimed a computer-assisted proof on 27 Sept 2026 with help from GPT-6 Astra. A second proof (arXiv, 6 Oct) says AI systems produced all of its new mathematics, its Lean proof and most of its text.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-10-13","updated":"2026-10-16","orgs":["Northwestern Polytechnical University","University of Toledo","OpenAI","Anthropic"]},"problem":"Erdős Problem #993 (Alavi, Malde, Schwenk, Erdős 1987): is the independent set sequence of every tree or forest unimodal?","short":"Erdős #993: forest independence sequences are unimodal","year_posed":1987,"years_open":39,"ai_system":["Claude Opus 5.5","Claude Fable 5.1","GPT-6 Astra","GPT-5.6 Sol","GPT-6 Sol","GPT-6.1 Sol"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"First proof: human strategy with GPT-6 Astra for computations. Second proof: AI-generated per the paper (all new mathematics, the Lean proof and most of the prose), in a project directed by Kevin Vallier and built on Zhang and Li's reductions","result":"Claimed: yes for every finite forest. Two proofs: Zhang–Li (27 Sept 2026, computer-assisted, GPT-6 Astra for computations) and Li–Vallier–Zhang (6 Oct 2026, new arguments and Lean 4 proof attributed to AI agents).","field":"mathematics"},{"entry":{"id":"2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding","url":"https://postcutoff.com/e/2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding/","date":"2026-10-14","date_precision":"day","short_title":"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","deck":null,"takeaway":"A week after the Li–Li network coding conjecture fell, two of its authors turned the refutation into an algorithm: a deterministic external-memory sort of Θ(log N)-bit integers that uses asymptotically fewer block transfers than any comparison sort in some parameter ranges. ChatGPT and Codex 'materially affected' the constructions and analyses.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-15","updated":"2026-10-15","orgs":["University of Toronto","Tsinghua University","OpenAI"]},"problem":"Aggarwal–Vitter (1988): can external-memory integer sorting beat the comparison-sorting I/O bound?","short":"Aggarwal–Vitter","year_posed":1988,"years_open":38,"ai_system":["ChatGPT","Codex"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-assisted: the authors (two of them the conjecture's original proposers) directed the work; ChatGPT/Codex 'materially affected' constructions, analyses and the Lean proofs","result":"Deterministic external-memory sort of Θ(log N)-bit integers with asymptotically fewer block transfers than the comparison bound in a range of parameters, via a new network-coding-over-routing construction","field":"computer-science"},{"entry":{"id":"2026-09-14-k-server-conjecture-proved","url":"https://postcutoff.com/e/2026-09-14-k-server-conjecture-proved/","date":"2026-09-14","date_precision":"day","short_title":"The k-server conjecture, the 'holy grail' of online algorithms, is proved at Oxford","deck":"ChatGPT 6 Astra generalised the authors' k = 3 proof to all k","takeaway":"The k-server conjecture is one of the best-known open problems in algorithms, and Koutsoupias co-proved the previous best bound in 1995.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":1,"filed":"2026-09-30","updated":"2026-09-30","orgs":["University of Oxford","Christian Coester","Elias Koutsoupias","Marek Zbysiński","OpenAI"]},"problem":"k-server conjecture","short":"k-server conjecture","year_posed":1988,"years_open":38,"ai_system":["GPT-6 Astra (ChatGPT)","ChatGPT 5.5 Pro","Gemini 3.1 Pro"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-designed potential and k = 3 proof; the AI generalised it to all k; humans adapted and verified the final proof","result":"Deterministic k-competitiveness of the work function algorithm on every metric space.","field":"computer-science"},{"entry":{"id":"2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu","url":"https://postcutoff.com/e/2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu/","date":"2026-10-14","date_precision":"day","short_title":"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","deck":null,"takeaway":"Three analytic number theorists at Shandong University claim the Möbius function cancels on shifted primes p + 2, a folklore case of the Hardy–Littlewood–Chowla conjecture and a textbook example of the sieve parity barrier. Their method follows an OpenAI Math Release preprint, which ChatGPT helped them understand.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-15","updated":"2026-10-15","orgs":["Shandong University","OpenAI"]},"problem":"Hardy–Littlewood–Chowla conjecture, case Σ_{p≤x} µ(p+h) = o(π(x)) (Möbius on shifted primes; parity-problem model case)","short":"Hardy–Littlewood–Chowla conjecture, case Σ_{p≤x} µ(p+h)","year_posed":1989,"years_open":37,"ai_system":["ChatGPT","OpenAI internal model (via the OpenAI Math Release preprint)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-written proof; ChatGPT helped the authors understand the OpenAI preprint's method, establish Appendix A and check the manuscript","result":"Claimed: Σ_{p≤x} µ(p+2) = o(π(x)), and each sign of µ(p+2) has density 1/2 among primes with p+2 squarefree","field":"mathematics"},{"entry":{"id":"2026-09-22-mumford-shah-conjecture-astra-claim","url":"https://postcutoff.com/e/2026-09-22-mumford-shah-conjecture-astra-claim/","date":"2026-09-22","date_precision":"day","short_title":"Mathematician posts an unchecked ChatGPT Astra proof of the planar Mumford–Shah conjecture (1989), citing OpenAI's '100 open problems' claim","deck":null,"takeaway":"If correct, it would settle one of the best-known open problems in the calculus of variations.","category":"science","category_label":"Science & math","importance":4,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-09-30","updated":"2026-10-19","orgs":["Francesco Deangelis","University of Münster","OpenAI"]},"problem":"Planar Mumford–Shah conjecture (regularity of minimizing discontinuity sets)","short":"Planar Mumford–Shah conjecture","year_posed":1989,"years_open":37,"ai_system":["GPT-6 Astra (ChatGPT, Work mode)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"The human supplied his thesis, unpublished results and a research protocol; the model produced the proof text; checking by the author is ongoing","result":"Claimed classification of planar Mumford–Shah generalized global minimizers, which would imply the conjecture.","field":"mathematics"},{"entry":{"id":"2026-11-03-alon-seymour-thomas-separator-conjecture-gpt-6-astra-two-proofs","url":"https://postcutoff.com/e/2026-11-03-alon-seymour-thomas-separator-conjecture-gpt-6-astra-two-proofs/","date":"2026-11-03","date_precision":"day","short_title":"K_h-minor-free separators O(h√n): two AI-assisted proofs","deck":null,"takeaway":"The separator theorem underlies many algorithms on minor-closed graph classes.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-03","updated":"2026-11-03","orgs":["OpenAI","Anthropic"]},"problem":"Alon–Seymour–Thomas conjecture: K_h-minor-free graphs have balanced separators of size O(h√n)","short":"Alon–Seymour–Thomas conjecture","year_posed":1990,"years_open":36,"ai_system":["GPT-6 Astra","Claude Opus 5.5","ChatGPT 6 Pro","GPT-6.1 Sol"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Conroy et al.: AI-originated ideas, humans verified and rewrote; Lin: human-guided AI-assisted proof checked by hand","result":"Claimed proved twice: 21h√n (Lin) and O(h√n) with Lean formalization (Conroy–Le–Spalding-Jamieson)","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-deligne-drinfeld-conjecture-proof","url":"https://postcutoff.com/e/2026-10-06-openai-deligne-drinfeld-conjecture-proof/","date":"2026-10-06","date_precision":"day","short_title":"Deligne–Drinfeld conjecture proved; experts simplify and reprove OpenAI's proof","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 maths release (family 008, dated 23 Sept) proves the Deligne–Drinfeld conjecture. It says that the Grothendieck–Teichmüller Lie algebra grt₁ is free on one generator in each odd weight 3, 5, 7, …. The question goes back to Deligne (1989) and Drinfeld (1990). Thomas Willwacher (ETH Zurich), whose July 2026 paper had extended Brown's half of the conjecture, wrote that 'the remaining part … was solved by OpenAI'. He posted a much shorter version of the proof, about eight pages instead of about forty. Minoru Hirose then gave an alternative proof through multiple zeta values. Specialists studied the proof at a Trinity College Dublin workshop in mid-October. OpenAI's Lean proof of the main statement was accepted in a community Comparator run. This is one of the best-confirmed famous results of the release.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":13,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","ETH Zurich","Kagoshima University"]},"problem":"Deligne–Drinfeld conjecture: the Grothendieck–Teichmüller Lie algebra grt₁ is freely generated by one element in each odd weight 3, 5, 7, …","short":"Deligne–Drinfeld conjecture","year_posed":1990,"years_open":36,"ai_system":["OpenAI internal model (openai/math release)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-generated original proof; Schneps and Enriquez reworked it, Willwacher simplified and re-proved it with AI help, Hirose gave an independent proof (discussing the mathematics with GPT-6 Astra Pro)","result":"Proved in full by OpenAI's internal model (dimension upper bound via characteristic two). An expert simplified the proof over 𝔽₂ and accepted that it settles the conjecture (Willwacher), and a second, independent proof identifies the motivic Galois group of mixed Tate motives over ℤ with GRT (Hirose)","field":"mathematics"},{"entry":{"id":"2026-11-11-band-matrix-que-critical-crossover-axiomprover-lean","url":"https://postcutoff.com/e/2026-11-11-band-matrix-que-critical-crossover-axiomprover-lean/","date":"2026-11-11","date_precision":"day","short_title":"Band-matrix QUE and critical crossover, 1.2M lines of Lean by AxiomProver","deck":null,"takeaway":"Random band matrices are the standard model for the Anderson localization transition. A Chicago–Axiom Math preprint proves delocalization (QUE) right down to W ≫ √N and the predicted critical law at W² ≍ N, and says AxiomProver formalized all of it in Lean, including the published results it relies on.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-13","updated":"2026-11-13","orgs":["Axiom Math","University of Chicago"]},"problem":"Delocalization (QUE) and the critical localization–delocalization crossover for 1-d random band matrices (Fyodorov–Mirlin 1991)","short":"Delocalization","year_posed":1991,"years_open":35,"ai_system":["AxiomProver","unnamed LLMs"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Authors provided the architecture; LLMs drafted proofs, edited by authors; AxiomProver autoformalized everything","result":"QUE for W²/N → ∞ and a DMPK-described critical law at W² ≍ N for the Gaussian periodic block model","field":"mathematics"},{"entry":{"id":"2026-10-28-laczkovich-lipschitz-mapping-problem-chatgpt-gaspar","url":"https://postcutoff.com/e/2026-10-28-laczkovich-lipschitz-mapping-problem-chatgpt-gaspar/","date":"2026-10-28","date_precision":"day","short_title":"Laczkovich's Lipschitz mapping problem: claimed answer, ChatGPT key step","deck":null,"takeaway":"In 1991 Miklós Laczkovich asked whether every measurable set of positive measure in ℝ^d can be mapped onto a cube by a Lipschitz map. Preiss and Matoušek answered it in the plane; dimension 3 and up stayed open. On 28 Oct Attila Gáspár posted a positive answer in every dimension. His disclosure says the proof of Theorem 1.8 'was obtained using ChatGPT on 26 October', and Proposition 2.8 with Lemma 2.7 is 'entirely due to ChatGPT'. He had known other parts and the lower bound since early 2026. It is a short preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI"]},"problem":"Laczkovich's Lipschitz mapping problem (1991): can every measurable set of positive measure in ℝ^d be mapped onto a cube by a Lipschitz map?","short":"Laczkovich's Lipschitz mapping problem","year_posed":1991,"years_open":35,"ai_system":["ChatGPT"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: ChatGPT produced the key proposition and lemma; the author had the framework and lower bound, rewrote the proof and wrote the paper","result":"Claimed positive answer for every d (previously only d ≤ 2), plus Kolmogorov's contraction problem for Jordan measurable sets","field":"mathematics"},{"entry":{"id":"2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim","url":"https://postcutoff.com/e/2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims local smoothing and Kakeya in all dimensions","deck":null,"takeaway":"A 59-page OpenAI manuscript dated 25 Oct claims Sogge's local smoothing conjecture for the wave equation in every dimension n ≥ 2. The stated corollaries include the Bochner–Riesz conjecture, Fourier restriction throughout the interior of the conjectured range, the Kakeya and Nikodym maximal conjectures, and full Hausdorff dimension for every Euclidean Kakeya set. That last one is the Kakeya set conjecture, proved in ℝ³ only in 2025 (Wang–Zahl). OpenAI lists Lean comparator statements for the main estimate and the corollaries. No expert has assessed it yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":5,"filed":"2026-10-26","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Sogge's local smoothing conjecture (1991) in all dimensions; consequences: Bochner–Riesz, Stein's restriction conjecture (interior of range), Kakeya maximal and set conjectures in ℝⁿ","short":"Sogge's local smoothing conjecture in all dimensions; consequences","year_posed":1991,"years_open":35,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Credited to 'OpenAI'; released as 'model output directly', with no named human authors and no designated reviewers","result":"Claimed proof of local smoothing for the Euclidean wave equation in every dimension n ≥ 2 with ε loss, with Bochner–Riesz, restriction in the interior of the range, Nikodym/Kakeya maximal bounds and full Hausdorff dimension of Kakeya sets as corollaries","field":"mathematics"},{"entry":{"id":"2026-11-05-z-diophantine-in-q-glazer-burungale","url":"https://postcutoff.com/e/2026-11-05-z-diophantine-in-q-glazer-burungale/","date":"2026-11-05","date_precision":"day","short_title":"ℤ is Diophantine in ℚ: Glazer (Astra, Claude) and Burungale claim Mazur's conjecture fails","deck":null,"takeaway":"OpenAI's October preprint claimed Hilbert's tenth problem over ℚ is undecidable without defining ℤ in ℚ. A month later two independent manuscripts claim the stronger statement: one polynomial equation picks out exactly the integers among the rationals. Glazer's was produced with Astra 6, Sol 6.1 and Opus 5.5; Burungale says ChatGPT assisted two sections. Neither is refereed.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":4,"filed":"2026-11-13","updated":"2026-11-13","orgs":["OpenAI","Anthropic","Principia Labs","University of Texas at Austin"]},"problem":"Is ℤ a Diophantine subset of ℚ? (Mazur's conjecture predicts no; Hilbert's tenth problem over ℚ)","short":"Is ℤ a Diophantine subset of ℚ?","year_posed":1992,"years_open":34,"ai_system":["GPT-6 Astra","GPT-6.1 Sol","Claude Opus 5.5","Claude Fable 5.1","Claude Fable 5.2","ChatGPT","Claude"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Glazer: manuscript produced with Astra, Sol and Opus, audited by Fable and by human-run AI reviews; Burungale: human proof, ChatGPT assisted Sections 4–5","result":"Two claimed proofs that ℤ is Diophantine in ℚ, with explicit quartics (Glazer: 222 variables; Burungale: degree at most four), refuting Mazur's conjecture on the topology of rational points","field":"mathematics"},{"entry":{"id":"2026-10-05-random-k-sat-satisfiability-conjecture-carenini-openai","url":"https://postcutoff.com/e/2026-10-05-random-k-sat-satisfiability-conjecture-carenini-openai/","date":"2026-10-05","date_precision":"day","short_title":"Random k-SAT threshold exists for every k ≥ 3: a human proof and an OpenAI Lean-checked proof","deck":null,"takeaway":"The satisfiability conjecture says that random k-SAT switches from satisfiable to unsatisfiable at one limiting clause density α_k. Friedgut proved a sharp but possibly n-dependent threshold in 1999. Ding, Sly and Sun proved the conjecture for large k (Annals, 2022). On 5 Oct 2026 Cambridge PhD student Gaia Carenini posted on ECCC a proof for every fixed k ≥ 3. Her report states that none of its arguments was generated by a language model. The next day OpenAI's maths release included a separate proof, dated 25 Sept, which credits Carenini with priority. Its Lean statement was accepted in the community Comparator run, and it adds Θ(n) hitting-time variance and computability of the 3-SAT threshold. Neither proof has had a published expert assessment as of 29 Oct. Neither determines the value of α_3.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-29","updated":"2026-10-29","orgs":["University of Cambridge","OpenAI"]},"problem":"Satisfiability (threshold) conjecture for random k-SAT: existence of a limiting threshold density α_k for every fixed k ≥ 3","short":"Satisfiability conjecture for random k-SAT","year_posed":1992,"years_open":34,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Carenini: human proof, no LLM-generated arguments by her statement. OpenAI: autonomous output released without named authors, crediting Carenini with priority","result":"Existence of α_k for every fixed k ≥ 3, proved twice: by Carenini via an O(n^{1/2+1/k}) scaling window plus Abbe–Montanari, and by OpenAI's models via a capped last-satisfiable-index argument; OpenAI adds Θ(n) hitting-time variance and computability of α_3","field":"computer-science"},{"entry":{"id":"2026-09-22-odlyzko-poonen-conjecture-proved","url":"https://postcutoff.com/e/2026-09-22-odlyzko-poonen-conjecture-proved/","date":"2026-09-22","date_precision":"day","short_title":"Odlyzko–Poonen conjecture (1993) proved unconditionally","deck":"'The proofs are due to GPT-6 Astra', which also wrote a 22,000-line Lean formalisation","takeaway":"The unconditional case had stayed open after Breuillard and Varjú's conditional proof in 2019.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-10-07","orgs":["Constantin Kogler","OpenAI"]},"problem":"Odlyzko–Poonen conjecture on irreducibility of random 0/1 polynomials","short":"Odlyzko–Poonen conjecture","year_posed":1993,"years_open":33,"ai_system":["GPT-6 Astra"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-generated proofs and formalisation; the author directed the work, rewrote and checked it; an expert (Breuillard) checked the proof","result":"Unconditional proof that random 0/1 polynomials are irreducible with probability tending to one, with the sharp asymptotic for the reducible probability.","field":"mathematics"},{"entry":{"id":"2026-10-07-lebrun-salamon-conjecture-proof","url":"https://postcutoff.com/e/2026-10-07-lebrun-salamon-conjecture-proof/","date":"2026-10-07","date_precision":"day","short_title":"Two claimed proofs of the LeBrun–Salamon conjecture","deck":"Hu, Liu & Wan (arXiv, ChatGPT as 'auxiliary tool') and OpenAI's catalogue (family 062)","takeaway":"The 1994 conjecture that every positive quaternion-Kähler manifold is a symmetric Wolf space now has two independent claimed proofs, one by human geometers who used ChatGPT and one in OpenAI's AI-generated catalogue. Neither has been refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-08","updated":"2026-10-13","orgs":["Chongqing University of Technology","OpenAI"]},"problem":"LeBrun–Salamon conjecture: every compact positive quaternion-Kähler manifold is a symmetric Wolf space","short":"LeBrun–Salamon conjecture","year_posed":1994,"years_open":32,"ai_system":["ChatGPT","OpenAI internal model (unreleased)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led with AI tools for Hu–Liu–Wan (ChatGPT as an 'auxiliary tool', role not specified); autonomous for OpenAI's family 062","result":"Claimed proof for all real dimensions 4n with n ≥ 2 (the 4-dimensional case was already known), via transverse rational curves on the twistor space; independently claimed in OpenAI's family 062.","field":"mathematics"},{"entry":{"id":"2026-11-04-glauber-dynamics-colorings-1-plus-1-over-e-gpt-6-astra-pro","url":"https://postcutoff.com/e/2026-11-04-glauber-dynamics-colorings-1-plus-1-over-e-gpt-6-astra-pro/","date":"2026-11-04","date_precision":"day","short_title":"Colourings Glauber dynamics above (1+1/e)Δ (GPT-6 Astra Pro)","deck":null,"takeaway":"If it holds, this is a major step towards the Δ+2 conjecture, and another case of a frontier model supplying the central idea of a theoretical computer science result.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-04","updated":"2026-11-04","orgs":["OpenAI"]},"problem":"Rapid mixing of Glauber dynamics for proper colourings (Jerrum 1995; conjecture q ≥ Δ+2)","short":"Rapid mixing of Glauber dynamics for proper colourings","year_posed":1995,"years_open":31,"ai_system":["GPT-6 Astra Pro"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-originated ideas; author simplified, verified and wrote the text","result":"Claimed spectral gap Ω(1/n) for q ≥ (1+1/e+γ)Δ, large Δ, any graph","field":"computer-science"},{"entry":{"id":"2026-10-05-claude-refutes-3sum-apsp-hypotheses","url":"https://postcutoff.com/e/2026-10-05-claude-refutes-3sum-apsp-hypotheses/","date":"2026-10-05","date_precision":"day","short_title":"Claude discovers an algorithm that refutes the 3SUM and APSP hypotheses","deck":"Truly subquadratic 3SUM and truly subcubic APSP (Alman & Vassilevska Williams, Lean-verified)","takeaway":null,"category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-06","updated":"2026-10-13","orgs":["Anthropic","Columbia University","MIT"]},"problem":"3SUM hypothesis (no O(n^(2−ε)) algorithm) and APSP hypothesis (no O(n^(3−ε)) algorithm), plus Exact Triangle and Zero-Weight k-Clique","short":"3SUM hypothesis","year_posed":1995,"years_open":31,"ai_system":["Claude (Anthropic internal research model)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-originated: the core algorithm was found autonomously by Claude (16M output tokens, no human input) during an unrelated cryptography task; Alman and Vassilevska Williams simplified, extended and wrote the paper","result":"Deterministic O(n^1.9992) 3SUM and O(n^2.9995) integer APSP, refuting the 3SUM, APSP, Exact Triangle and Zero-Weight k-Clique hypotheses and three hinted OMv conjectures","field":"computer-science"},{"entry":{"id":"2026-10-02-kls-conjecture-proved-ai-assisted","url":"https://postcutoff.com/e/2026-10-02-kls-conjecture-proved-ai-assisted/","date":"2026-10-02","date_precision":"day","short_title":"Kannan–Lovász–Simonovits conjecture proved in two AI-assisted preprints","deck":"Song–Zhang's O(1) bound, then Bizeul–Klartag–Lehec ('most proofs … found by ChatGPT')","takeaway":"KLS was among the best-known open problems in high-dimensional geometry, with consequences for sampling algorithms on convex bodies.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-06","updated":"2026-10-13","orgs":["Microsoft Research","Weizmann Institute of Science","OpenAI","Anthropic"]},"problem":"Kannan–Lovász–Simonovits conjecture (universal Cheeger/Poincaré constant for isotropic log-concave measures)","short":"Kannan–Lovász–Simonovits conjecture","year_posed":1995,"years_open":31,"ai_system":["GPT-6 Astra","GPT-5.6 Sol","Claude Fable 5","Claude Fable 5.1","ChatGPT"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted (Song–Zhang: human-chosen approaches, AI co-developed); largely AI-generated (Bizeul–Klartag–Lehec: authors mainly understood and exposited ChatGPT's proofs)","result":"Universal O(1) bound on the KLS constant ψ_n (Song–Zhang); a second, AI-generated proof via Song–Zhang's criterion (Bizeul–Klartag–Lehec)","field":"mathematics"},{"entry":{"id":"2026-11-05-quasiplanar-graphs-linear-edges-chatgpt-6-pach-tardos-zeng","url":"https://postcutoff.com/e/2026-11-05-quasiplanar-graphs-linear-edges-chatgpt-6-pach-tardos-zeng/","date":"2026-11-05","date_precision":"day","short_title":"Quasiplanar graph conjecture proved by ChatGPT-6 (Pach, Tardos, Zeng)","deck":null,"takeaway":"János Pach, one of the conjecture's authors, with Gábor Tardos and Ji Zeng, writes that ChatGPT-6 proved that graphs drawn with no k pairwise crossing edges have linearly many edges, a conjecture known only for k ≤ 4. Their note gives a self-contained combinatorial version; an independent human preprint by Ankan Sadhu (1 Nov) claims a similar proof.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-11-08","updated":"2026-11-08","orgs":["OpenAI","Rényi Institute","ISTA"]},"problem":"Pach–Shahrokhi–Szegedy conjecture: k-quasiplanar graphs have O_k(n) edges","short":"Pach–Shahrokhi–Szegedy conjecture","year_posed":1996,"years_open":30,"ai_system":["ChatGPT-6"],"status":"pending","verification":{"key":"expert","label":"Expert-checked"},"human_role":"AI produced the proof (relying on Beeker–Lazarovich); human experts rewrote and simplified it","result":"Proof produced by ChatGPT-6; simplified self-contained proof by Pach, Tardos and Zeng","field":"mathematics"},{"entry":{"id":"2026-10-29-online-steiner-forest-greedy-log-k-loglog-k-gpt-astra-lean","url":"https://postcutoff.com/e/2026-10-29-online-steiner-forest-greedy-log-k-loglog-k-gpt-astra-lean/","date":"2026-10-29","date_precision":"day","short_title":"Online Steiner forest: greedy claimed Θ(log k log log k), with GPT Astra","deck":null,"takeaway":"In online Steiner forest, terminal pairs arrive one by one and must be connected at once. Awerbuch, Azar and Bartal proved in 1996 that the greedy algorithm is O(log² k)-competitive and conjectured O(log k). On 29 Oct a University of Maryland group led by MohammadTaghi Hajiaghayi posted matching bounds of Θ(log k log log k). If correct, that refutes the conjecture and separates greedy from the best online algorithm by a log log k factor. Some proofs 'were developed with the help of OpenAI's GPT Astra', with Gemini Argon assisting, and Lean files for the main theorem are on GitHub. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Google DeepMind","University of Maryland"]},"problem":"Competitive ratio of greedy for online Steiner forest; Awerbuch–Azar–Bartal (1996) conjectured O(log k)","short":"Competitive ratio of greedy for online Steiner forest; Awerbuch–Azar–Bartal conjectured O(log k)","year_posed":1996,"years_open":30,"ai_system":["GPT-6 Astra","Gemini 4 Argon"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-assisted: GPT Astra helped develop some proofs and the write-up, Gemini Argon assisted in parts; the authors take responsibility","result":"Claimed tight bound Θ(log k log log k), so the O(log k) conjecture is false","field":"computer-science"},{"entry":{"id":"2026-10-07-exact-overlaps-conjecture-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-07-exact-overlaps-conjecture-gpt-6-astra/","date":"2026-10-07","date_precision":"day","short_title":"Exact overlaps conjecture for self-similar measures on the line proved","deck":"GPT-6 Astra found the first proof (Kittle & Kogler), 47,000-line Lean formalization; OpenAI's catalogue claims it independently","takeaway":"A central conjecture of fractal geometry, open in general since Hochman's 2014 breakthrough, now has a proof whose key part came from a GPT-6 Astra session. The authors rewrote and formalized it in Lean, and OpenAI's own release claims the same theorem.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-08","updated":"2026-10-13","orgs":["University College London","Institute for Advanced Study","OpenAI"]},"problem":"Exact overlaps conjecture for self-similar measures on ℝ (Simon 1996 for sets; Hochman 2014 programme), and the generalized entropy–Lyapunov dimension formula","short":"Exact overlaps conjecture (dimension one)","year_posed":1996,"years_open":30,"ai_system":["GPT-6 Astra"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-found, human-rewritten: GPT-6 Astra produced the first full proof in a ~2.5-hour session and the Lean formalization; Kittle found a new deduction of the main theorem and the authors rewrote the paper and checked it","result":"Proof that every self-similar measure on the line has dim ν = min{1, h_μ/|χ_μ|}, so dimension drop occurs only with exact overlaps; no separation or algebraicity assumptions.","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-periodic-tiling-conjecture-dimension-three","url":"https://postcutoff.com/e/2026-10-06-openai-periodic-tiling-conjecture-dimension-three/","date":"2026-10-06","date_precision":"day","short_title":"Periodic tiling conjecture fails in dimension 3 (OpenAI claim, Lean-checked)","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 maths release (family 155, dated 23 Sept) claims a finite set of cubes in ℤ³ whose translates tile space, while no tiling by them is periodic. The same holds for its solid version in ℝ³. Greenfeld and Tao had disproved the periodic tiling conjecture only in some very high dimension. In one and two dimensions the conjecture is true, so three would be the lowest dimension where it fails. The Lean proof, which also covers the lower dimensions, was accepted in a community Comparator run. Erik Demaine and Stefan Langerman reused the paper's gadgets to prove that deciding whether one connected polycube tiles 3D space is undecidable. Chaim Goodman-Strauss said the paper has novel ideas but is 'unnecessarily difficult to read'. No published expert check of the whole proof was found as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":9,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Periodic tiling conjecture (explicit Euclidean formulation by Lagarias and Wang, 1996): every finite subset of ℤᵈ that tiles it by translations also tiles it periodically; likewise for bounded tiles of ℝᵈ","short":"Periodic tiling conjecture","year_posed":1996,"years_open":30,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Autonomous output released without named authors; method adapted from Greenfeld and Tao's Sudoku construction","result":"Claimed: a finite tile of ℤ³, and its unit-cube thickening in ℝ³, that tiles but has no fully periodic tiling, the lowest possible lattice dimension (Greenfeld–Tao had a counterexample only in very high dimension)","field":"mathematics"},{"entry":{"id":"2026-09-24-benjamini-schramm-pc-pu-nonamenable-openai-lean","url":"https://postcutoff.com/e/2026-09-24-benjamini-schramm-pc-pu-nonamenable-openai-lean/","date":"2026-09-24","date_precision":"day","short_title":"Benjamini–Schramm p_c < p_u: OpenAI bond proof in Lean, Claude site form","deck":null,"takeaway":"An openai/math manuscript dated 24 Sep 2026 claims p_c < p_u for bond percolation on every nonamenable quasi-transitive graph, with Comparator-checked Lean statements; on 10 Nov Yan Ru Pei posted a Claude Opus 5.5-assisted extension to site percolation.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-11-10","updated":"2026-11-10","orgs":["OpenAI","Anthropic"]},"problem":"Benjamini–Schramm conjecture: p_c < p_u on nonamenable quasi-transitive graphs","short":"Benjamini–Schramm conjecture","year_posed":1996,"years_open":30,"ai_system":["OpenAI internal model","Claude Opus 5.5"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"OpenAI: not stated per paper; Pei: chose problem and strategy, reviewed key lemmas, AI found and wrote proofs","result":"Claimed for bond percolation (OpenAI, Lean) and for site percolation (Pei, Claude agents)","field":"mathematics"},{"entry":{"id":"2026-09-29-list-total-colouring-conjecture-false-astra","url":"https://postcutoff.com/e/2026-09-29-list-total-colouring-conjecture-false-astra/","date":"2026-09-29","date_precision":"day","short_title":"List Total Colouring Conjecture (late 1990s) disproved","deck":"Counterexample found by ChatGPT 6 Astra Ultra 'with little input'","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-05","updated":"2026-10-07","orgs":["OpenAI","University of Victoria"]},"problem":"List Total Colouring Conjecture (Borodin–Kostochka–Woodall; Juvan–Mohar–Škrekovski; Hilton–Johnson)","short":"List total colouring conjecture","year_posed":1997,"years_open":29,"ai_system":["GPT-6 Astra Ultra"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-found counterexample: the author prompted the model to disprove the conjecture; he checked the proof and rewrote the drafts","result":"Disproved: an explicit 20-vertex cubic graph with total chromatic number 4 and list total chromatic number 5.","field":"mathematics"},{"entry":{"id":"2026-10-30-alon-krivelevich-conjecture-disproved-gpt-6-astra-xean","url":"https://postcutoff.com/e/2026-10-30-alon-krivelevich-conjecture-disproved-gpt-6-astra-xean/","date":"2026-10-30","date_precision":"day","short_title":"Alon–Krivelevich conjecture disproved with GPT-6 Astra","deck":null,"takeaway":"The conjecture, from Noga Alon and Michael Krivelevich's 1998 paper, was a central open question on list colouring of bipartite graphs (the best known upper bound is Δ/log Δ-type).","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-02","updated":"2026-11-02","orgs":["OpenAI","Nanjing Normal University","William & Mary"]},"problem":"Alon–Krivelevich conjecture: bipartite graphs of maximum degree Δ are O(log Δ)-choosable","short":"Alon–Krivelevich conjecture","year_posed":1998,"years_open":28,"ai_system":["GPT-6 Astra","xean","Claude Opus 5.5"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-found: GPT-6 Astra found the counterexample and proof; authors verified it; Claude rewrote the text","result":"Disproved: bipartite graphs with ch(G) ≥ (1/32 − o(1))(ln Δ)²/(ln ln Δ)³","field":"mathematics"},{"entry":{"id":"2026-10-09-smale-sixth-problem-five-bodies-lean-tomar","url":"https://postcutoff.com/e/2026-10-09-smale-sixth-problem-five-bodies-lean-tomar/","date":"2026-10-09","date_precision":"day","short_title":"Smale's sixth problem claimed for five bodies in the plane","deck":"Finitely many central configurations for all positive masses, AI-found and proved in Lean (Tomar)","takeaway":"An independent researcher claims the planar five-body case of one of Smale's 18 problems for the 21st century, removing the exceptional masses left by Albouy and Kaloshin in 2012. He states that generative AI found and proved the result and wrote the Lean formalization; the paper does not name the models.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-12","updated":"2026-10-19","orgs":["independent"]},"problem":"Smale's sixth problem: is the number of central configurations (relative equilibria) of the n-body problem finite for every choice of positive masses? Case n = 5 in the plane","short":"Smale's 6th problem, five bodies in the plane","year_posed":1998,"years_open":28,"ai_system":["unnamed generative AI tools"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"AI-generated per the author's disclosure (AI found and proved the results, wrote the programs, the Lean formalization and the draft); the human role is not described","result":"Claimed: finitely many planar central configurations for every five positive masses, removing the codimension-two exceptional set of Albouy–Kaloshin.","field":"mathematics"},{"entry":{"id":"2026-10-29-q-colorability-threshold-existence-aronow-lopatto-ai","url":"https://postcutoff.com/e/2026-10-29-q-colorability-threshold-existence-aronow-lopatto-ai/","date":"2026-10-29","date_precision":"day","short_title":"q-colourability threshold of G(n, d/n) exists, says a machine-written paper","deck":null,"takeaway":"Achlioptas and Friedgut proved in 1999 that q-colourability of the random graph G(n, d/n) has a sharp threshold, but only along a sequence that might depend on n. Whether that sequence converges to a constant d_q was recorded as an open problem and is widely called well known. On 29 Oct P. M. Aronow (Yale) and Patrick Lopatto (UNC Chapel Hill) posted on Hexagon a proof that it converges for every q ≥ 3, with a window of O(n(log n)^(−3/2)) steps for the hitting time. The authors' note says the whole document except that note is machine-written. The provenance names Opus 5.5, GPT 6 Astra, GPT Luna 6 and GPT 6.1 Sol, and says the results were formalised in Lean, but the paper links no Lean repository. Human understanding: 'some parts'. Two companion papers on the dynamics of the spherical Sherrington–Kirkpatrick model came the same day.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","OpenAI","Yale University","University of North Carolina at Chapel Hill"]},"problem":"Existence of the q-colourability threshold of G(n, d/n) (Achlioptas–Friedgut, 1999)","short":"Existence of the q-colourability threshold of G(n, d/n)","year_posed":1999,"years_open":27,"ai_system":["Claude Opus 5.5","GPT-6 Astra","GPT-6 Luna","GPT-6.1 Sol"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Authors supervised; the models did proof discovery, verification, Lean formalization and all writing; human understanding 'some parts'","result":"Claimed: for every q ≥ 3 a constant threshold d_q exists for G(n, d/n) (d_q/2 for G(n, ⌊cn⌋)), with an O(n(log n)^(−3/2)) hitting-time window","field":"mathematics"},{"entry":{"id":"2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential","url":"https://postcutoff.com/e/2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential/","date":"2026-10-27","date_precision":"day","short_title":"Anthropic's Claude also refutes ETH","deck":null,"takeaway":"Hours after OpenAI's ETH announcement, Anthropic posted 'A 2^o(n) Time Algorithm for 3-SAT', credited to Claude. It is a deterministic #k-SAT algorithm built on tensor-network contraction, with a different approach from OpenAI's paper, a near-identical bound and a Lean 4 proof that uses only the standard axioms. The paper says the core algorithm was found on 16 Oct with 14M output tokens and no human input. OpenAI reports its model found its disproof that same day. Neither proof has had expert review.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":10,"official":8,"filed":"2026-10-27","updated":"2026-11-06","orgs":["Anthropic"]},"problem":"Exponential Time Hypothesis (Impagliazzo–Paturi 1999/2001) and SETH","short":"Exponential Time Hypothesis","year_posed":1999,"years_open":27,"ai_system":["internal Claude research model (Anthropic)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous discovery (14M output tokens, 'no human input'); Anthropic researchers worked with Claude on the presentation; Lean formalization written by AI models","result":"Claimed deterministic 2^O(n log log n/√log n)-time (fixed k) algorithm for #k-SAT, refuting ETH and SETH, found independently of OpenAI's","field":"computer-science"},{"entry":{"id":"2026-10-26-openai-eth-false-subexponential-sat-claim","url":"https://postcutoff.com/e/2026-10-26-openai-eth-false-subexponential-sat-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims ETH is false","deck":null,"takeaway":"Anthropic independently posted a different Lean-checked refutation the next day (separate entry). OpenAI's model output claims a deterministic algorithm that counts the satisfying assignments of any k-CNF in time 2^(n/(log n)^(1/2+o(1))). If correct, it refutes ETH, SETH, Gap-ETH and NSETH, the assumptions behind hundreds of fine-grained hardness results. OpenAI says the main theorem is formalized in Lean. A working group led by ETH's proposers, Russell Impagliazzo and Ramamohan Paturi, with Ryan Williams, will examine the proof. No expert has assessed it in public yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":35,"official":11,"filed":"2026-10-26","updated":"2026-11-08","orgs":["OpenAI"]},"problem":"Exponential Time Hypothesis (Impagliazzo–Paturi 1999/2001), Strong ETH, Gap-ETH, NSETH","short":"Exponential Time Hypothesis","year_posed":1999,"years_open":27,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Credited to 'OpenAI' with no named human authors; OpenAI says it is model output. Human examination is assigned to an outside working group (Impagliazzo, Paturi, Williams)","result":"Claimed deterministic (n+m)^O_k(1)·2^(n/(log n)^(1/2+o(1)))-time, polynomial-space algorithm for #k-SAT, every fixed k ≥ 2, which would refute ETH and its variants","field":"computer-science"},{"entry":{"id":"2026-09-04-reed-solomon-list-decoding-capacity-ai","url":"https://postcutoff.com/e/2026-09-04-reed-solomon-list-decoding-capacity-ai/","date":"2026-09-04","date_precision":"day","short_title":"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)","deck":null,"takeaway":"In early September 2026 two preprints settled long-standing problems about Reed–Solomon codes, the most widely used error-correcting codes.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-07","updated":"2026-10-29","orgs":["Simons Institute","OpenAI","Ethereum Foundation"]},"problem":"Efficient list decoding of Reed–Solomon codes beyond the Johnson radius up to capacity; proximity gaps / mutual correlated agreement for RS codes","short":"Reed–Solomon list decoding up to capacity","year_posed":1999,"years_open":27,"ai_system":["GPT-5.6 Sol","ChatGPT 5.6 Pro","ChatGPT 6 Pro"],"status":"pending","verification":{"key":"expert","label":"Expert-checked"},"human_role":"AI-assisted: humans extracted the idea from a crowd-sourced prize submission and directed the work; GPT-5.6 Sol helped produce most of the remaining results (BCPZZ); ChatGPT 5.6 Pro interactions were 'crucial' (Jeronimo); authors verified everything","result":"Deterministic polynomial-time list decoding of RS codes over prime fields up to capacity for every constant rate and evaluation set, and near-optimal mutual correlated agreement (proximity gaps) up to capacity with fixed slack.","field":"computer-science"},{"entry":{"id":"2026-10-22-simon-l2-conjecture-disproved-chatgpt","url":"https://postcutoff.com/e/2026-10-22-simon-l2-conjecture-disproved-chatgpt/","date":"2026-10-22","date_precision":"day","short_title":"Barry Simon's L² conjecture (Problem 8 of his 2000 list) disproved","deck":"Becker and Yang build a counterexample largely found with ChatGPT","takeaway":"One of Barry Simon's fifteen problems for Schrödinger operators, open since 2000, is false: a small, smooth potential satisfying his L² condition can leave no continuous spectrum at all. The authors say ChatGPT produced the key mechanism and, 'very much to our surprise', extended it to the full spectrum.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-23","updated":"2026-10-23","orgs":["OpenAI"]},"problem":"Barry Simon's multidimensional L² conjecture (Problem 8, 'Schrödinger operators in the twenty-first century')","short":"Barry Simon's multidimensional L² conjecture","year_posed":2000,"years_open":26,"ai_system":["ChatGPT"],"status":"pending","verification":{"key":"expert","label":"Expert-checked"},"human_role":"AI-assisted: authors chose the problem and suspected falsity; ChatGPT developed the construction and the extension to the full spectrum; authors rewrote the argument","result":"False in every dimension d ≥ 2: smooth, bounded, nonnegative, arbitrarily small potentials satisfying the (even weighted) L² condition for which −Δ + V has only pure point spectrum on (0, ∞).","field":"mathematics"},{"entry":{"id":"2026-10-06-gch-above-strongly-compact-woodin-question","url":"https://postcutoff.com/e/2026-10-06-gch-above-strongly-compact-woodin-question/","date":"2026-10-06","date_precision":"day","short_title":"Woodin's question answered","deck":"GCH below a strongly compact cardinal implies GCH everywhere (Zhixing You, 4-page proof with GPT-6 Astra)","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-07","updated":"2026-10-13","orgs":["Hebrew University of Jerusalem","OpenAI"]},"problem":"Woodin's question: if κ is strongly compact and GCH holds below κ, does GCH hold everywhere?","short":"Woodin's question","year_posed":2000,"years_open":26,"ai_system":["GPT-6 Astra (ultra reasoning)","GPT-6.1 Sol"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted; the author acknowledges GPT-6 Astra and GPT-6.1 Sol 'during the investigation' without detailing their contribution and takes responsibility","result":"Yes, in ZFC; the argument also works for ω₁-strongly compact cardinals.","field":"mathematics"},{"entry":{"id":"2026-09-08-openai-navier-stokes-blowup","url":"https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/","date":"2026-09-08","date_precision":"day","short_title":"OpenAI claims a Millennium Prize problem","deck":"10,000 AI agents prove forced Navier–Stokes blow-up; priority dispute erupts","takeaway":"It is the first credible AI claim on a Clay Millennium Prize problem, even if only a technically permitted variant.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"disputed","labels":["Disputed"]},"sources":42,"official":4,"filed":"2026-09-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Navier–Stokes existence and smoothness (Clay Millennium Prize problem), forced-breakdown case","short":"Navier–Stokes existence and smoothness","year_posed":2000,"years_open":26,"ai_system":["OpenAI internal model (≈10,000 parallel agents)"],"status":"disputed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Largely autonomous agent swarm; builds on human techniques of Córdoba and Martínez-Zoroa","result":"Claimed proof, formalised in Lean, of finite-time blow-up for 3D incompressible Navier–Stokes with smooth external forcing.","field":"mathematics"},{"entry":{"id":"2026-11-12-binary-search-tree-unified-bound-ai","url":"https://postcutoff.com/e/2026-11-12-binary-search-tree-unified-bound-ai/","date":"2026-11-12","date_precision":"day","short_title":"A BST meets Iacono's 2001 unified bound, with Claude and GPT","deck":null,"takeaway":"Since 2001 it was open whether any binary search tree, even offline, achieves Iacono's unified bound, which combines the working-set and dynamic-finger bounds. A new preprint gives an online deterministic one, so dynamic optimality would imply Iacono's conjecture for splay trees. Five models from Anthropic and OpenAI helped design and analyse it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-13","updated":"2026-11-13","orgs":["Anthropic","OpenAI","Pennsylvania State University","Jilin University","Tsinghua University"]},"problem":"Is there a binary search tree satisfying Iacono's unified bound? (open since 2001; Iacono's splay-tree conjecture)","short":"Is there a binary search tree satisfying Iacono's unified bound?","year_posed":2001,"years_open":25,"ai_system":["Claude Opus 5.5","Claude Fable 5.1","Claude Fable 5.2","GPT-6 Astra","GPT-5.6 Sol"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted design, analysis, proof checking and drafting; authors take responsibility","result":"A deterministic online BST with cost O(m + UB(X)); dynamic optimality now implies the unified bound for splay trees","field":"computer-science"},{"entry":{"id":"2026-10-30-mezard-parisi-formula-panchenko-proof-found-by-chatgpt-astra","url":"https://postcutoff.com/e/2026-10-30-mezard-parisi-formula-panchenko-proof-found-by-chatgpt-astra/","date":"2026-10-30","date_precision":"day","short_title":"Mézard–Parisi formula: Panchenko's proof 'found by ChatGPT Astra'","deck":null,"takeaway":"Dmitry Panchenko (Toronto), a leading probabilist of spin glasses, posted a proof of the Mézard–Parisi formula on 29–30 Oct 2026. The formula gives the free energy of random K-SAT and diluted p-spin models with even p. Franz and Leone had proved the upper bound; Panchenko's paper claims the matching lower bound. His AI statement says: 'The proof in this paper was found by ChatGPT Astra.' A companion note was 'written by ChatGPT Astra'. He adds that OpenAI released another proof a few days later, which 'appears to follow a different approach'. OpenAI's version is family 221 of its maths repository. Both are unrefereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","University of Toronto"]},"problem":"Mézard–Parisi formula (2001) for the free energy of diluted spin-glass models, including random K-SAT","short":"Mézard–Parisi formula for the free energy of diluted spin-glass models, including random K-SAT","year_posed":2001,"years_open":25,"ai_system":["ChatGPT Astra","Claude Fable"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-found proof directed by the field's main expert: Panchenko asked for a new perturbation in his earlier argument; ChatGPT Astra found the proof; Panchenko rewrote it with ChatGPT Astra and Claude Fable","result":"Claimed matching lower bound for diluted p-spin models with even p and random K-sat under symmetry and integrability assumptions, completing the formula (upper bound: Franz–Leone)","field":"mathematics"},{"entry":{"id":"2026-08-04-period-index-conjecture-false","url":"https://postcutoff.com/e/2026-08-04-period-index-conjecture-false/","date":"2026-08-04","date_precision":"day","short_title":"Alexander Perry disproves the period-index conjecture","deck":"A flawed ChatGPT example was the starting point","takeaway":"The period-index conjecture was a central open question on Brauer groups of function fields.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Alexander Perry","OpenAI"]},"problem":"Period-index conjecture","short":"Period-index conjecture","year_posed":2001,"years_open":25,"ai_system":["ChatGPT"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led: a wrong AI example inspired the construction; the human found and proved the correct one, with AI help on literature search and lattice calculations","result":"Varieties of every dimension d ≥ 3 with a Brauer class of period 2 and index 2^d, contradicting the conjecture; a corrected conjecture is proposed.","field":"mathematics"},{"entry":{"id":"2026-10-29-spectral-isometry-conjecture-c-star-algebras-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-29-spectral-isometry-conjecture-c-star-algebras-gpt-6-astra/","date":"2026-10-29","date_precision":"day","short_title":"Spectral isometry conjecture for C*-algebras: GPT-6 Astra proof, Lean","deck":null,"takeaway":"Mathieu's spectral isometry conjecture asks whether every unital surjective linear map between unital C*-algebras that preserves the spectral radius is a Jordan isomorphism. Only special classes were known. On 29 Oct Matthew Colbrook and George Stepaniants (Cambridge) posted what they call 'a human digestion of a proof … found by GPT-6 Astra'. The theorem is registered in Lean on Palomar. Their AI declaration says the proof came 'after a sequence of simple prompts'. This is the second such digest from Colbrook in a week, after Smale's mean value conjecture. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","University of Cambridge"]},"problem":"Spectral isometry conjecture (Mathieu–Schick 2002): unital surjective spectral isometries between unital C*-algebras are Jordan isomorphisms","short":"Spectral isometry conjecture","year_posed":2002,"years_open":24,"ai_system":["GPT-6 Astra","ChatGPT 6"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI found the proof; the authors studied it, rewrote it as a human-readable paper and checked it","result":"Claimed full proof for arbitrary unital C*-algebras, plus the non-unital-map variant; Lean-checked per the authors (Palomar)","field":"mathematics"},{"entry":{"id":"2026-10-06-finite-index-over-treeable-jkl-problem","url":"https://postcutoff.com/e/2026-10-06-finite-index-over-treeable-jkl-problem/","date":"2026-10-06","date_precision":"day","short_title":"Chen, Tserunyan & Tucker-Drob solve the Jackson–Kechris–Louveau 'finite index over treeable' problem (2002); ChatGPT 6 Astra Ultra supplied a 'crucial observation'","deck":null,"takeaway":"A countable Borel equivalence relation that contains a treeable subrelation of finite index is itself treeable. This answers Open Problem 6.4(B) of Jackson–Kechris–Louveau (2002), open even in the index-2 measure-preserving case. The authors had the strategy; GPT supplied the observation that made the stages fit together.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-09","updated":"2026-10-13","orgs":["University of Warsaw","McGill University","University of Florida","OpenAI"]},"problem":"Jackson–Kechris–Louveau Open Problem 6.4(B): is a countable Borel equivalence relation with a finite-index treeable subrelation treeable?","short":"Finite index over treeable (JKL 2002)","year_posed":2002,"years_open":24,"ai_system":["ChatGPT 6 Astra Ultra"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led: the authors designed the staged construction; GPT contributed one key coherence observation; the paper was written by the authors","result":"Yes, in the purely Borel setting (hence also measurably); with consequences for strongly treeable groups.","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-abelian-zilber-pink-claim","url":"https://postcutoff.com/e/2026-10-06-openai-abelian-zilber-pink-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims abelian Zilber–Pink","deck":null,"takeaway":"Family 016 of OpenAI's 6 Oct 2026 release claims the abelian case of the Zilber–Pink conjecture on unlikely intersections: every subvariety of an abelian variety over a number field has only finitely many maximal atypical subvarieties. By a 2022 theorem of Barroero and Dill this would give Zilber–Pink for abelian varieties over any field of characteristic zero. Three more papers claim the curve case in the Siegel threefold A2. Nothing is formalized; the first specialist papers cite it as an 'announced' AI-generated proof, and one group says the release pushed them to post their own related work a day later.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Zilber–Pink conjecture for abelian varieties (Zilber 2002, Pink 2005); curve case in the Siegel threefold A2","short":"Zilber–Pink conjecture for abelian varieties","year_posed":2002,"years_open":24,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","result":"Claimed: finiteness of maximal atypical subvarieties for every subvariety of an abelian variety over Q̄ (hence over any characteristic-zero field by Barroero–Dill); Zilber–Pink for Hodge-generic curves in A2","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e","url":"https://postcutoff.com/e/2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims tight 1 + 2/e approximation for k-median","deck":null,"takeaway":"Family 125 of OpenAI's 722-manuscript release claims a deterministic polynomial-time (1 + 2/e + ε)-approximation for metric k-median, which would settle the problem's approximability under P ≠ NP. OpenAI lists Lean comparator statements for it; no independent expert check has been published yet.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-15","updated":"2026-10-15","orgs":["OpenAI"]},"problem":"Exact polynomial-time approximability of metric k-median (gap between the 1 + 2/e hardness and the ~2 upper bound)","short":"Exact polynomial-time approximability of metric k-median","year_posed":2002,"years_open":24,"ai_system":["unreleased internal OpenAI model"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous per OpenAI: produced by the model in OpenAI's open-problem evaluation; humans curated and released","result":"Claimed: deterministic polynomial-time (1 + 2/e + ε)-approximation for every fixed ε > 0, matching the NP-hardness bound, so the threshold is exactly 1 + 2/e; plus a randomized (2 − σ)-approximation","field":"computer-science"},{"entry":{"id":"2026-10-06-unique-games-conjecture-openai","url":"https://postcutoff.com/e/2026-10-06-unique-games-conjecture-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims a proof of Khot's Unique Games Conjecture, with Lean formalization","deck":null,"takeaway":"If OpenAI's claimed proof holds, it settles the Unique Games Conjecture, a central open problem in hardness of approximation since 2002. It comes with a Lean formalization, but experts have not yet checked it.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":16,"official":5,"filed":"2026-10-07","updated":"2026-11-12","orgs":["OpenAI"]},"problem":"Unique Games Conjecture (Khot 2002)","short":"Unique Games Conjecture","year_posed":2002,"years_open":24,"ai_system":["OpenAI internal model (unreleased)"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"Autonomous by OpenAI's fixed procedure (per README)","result":"Claimed: NP-hardness of (1−ε, δ)-gap Unique Games for every fixed ε, δ; hence optimal NP-hardness for Max-Cut (Goemans–Williamson) and Vertex Cover (factor 2).","field":"computer-science"},{"entry":{"id":"2026-10-07-multiple-unicast-conjecture-false-gpt-6","url":"https://postcutoff.com/e/2026-10-07-multiple-unicast-conjecture-false-gpt-6/","date":"2026-10-07","date_precision":"day","short_title":"Braverman & He refute the 2004 undirected multiple-unicast (network coding) conjecture with a GPT-6-found counterexample on PG(2, 9)","deck":null,"takeaway":"A 22-year-old network coding conjecture is false: on undirected networks, coding can beat routing. The counterexample came from GPT-6, and the Princeton authors checked it with a standalone symbolic verifier.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-08","updated":"2026-10-15","orgs":["Princeton University","OpenAI"]},"problem":"Undirected multiple-unicast conjecture (Li & Li 2004; Harvey, Kleinberg & Lehman 2004)","short":"Multiple-unicast network coding conjecture","year_posed":2004,"years_open":22,"ai_system":["GPT-6 (ChatGPT)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI found the counterexample to a strengthened version the authors formulated; humans steered the search, converted it to a causal code and checked the proof","result":"Disproved: an explicit linear network code over 𝔽₉ on a 182-vertex PG(2, 9) incidence subgraph beats fractional multicommodity flow (rate 1 vs ≤ 147/157), giving Ω((log n)^ε) coding gaps by amplification.","field":"computer-science"},{"entry":{"id":"2026-07-27-crouzeix-conjecture-proved","url":"https://postcutoff.com/e/2026-07-27-crouzeix-conjecture-proved/","date":"2026-07-27","date_precision":"day","short_title":"Neurosurgery resident uses GPT-5.6 Sol to prove Crouzeix's conjecture in a 16-hour autonomous run","deck":null,"takeaway":"Along with #1196, it showed that frontier models let amateurs resolve famous problems, which upended assumptions about who can do research mathematics.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":1,"filed":"2026-09-29","updated":"2026-09-30","orgs":["OpenAI"]},"problem":"Crouzeix's conjecture","short":"Crouzeix's conjecture","year_posed":2004,"years_open":22,"ai_system":["GPT-5.6 Sol"],"status":"confirmed","verification":{"key":"expert","label":"Expert-checked"},"human_role":"Autonomous: non-specialist prompted; experts verified","result":"Proof that the numerical range W(A) is a 2-spectral set for every matrix A.","field":"mathematics"},{"entry":{"id":"2026-11-10-explicit-rip-matrices-two-thirds-gpt-6-astra","url":"https://postcutoff.com/e/2026-11-10-explicit-rip-matrices-two-thirds-gpt-6-astra/","date":"2026-11-10","date_precision":"day","short_title":"Explicit RIP matrices reach k = m^(2/3): main theorem by GPT-6 Astra","deck":null,"takeaway":"Since 2011 explicit compressed-sensing matrices had only beaten the square-root bottleneck by a tiny ε. A GPT-6 Astra proof, digested and written up by two specialists and formalized in Lean, reaches exponent 2/3.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":1,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","University of Central Florida","Ohio State University"]},"problem":"Explicit (deterministic) matrices with the restricted isometry property beyond the square-root bottleneck (Tao 2007)","short":"Explicit matrices with the restricted isometry property beyond the square-root bottleneck","year_posed":2007,"years_open":19,"ai_system":["GPT-6 Astra","ChatGPT","Claude Opus 5.5"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-generated main proof; humans digested it, found the 2/3 barrier with ChatGPT and rewrote the paper","result":"Explicit real RIP matrices with k = m^(2/3−o(1)), plus a proof that absolute-value Gram-matrix methods cannot exceed exponent 2/3","field":"mathematics"},{"entry":{"id":"2026-10-16-openai-2d-area-law-lean-formalized-axiom","url":"https://postcutoff.com/e/2026-10-16-openai-2d-area-law-lean-formalized-axiom/","date":"2026-10-16","date_precision":"day","short_title":"OpenAI's 2D area law proof Lean-verified by Axiom","deck":null,"takeaway":"One of the physics claims in OpenAI's October maths release, an entropy area law for gapped 2D lattice ground states (open since Hastings' 1D proof in 2007), now has a complete Lean 4 proof with a frozen statement layer and comparator check, produced by AI agents under a human auditor.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Axiom Math","OpenAI"]},"problem":"Area law for entanglement entropy of ground states of gapped local Hamiltonians in two dimensions","short":"Area law for entanglement entropy of ground states of gapped local Hamiltonians in two dimensions","year_posed":2007,"years_open":19,"ai_system":["OpenAI internal model (original proof)","Axiom agent tooling","OpenAI Codex"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-generated proof (OpenAI release); formalization by AI agents directed by Ruiz Chen (Axiom), who froze and audited the Lean statement layer and checked each result","result":"Lean 4 proof, with no sorry, of OpenAI's theorem that the unique ground state of a globally gapped finite-range Hamiltonian on any finite induced square-lattice domain has entanglement entropy at most C × (number of boundary edges), with C independent of domain size and shape.","field":"physics"},{"entry":{"id":"2026-09-13-matroid-secretary-conjecture-resolved-gpt-astra","url":"https://postcutoff.com/e/2026-09-13-matroid-secretary-conjecture-resolved-gpt-astra/","date":"2026-09-13","date_precision":"day","short_title":"Matroid secretary conjecture resolved, with GPT Astra","deck":null,"takeaway":"The matroid secretary problem was one of the best-known open problems in online algorithms.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":5,"filed":"2026-11-09","updated":"2026-11-09","orgs":["OpenAI","various researchers"]},"problem":"Matroid secretary conjecture (Babaioff–Immorlica–Kleinberg) and its strong form","short":"Matroid secretary conjecture","year_posed":2007,"years_open":19,"ai_system":["GPT-6 Astra","GPT-5.6 Sol","ChatGPT Pro"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: authors chose approaches and ran the LP computations; GPT Astra used in obtaining the main theorems; authors checked the proofs","result":"O(1) (4-probability-competitive) secretary algorithm for every matroid; merged paper claims the optimal ratio e and a ratio-2 single-sample prophet inequality","field":"computer-science"},{"entry":{"id":"2026-11-06-psi-aaronson-ambainis-conjecture-proof","url":"https://postcutoff.com/e/2026-11-06-psi-aaronson-ambainis-conjecture-proof/","date":"2026-11-06","date_precision":"day","short_title":"PSI claims Lean-checked proof of the Aaronson–Ambainis conjecture","deck":null,"takeaway":"The conjecture is the main missing step for showing that quantum computers can't get more than a polynomial speedup on unstructured problems for most inputs.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":4,"filed":"2026-11-07","updated":"2026-11-12","orgs":["Physical Superintelligence","Anthropic"]},"problem":"Aaronson–Ambainis conjecture (2008): bounded low-degree polynomials have an influential variable","short":"Aaronson–Ambainis conjecture","year_posed":2008,"years_open":18,"ai_system":["non-public Anthropic model","PSI research system"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI found the proof in an autonomous run; a human scoped the problem, reviewed the AI-drafted paper and advisers audited it","result":"Max influence ≥ c·v^(5/2)/d^6 (≥ (v/d)^7) for bounded degree-d polynomials of variance v, claimed with a Lean-checked proof","field":"computer-science"},{"entry":{"id":"2026-11-03-mixed-volume-fpras-gurvits-conjecture-disproved-gpt-6-astra","url":"https://postcutoff.com/e/2026-11-03-mixed-volume-fpras-gurvits-conjecture-disproved-gpt-6-astra/","date":"2026-11-03","date_precision":"day","short_title":"FPRAS for mixed volumes, ideas from GPT-6 Astra","deck":null,"takeaway":"Mixed volumes generalize permanents and volumes; Gurvits had conjectured that no such scheme exists.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-03","updated":"2026-11-03","orgs":["OpenAI"]},"problem":"Gurvits's conjecture (2009) that mixed volumes admit no FPRAS in the oracle model","short":"Gurvits's conjecture that mixed volumes admit no FPRAS in the oracle model","year_posed":2009,"years_open":17,"ai_system":["GPT-6 Astra"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-originated: main ideas from GPT-6 Astra, humans simplified and wrote the exposition","result":"FPRAS for mixed volumes of n convex bodies in R^n","field":"computer-science"},{"entry":{"id":"2026-10-07-toms-winter-conjecture-completed-chatgpt-astra-lemma","url":"https://postcutoff.com/e/2026-10-07-toms-winter-conjecture-completed-chatgpt-astra-lemma/","date":"2026-10-07","date_precision":"day","short_title":"Toms–Winter conjecture on C*-algebras completed by two preprints in one week","deck":"Both credit AI with a key lemma (Thiel; Evington, Hua, Schafhauser, Seth, Tikuisis, White)","takeaway":"The Toms–Winter conjecture (2000s) says three very different regularity properties of simple nuclear C*-algebras are equivalent; it was the central structural question of the classification programme. Hannes Thiel proved one missing implication on 5 Oct 2026 and a six-author team the last one on 7 Oct. The team says an initial version of an essential lemma came from ChatGPT 6 Astra in 27 minutes; Thiel says AI collaboration led to his key idea.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-13","updated":"2026-10-13","orgs":["University of Oxford","University of Ottawa","Chalmers University of Technology","OpenAI"]},"problem":"Toms–Winter conjecture: for simple separable nuclear non-elementary C*-algebras, finite nuclear dimension, Z-stability and almost unperforated Cuntz semigroup (strict comparison) are equivalent","short":"Toms–Winter conjecture (C*-algebras)","year_posed":2009,"years_open":17,"ai_system":["ChatGPT 6 Astra (GPT-6 Astra via Codex)","unnamed AI (Thiel)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led with AI input: one prompt gave a 4-page proof of a lemma in a special case (W*-bundles); the authors built the general theory and the 70-page proof without further AI use. Thiel: AI collaboration at an initial stage led to the key idea of two lemmas","result":"Claimed complete: Thiel proves strict comparison implies pureness (5 Oct 2026); Evington, Hua, Schafhauser, Seth, Tikuisis and White prove pureness implies Z-stability (7 Oct 2026).","field":"mathematics"},{"entry":{"id":"2026-09-21-courtade-kumar-conjecture-proved","url":"https://postcutoff.com/e/2026-09-21-courtade-kumar-conjecture-proved/","date":"2026-09-21","date_precision":"day","short_title":"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","deck":null,"takeaway":"A central 2013 conjecture of information theory, that one input bit (a dictator) keeps the most information through noise, fell to three independent proofs on 21–22 Sep 2026. All three teams disclose AI help; Google's 250-page proof says 'the overwhelming majority of the novel ideas' came from AI and is checked end-to-end in Lean.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":8,"official":6,"filed":"2026-10-09","updated":"2026-10-13","orgs":["Google","Chinese University of Hong Kong","FPT University","OpenAI"]},"problem":"Courtade–Kumar (most informative Boolean function) conjecture: dictators maximise I(f(X); Y) under binary symmetric noise","short":"Courtade–Kumar conjecture","year_posed":2013,"years_open":13,"ai_system":["Gemini (Stellar Colosseum)","ChatGPT","unnamed generative AI systems"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Google–CUHK: humans supplied the reduction and the Bellman function, AI generated most new ideas and the region-by-region proof; Ky–Tran: human approach, ChatGPT developed the proofs; Mahdavifar–Beirami: AI generated the single-bit proof under human direction","result":"Proved in full (all n, all noise levels, no balance assumption) by three independent teams on 21–22 Sep 2026; the Google–CUHK proof is computer-assisted and verified end-to-end in Lean.","field":"mathematics"},{"entry":{"id":"2026-08-30-bounded-prime-gaps-186","url":"https://postcutoff.com/e/2026-08-30-bounded-prime-gaps-186/","date":"2026-08-30","date_precision":"day","short_title":"GPT-6 Astra lowers the bounded prime gaps record from 246 to 186","deck":null,"takeaway":"Bounded prime gaps were one of the celebrated stories of 2013–14.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"]},"problem":"Bounded gaps between primes (toward the twin prime conjecture)","short":"Bounded gaps between primes","year_posed":2014,"years_open":12,"ai_system":["GPT-6 Astra"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"Largely AI-generated per OpenAI","result":"Claimed proof that infinitely many pairs of primes differ by at most 186.","field":"mathematics"},{"entry":{"id":"2026-11-03-kronheimer-mrowka-dodecahedron-web-instanton-homology-64-chatgpt","url":"https://postcutoff.com/e/2026-11-03-kronheimer-mrowka-dodecahedron-web-instanton-homology-64-chatgpt/","date":"2026-11-03","date_precision":"day","short_title":"Kronheimer–Mrowka: dodecahedron instanton homology is 64 (ChatGPT Astra 6)","deck":null,"takeaway":"Kronheimer and Mrowka built J♯ in 2015 as a possible route to the four-colour theorem, asking whether its dimension counts Tait colourings for planar webs.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-03","updated":"2026-11-03","orgs":["OpenAI","Anthropic","Harvard University","MIT"]},"problem":"Dimension of the SO(3) instanton homology J♯ of the dodecahedron web: 60 (number of Tait colourings) or 64","short":"Dimension of the SO(3) instanton homology J♯ of the dodecahedron web","year_posed":2015,"years_open":11,"ai_system":["ChatGPT Astra 6","ChatGPT","Claude"],"status":"pending","verification":{"key":"none","label":"Unverified claim"},"human_role":"Leading gauge theorists posed the question and supplied earlier AI-computed data; the text is primarily AI-generated and only partly checked","result":"Proposed proof that the dimension is 64","field":"mathematics"},{"entry":{"id":"2026-11-05-3-uniform-hyperclique-hypothesis-false-chatgpt-sol","url":"https://postcutoff.com/e/2026-11-05-3-uniform-hyperclique-hypothesis-false-chatgpt-sol/","date":"2026-11-05","date_precision":"day","short_title":"3-uniform hyperclique hypothesis refuted with ChatGPT Sol 6.1","deck":null,"takeaway":"The h-uniform hyperclique hypothesis was a standard assumption for conditional lower bounds.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-05","updated":"2026-11-05","orgs":["OpenAI","University of Illinois Urbana-Champaign"]},"problem":"3-Uniform Hyperclique Hypothesis (Lincoln–Vassilevska Williams–Williams 2018)","short":"3-Uniform Hyperclique Hypothesis","year_posed":2018,"years_open":8,"ai_system":["ChatGPT Sol 6.1"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: authors prompted the model with different suggestions; both proofs came from the model, the authors verified and rewrote them","result":"Truly subquartic (O(n^3.957)) algorithm for detecting or counting 4-hypercliques in 3-uniform hypergraphs; the hypothesis is false","field":"computer-science"},{"entry":{"id":"2026-11-11-benson-etingof-ostrik-conjecture-gpt-6-astra","url":"https://postcutoff.com/e/2026-11-11-benson-etingof-ostrik-conjecture-gpt-6-astra/","date":"2026-11-11","date_precision":"day","short_title":"Benson–Etingof–Ostrik conjecture: full proof generated by GPT-6 Astra","deck":null,"takeaway":"A month after the finite case, the full 2023 conjecture on symmetric tensor categories in positive characteristic is claimed in a paper 'generated by GPT-6 Astra', posted by a team that includes Etingof and Ostrik, who posed it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","MIT","University of Sydney","University of Bonn","University of Oregon"]},"problem":"Benson–Etingof–Ostrik conjecture (2023) on symmetric tensor categories of moderate growth in positive characteristic","short":"Benson–Etingof–Ostrik conjecture on symmetric tensor categories of moderate growth in positive characteristic","year_posed":2023,"years_open":3,"ai_system":["GPT-6 Astra"],"status":"pending","verification":{"key":"expert","label":"Expert-checked"},"human_role":"AI-generated paper; five specialists (including two of the conjecture's authors) directed it with questions, reductions and ideas","result":"Claimed full proof: every such category admits a symmetric tensor functor to a higher Verlinde category, possibly of infinite height","field":"mathematics"},{"entry":{"id":"2026-11-11-hyperelliptic-schottky-problem-huang-jiang-ai","url":"https://postcutoff.com/e/2026-11-11-hyperelliptic-schottky-problem-huang-jiang-ai/","date":"2026-11-11","date_precision":"day","short_title":"Hyperelliptic Schottky problem solved; AI suggested the final step","deck":null,"takeaway":"Which abelian varieties are Jacobians of hyperelliptic curves? Huang and Jiang say: exactly those whose theta divisor has singular locus of codimension three. Their acknowledgement says AI tools suggested completing the proof of the main theorem with a known result and constructed the paper's sharpness example.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-13","updated":"2026-11-13","orgs":["Fudan University","various researchers"]},"problem":"Hyperelliptic Schottky problem (codim-3 theta singularities); Pareschi–Popa Conjecture A; Debarre's conjecture on base loci","short":"Hyperelliptic Schottky problem","year_posed":null,"years_open":null,"ai_system":["unnamed AI tools"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led; AI suggested the final step of the main proof and constructed the sharpness example","result":"Codim Sing(Θ) = 3 characterizes hyperelliptic Jacobians; IT₀ criterion for GV-subschemes; generic reducedness of codim-2 base loci","field":"mathematics"},{"entry":{"id":"2026-11-11-talagrand-ultimate-matching-conjecture-gpt-6","url":"https://postcutoff.com/e/2026-11-11-talagrand-ultimate-matching-conjecture-gpt-6/","date":"2026-11-11","date_precision":"day","short_title":"Talagrand's Ultimate Matching Conjecture: GPT-6 proof (Hexagon)","deck":null,"takeaway":"Three specialists in random matching say GPT-6 found the critical-exponent proof of a research problem from Talagrand's book and then the endpoint answer; the text is mostly model-written and the authors promise a digested version.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","CNRS","University of Pisa"]},"problem":"Talagrand's Ultimate Matching Conjecture (Research Problem 17.1.2 in 'Upper and Lower Bounds for Stochastic Processes')","short":"Talagrand's Ultimate Matching Conjecture","year_posed":null,"years_open":null,"ai_system":["GPT-6"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: specialists steered weeks of LLM interaction; GPT-6 found the critical and endpoint proofs; text mostly LLM-written","result":"Claimed proof for all finite critical exponent pairs α₁, α₂ > 2 with 1/α₁ + 1/α₂ = 1/2; claimed failure at the endpoints (2, ∞), (∞, 2)","field":"mathematics"},{"entry":{"id":"2026-11-09-non-split-cartan-121-two-ai-proofs","url":"https://postcutoff.com/e/2026-11-09-non-split-cartan-121-two-ai-proofs/","date":"2026-11-09","date_precision":"day","short_title":"X_ns+(121): two AI-generated proofs in two days","deck":null,"takeaway":"Arango-Piñeros, Najman and Zureick-Brown (Claude Opus 5.5, GPT-6.1 Sol, GPT-6 Astra) and, separately, Manvir Jaswal (an agent system on Claude Opus 5.5) posted proofs on Hexagon that X_ns+(121) has exactly 7 rational points, all CM. Neither is fully human-checked.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-11-10","updated":"2026-11-10","orgs":["Anthropic","OpenAI","University of Zagreb","UMass Amherst","Amherst College"]},"problem":"Rational points on X_ns+(121); the level-121 case of the Rouse–Sutherland–Zureick-Brown conjecture on ℓ-adic images","short":"Rational points on X_ns+(121); the level-121 case of the Rouse–Sutherland–Zureick-Brown conjecture on ℓ-adic images","year_posed":null,"years_open":null,"ai_system":["Claude Opus 5.5","GPT-6.1 Sol","GPT-6 Astra","Claude Code"],"status":"pending","verification":{"key":"none","label":"Unverified claim"},"human_role":"Humans chose the problem and directed; AI produced almost all the mathematics, computations and text; partial (2611.00231) or no (2611.00255) human checking","result":"Claimed: X_ns+(121)(Q) consists of 7 CM points, so the 11-adic images of non-CM elliptic curves over Q are determined","field":"mathematics"},{"entry":{"id":"2026-11-07-anthropic-claude-hawking-page-modular-bootstrap-hexagon","url":"https://postcutoff.com/e/2026-11-07-anthropic-claude-hawking-page-modular-bootstrap-hexagon/","date":"2026-11-07","date_precision":"day","short_title":"Anthropic's Claude physics papers: Hawking–Page and modular bootstrap","deck":null,"takeaway":"Anthropic's own Hexagon account posted two 'primarily AI-generated' hep-th papers on 7 Nov: an internal Claude research model computed a three-loop coefficient unknown for two decades (b = −1.48×10⁻³ λ²), and found the optimal slope of the spinless modular bootstrap, later reproduced by public Claude Opus 5.5. A physicist checked the results.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-11-08","updated":"2026-11-09","orgs":["Anthropic"]},"problem":"Order of the weak-coupling Hawking–Page/deconfinement transition in N=4 SYM; optimal large-c spinless modular bootstrap bound","short":"Order of the weak-coupling Hawking–Page/deconfinement transition in N=4 SYM; optimal large-c spinless modular bootstrap bound","year_posed":null,"years_open":null,"ai_system":["internal Claude research model","Claude Opus 5.5"],"status":"pending","verification":{"key":"none","label":"Unverified claim"},"human_role":"Autonomous AI result with a physicist prompting and checking","result":"b = −1.48×10⁻³ λ² (first-order transition); optimal slope Λ_V = 12π²/(π²+4 arcsinh²1) ≈ 9.1266","field":"physics"},{"entry":{"id":"2026-11-06-tusnady-problem-settled-two-ai-assisted-preprints","url":"https://postcutoff.com/e/2026-11-06-tusnady-problem-settled-two-ai-assisted-preprints/","date":"2026-11-06","date_precision":"day","short_title":"Tusnády's problem claimed settled twice, with AI help","deck":null,"takeaway":"Tusnády's problem is a standard open problem of geometric discrepancy theory, and the gap in dimensions d ≥ 3 had been open for decades.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-06","updated":"2026-11-06","orgs":["Anthropic","OpenAI"]},"problem":"Tusnády's problem: combinatorial discrepancy of points with respect to axis-parallel boxes in R^d","short":"Tusnády's problem","year_posed":null,"years_open":null,"ai_system":["Claude","Codex","ChatGPT"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: authors credit the tools with proof development and checking (Wengiel) and with generating ideas (Lei)","result":"Claimed Θ_d((log N)^(d−1/2)) in every fixed dimension d ≥ 2 (two independent proofs)","field":"mathematics"},{"entry":{"id":"2026-11-04-axiom-stacks-project-lean-formalization","url":"https://postcutoff.com/e/2026-11-04-axiom-stacks-project-lean-formalization/","date":"2026-11-04","date_precision":"day","short_title":"Axiom: the Stacks Project formalized in Lean","deck":null,"takeaway":"On 4 Nov 2026 Axiom Math released AxiomMath/Stacks, a Lean 4 formalization of 'every tag of the Stacks Project with formalizable content' (about 63,000 modules), with an agent-written StacksFrontier tier and a human-reviewed StacksCore tier. Axiom's X post had 6.9M views.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-11-07","updated":"2026-11-09","orgs":["Axiom Math"]},"problem":"Formalizing the Stacks Project (de Jong et al.) in Lean","short":"Formalizing the Stacks Project in Lean","year_posed":null,"years_open":null,"ai_system":["Axiom agents"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI agents wrote StacksFrontier; humans reviewed StacksCore","result":"Claimed Lean 4 formalization of every formalizable Stacks tag, in human-reviewed and agent-written tiers","field":"mathematics"},{"entry":{"id":"2026-11-04-elliptic-curve-torsion-degrees-6-9-gpt-6-1-sol-najman","url":"https://postcutoff.com/e/2026-11-04-elliptic-curve-torsion-degrees-6-9-gpt-6-1-sol-najman/","date":"2026-11-04","date_precision":"day","short_title":"Elliptic-curve torsion in degrees 6–9, mostly by GPT-6.1 Sol","deck":null,"takeaway":"Filip Najman deposited on Hexagon a paper that determines every group occurring as the torsion of an elliptic curve over a number field of degree 6, 7, 8 or 9, including three groups not previously known to occur.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-05","updated":"2026-11-06","orgs":["OpenAI","Anthropic","University of Zagreb"]},"problem":"Torsion groups of elliptic curves over number fields of fixed degree d = 6, 7, 8, 9","short":"Torsion groups of elliptic curves over number fields of fixed degree d","year_posed":null,"years_open":null,"ai_system":["GPT-6.1 Sol","Claude Opus 5.5","ChatGPT-6 Astra"],"status":"pending","verification":{"key":"none","label":"Unverified claim"},"human_role":"Human posed questions and guided; AI produced the mathematics, computations and manuscript; human checked degree 6 only","result":"Complete list of torsion groups for d = 6–9, with three groups new","field":"mathematics"},{"entry":{"id":"2026-10-29-deterministic-factoring-n-o1-lean-claude-rudich-rousseau","url":"https://postcutoff.com/e/2026-10-29-deterministic-factoring-n-o1-lean-claude-rudich-rousseau/","date":"2026-10-29","date_precision":"day","short_title":"Deterministic factoring in N^o(1) steps, Lean-checked, ideas by Claude","deck":null,"takeaway":"The best proven deterministic factoring algorithms ran in time N^(1/5+o(1)) (Harvey; Harvey–Hittmeir). On 29 Oct Isaac Rudich and Louis-Martin Rousseau (Polytechnique Montréal) posted on Hexagon a Lean 4 proof that one deterministic Turing machine finds the smallest prime factor of every N within C_k·2^(n/k) steps for every k, i.e. N^o(1). The 13-line statement uses only Mathlib and is registered in Palomar; the proof is about 36,000 lines. The authors say every idea and all Lean code came from Claude agents, which they asked to apply the ETH refutation to factoring; they audited the statement, not the proof. The algorithm is 'galactic' and slower than the randomised and heuristic methods used in practice; what is new is a proven deterministic N^o(1) bound.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-29","updated":"2026-10-31","orgs":["Anthropic","Polytechnique Montréal"]},"problem":"Best proven running time of deterministic integer factoring (record N^(1/5+o(1)), Harvey 2021; Harvey–Hittmeir)","short":"Best proven running time of deterministic integer factoring","year_posed":null,"years_open":null,"ai_system":["Claude Opus 5.5"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Authors posed the question and designed and audited the 13-line Lean statement; Claude agents produced the reduction, method, proof plan and all Lean code; the authors did not read the proof","result":"Lean-proved: deterministic factoring in time N^o(1) (C_k·2^(n/k) for every k)","field":"computer-science"},{"entry":{"id":"2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev","url":"https://postcutoff.com/e/2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev/","date":"2026-10-29","date_precision":"day","short_title":"Kervaire's conjecture: OpenAI proof confirmed in the Kourovka Notebook","deck":null,"takeaway":"Kervaire's conjecture says that adding one generator and one relation to a nontrivial group never gives the trivial group. It was known for torsion-free groups (Klyachko 1993) and residually finite groups (Gerstenhaber–Rothaus 1962). OpenAI's 23–24 Sept release (family 256) claimed it in general, and also the Kervaire–Laudenbach conjecture on independent systems of equations (Howie's form). The Kourovka Notebook's update file now marks problems 17.94 and 18.87(a) solved: 'AI solution confirmed by N. V. Bogachev'. The same file lists 48 AI solutions confirmed by named mathematicians and 35 still unverified, including OpenAI's claim of infinite finitely presented periodic groups (8.52).","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":6,"official":6,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Kourovka Notebook"]},"problem":"Kervaire's conjecture (1960s; Kourovka 17.94) and the Kervaire–Laudenbach conjecture (Kourovka 18.87(a))","short":"Kervaire's conjecture","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (unreleased, unnamed)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Proofs from OpenAI's internal model run; human confirmation by N. V. Bogachev for the Notebook","result":"OpenAI's proofs accepted as solutions in the Kourovka Notebook after confirmation by N. V. Bogachev","field":"mathematics"},{"entry":{"id":"2026-10-29-petty-equilateral-set-conjecture-dimension-five-tencent-hyprover","url":"https://postcutoff.com/e/2026-10-29-petty-equilateral-set-conjecture-dimension-five-tencent-hyprover/","date":"2026-10-29","date_precision":"day","short_title":"Petty's conjecture in dimension 5: proof by Tencent's hyprover agent","deck":null,"takeaway":"Petty conjectured that every n-dimensional normed space contains n + 1 points at equal distances. Petty proved dimension 3 and Makeev dimension 4. On 29 Oct Kaiwen Yang (Tongji University and Tencent Hunyuan) posted a proof for dimension 5. Its AI declaration says the proof 'was generated by Tencent Hunyuan's AI agent hyprover on September 29, 2026', and that the author organised, verified and rewrote it. The paper prints the full prompt, which points the agent to Lu–Yang's solution of Banach's isometric conjecture. Few research results in our dataset credit a Chinese lab's agent. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-30","updated":"2026-10-30","orgs":["Tencent","Tongji University"]},"problem":"Petty's equilateral-set conjecture (1971): every n-dimensional normed space contains n + 1 equidistant points","short":"Petty's equilateral-set conjecture","year_posed":null,"years_open":null,"ai_system":["Tencent Hunyuan hyprover"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Author wrote the prompt with a methodological hint; the agent generated the proof; the author organised, verified and rewrote it","result":"Claimed proof in dimension 5 (dimensions 3 and 4 were known)","field":"mathematics"},{"entry":{"id":"2026-10-29-pure-semisimplicity-conjecture-counterexample-gpt-6-astra-lean","url":"https://postcutoff.com/e/2026-10-29-pure-semisimplicity-conjecture-counterexample-gpt-6-astra-lean/","date":"2026-10-29","date_precision":"day","short_title":"Pure semisimplicity conjecture: claimed counterexample, GPT-6 Astra, Lean","deck":null,"takeaway":"The pure semisimplicity conjecture says a ring over which every module is a direct sum of finitely generated ones must have finite representation type. It holds for Artin algebras (Auslander) and PI rings (Simson, Herzog). Simson reduced a counterexample to a 'generalized Artin problem' about division ring extensions. On 29 Oct Junpeng Ren posted a division ring extension that solves it. The AI declaration says GPT-6 Astra helped choose the criterion and develop the proof, 'in particular the construction of the division rings'. The main theorem is fully formalised in Lean 4 in a public repository. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Anthropic"]},"problem":"Pure semisimplicity conjecture: every right pure semisimple ring has finite representation type","short":"Pure semisimplicity conjecture","year_posed":null,"years_open":null,"ai_system":["GPT-6 Astra","GPT-6.1 Sol","Claude Opus 5.5"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-assisted: GPT-6 Astra helped choose the criterion and construct the division rings; the author verified the proofs and completed the Lean formalisation with GPT-6.1 Sol and Claude Opus 5.5","result":"Claimed counterexample via a division ring extension with right dimension sequence (1, 2, 2, 2, …), through Simson's reduction","field":"mathematics"},{"entry":{"id":"2026-10-26-three-coloring-n-0-1537-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-26-three-coloring-n-0-1537-gpt-6-astra/","date":"2026-10-26","date_precision":"day","short_title":"3-colorable graphs: O(n^0.1537) colors with GPT-6 Astra","deck":null,"takeaway":"Ken-ichi Kawarabayashi, Kaisei Deguchi, Hirotaka Yoneda and Masataka Yoneda (NII / University of Tokyo) give a polynomial-time algorithm that colors any 3-colorable graph with O(n^0.1537) colors. The previous best was O(n^0.1954) (Bansal–Huang–Lee, SODA '27). By the paper's account, GPT-6 Astra and GPT-6 Pro generated each intermediate algorithm (n^0.1736, n^0.1712, n^(1/6+ε), n^0.16, then n^0.1537), with the authors suggesting ideas and simplifying the analyses. The humans wrote the text.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-27","updated":"2026-10-27","orgs":["National Institute of Informatics","University of Tokyo","OpenAI"]},"problem":"Coloring 3-colorable graphs with few colors in polynomial time (best exponent of n)","short":"Coloring 3-colorable graphs with few colors in polynomial time","year_posed":null,"years_open":null,"ai_system":["GPT-6 Astra","GPT-6 Pro"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human–AI collaboration: the models generated each successive algorithm; the authors studied the analyses, suggested ideas (incl. the generalization to vector k-coloring), simplified the arguments and wrote the paper","result":"O(n^0.1537) colors, improving O(n^0.1954); also an n^(1/6+o(1)) framework","field":"computer-science"},{"entry":{"id":"2026-10-26-three-dimensional-su2-yang-mills-continuum-limit-aronow-lopatto-ai","url":"https://postcutoff.com/e/2026-10-26-three-dimensional-su2-yang-mills-continuum-limit-aronow-lopatto-ai/","date":"2026-10-26","date_precision":"day","short_title":"3D SU(2) Yang–Mills continuum limit claimed in five AI-written papers","deck":null,"takeaway":"Building Yang–Mills theory as a rigorous continuum measure is open even in three dimensions, where the theory is superrenormalizable. On 26 Oct P. M. Aronow (Yale) and Patrick Lopatto (UNC Chapel Hill) posted five Hexagon papers. They claim that flowed Wilson loops of 3D SU(2) lattice Yang–Mills on a torus converge as the lattice spacing goes to zero, to a unique measure on the Cao–Chatterjee state space. The proof is a Bałaban-style multiscale renormalization-group analysis. The provenance statement says proof discovery, verification and writing were done by Opus 5.5, GPT 6 Astra and GPT 6.1 Sol, with only 'limited background formalizations' in Lean. Hexagon records human understanding as 'no parts'. It is not the four-dimensional Clay problem. We found no expert check as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","OpenAI","Yale University","University of North Carolina at Chapel Hill"]},"problem":"Rigorous continuum limit of three-dimensional Yang–Mills theory (here SU(2) on a torus, via Wilson-flowed observables)","short":"Rigorous continuum limit of three-dimensional Yang–Mills theory","year_posed":null,"years_open":null,"ai_system":["Claude Opus 5.5","GPT-6 Astra","GPT-6.1 Sol"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Contributors posted the work; by the Hexagon label no part is understood by a human; proof discovery, verification and writing by the models","result":"Claimed convergence of flowed Wilson-loop laws of 3D SU(2) lattice Yang–Mills on a torus to a unique continuum measure on the Cao–Chatterjee state space, with short-distance and weak-coupling asymptotics","field":"physics"},{"entry":{"id":"2026-10-22-critical-2d-kpz-process-chatgpt","url":"https://postcutoff.com/e/2026-10-22-critical-2d-kpz-process-chatgpt/","date":"2026-10-22","date_precision":"day","short_title":"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","deck":null,"takeaway":"A long-sought object of stochastic analysis, a non-trivial solution of the two-dimensional KPZ equation at criticality, was built by a team of leading probabilists who say ChatGPT contributed the tightness argument, the convergence proof and the proof of non-Gaussianity under their direction.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-23","updated":"2026-10-23","orgs":["OpenAI"]},"problem":"Constructing a solution of the two-dimensional KPZ equation at criticality","short":"Constructing a solution of the two-dimensional KPZ equation at criticality","year_posed":null,"years_open":null,"ai_system":["ChatGPT 5.5","ChatGPT 5.6","ChatGPT 6"],"status":"pending","verification":{"key":"expert","label":"Expert-checked"},"human_role":"Human-led with substantial AI input: authors supplied structure, literature and judgement; ChatGPT developed key estimates and proofs, some of which failed audit","result":"Construction of the Critical 2D KPZ Process as the unique limit of centred log partition functions of critical 2D directed polymers, a functional of the Critical 2D Stochastic Heat Flow; proved non-Gaussian.","field":"mathematics"},{"entry":{"id":"2026-10-19-integer-programming-linear-exponent-gpt-astra-ultra","url":"https://postcutoff.com/e/2026-10-19-integer-programming-linear-exponent-gpt-astra-ultra/","date":"2026-10-19","date_precision":"day","short_title":"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)","deck":null,"takeaway":"Ten days after their Subset Sum claim, the same Zhejiang trio claim to settle a basic question in integer programming. With d constraints and entries bounded by N, the running time exponent drops from O(d²) to linear in d. As before, the authors give the idea and say GPT Astra Ultra wrote the complete proofs.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["Zhejiang University","OpenAI"]},"problem":"Can integer programs in standard form with d constraints and entries ≤ N be solved in N^O(d) time instead of N^O(d²)?","short":"Integer programming with exponent linear in d","year_posed":null,"years_open":null,"ai_system":["GPT-6 Astra Ultra"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Human-led: the authors designed the proof route and core idea; GPT Astra Ultra wrote the complete proofs and a Lean check","result":"Claimed deterministic f(d)·N^O(d)·poly-log(c_max) + poly(input) algorithm for bounded standard-form integer programming.","field":"computer-science"},{"entry":{"id":"2026-10-19-perfect-matchings-fpras-general-graphs-openai-liu-meng","url":"https://postcutoff.com/e/2026-10-19-perfect-matchings-fpras-general-graphs-openai-liu-meng/","date":"2026-10-19","date_precision":"day","short_title":"Counting perfect matchings in general graphs: OpenAI FPRAS, simplified proof","deck":null,"takeaway":"Approximately counting perfect matchings is easy for bipartite graphs (Jerrum–Sinclair–Vigoda, 2004) but had no polynomial-time scheme for general graphs. OpenAI's 23 Sept release (family 113) claimed a fully polynomial randomized approximation scheme for all graphs, built on a restricted Poincaré inequality for pairs of matchings, and a Lean development. On 19 Oct Chenghua Liu and Boning Meng posted 'a simplified, self-contained proof of OpenAI's FPRAS'. It keeps the same parameter bounds and avoids the original auxiliary vertices and tree constructions. The same release also claimed the Anari–Oveis Gharan–Vinzant perfect-matching entropy conjecture.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":4,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI"]},"problem":"FPRAS for counting perfect matchings in general graphs (bipartite case: Jerrum–Sinclair–Vigoda 2004); Anari–Oveis Gharan–Vinzant entropy conjecture","short":"FPRAS for counting perfect matchings in general graphs","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (unreleased, unnamed)"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Proof from OpenAI's model run; independent human re-proof and simplification by Liu and Meng","result":"Claimed FPRAS for all finite simple graphs (OpenAI), re-proved in simplified form by Liu and Meng","field":"computer-science"},{"entry":{"id":"2026-10-17-non-sofic-hyperbolic-group-sandhu-thom","url":"https://postcutoff.com/e/2026-10-17-non-sofic-hyperbolic-group-sandhu-thom/","date":"2026-10-17","date_precision":"day","short_title":"A non-sofic word-hyperbolic group","deck":"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","takeaway":"Whether every hyperbolic group is sofic was a well-known open question in group theory. Two groups now say no. Both build on the torsion-free, non-residually-finite hyperbolic group in OpenAI's 6 October release. One is co-written by Andreas Thom, an expert in sofic groups. The other, by Douglas and Mghirbi, comes with a Lean formalization.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-20","updated":"2026-10-20","orgs":["TU Dresden","OpenAI"]},"problem":"Is every word-hyperbolic group sofic? (Hyperbolic groups were the leading candidates for sofic counterexamples after residual finiteness)","short":"A non-sofic hyperbolic group","year_posed":null,"years_open":null,"ai_system":["GPT Astra","OpenAI release model (family 252)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"AI-assisted: AI-generated construction (Sandhu, 7 Oct) built on OpenAI's AI result; Thom reworked and checked the proof; GPT Astra helped draft","result":"Claimed construction of a torsion-free two-dimensional word-hyperbolic group that is not sofic; independently, a claimed proof with Lean that OpenAI's non-residually-finite hyperbolic group is not sofic.","field":"mathematics"},{"entry":{"id":"2026-10-16-gauss-circle-problem-209-670-bound","url":"https://postcutoff.com/e/2026-10-16-gauss-circle-problem-209-670-bound/","date":"2026-10-16","date_precision":"day","short_title":"Gauss circle problem: exponent 209/670, below 5/16","deck":null,"takeaway":"The error term in counting lattice points in a circle gets its first exponent below 5/16, a threshold Huxley had suggested might be the limit of the Bombieri–Iwaniec method; the authors disclose ChatGPT and Codex help with strategy and checking.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-17","updated":"2026-10-17","orgs":["OpenAI"]},"problem":"Gauss circle problem: the exponent θ in E(X) = O(X^θ)","short":"Gauss circle problem","year_posed":null,"years_open":null,"ai_system":["ChatGPT","Codex"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led with AI tools: authors used ChatGPT and Codex to discuss proof strategies and to draft and check parts of the arguments; final proofs written and verified by the authors","result":"θ ≤ 209/670 + ε ≈ 0.31194, improving Li–Yang's 0.314483 and going below Huxley's suggested 5/16 barrier for the Bombieri–Iwaniec method.","field":"mathematics"},{"entry":{"id":"2026-10-16-smith-toda-v4-angeltveit-improves-openai","url":"https://postcutoff.com/e/2026-10-16-smith-toda-v4-angeltveit-improves-openai/","date":"2026-10-16","date_precision":"day","short_title":"Smith–Toda V(4) for p ≥ 11, after OpenAI's p = 1009","deck":null,"takeaway":"A homotopy theorist checked two Smith–Toda papers from OpenAI's maths release, found no errors but 'strange choices' (a suspiciously high prime, needless ultrafilters), and widened the result from one prime to all primes p ≥ 11.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Australian National University"]},"problem":"Existence of the Smith–Toda complex V(4); V(3) exists for p ≥ 7 (Toda)","short":"Existence of the Smith–Toda complex V(4); V(3) exists for p ≥ 7","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (openai/math release)","ChatGPT"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"OpenAI's model wrote the original papers; Angeltveit checked, simplified and extended them, using ChatGPT as an interactive assistant; he checked all arguments himself","result":"V(4) exists for every prime p ≥ 11. OpenAI's internal model had shown it only for p = 1009, plus a V(n) at some prime for each n","field":"mathematics"},{"entry":{"id":"2026-10-16-yang-mills-claude-balaban-program-alpoge-sussman","url":"https://postcutoff.com/e/2026-10-16-yang-mills-claude-balaban-program-alpoge-sussman/","date":"2026-10-16","date_precision":"day","short_title":"Alpöge and Sussman say Claude has an (unreadable) existence argument for 4D Yang–Mills on ℝ⁴, by finishing Bałaban's 1980s program","deck":null,"takeaway":"A public repo of raw Claude output claims a construction of four-dimensional SU(2) Yang–Mills theory, the existence half of the Millennium problem but not the mass gap; its authors call it 'slop in progress' that nobody can yet read or check.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Anthropic","Northwestern University"]},"problem":"Construction of four-dimensional quantum Yang–Mills theory (existence part of the Yang–Mills existence and mass gap Millennium Prize Problem), via Bałaban's renormalisation-group program (1985–1989)","short":"Yang–Mills existence (YM4)","year_posed":null,"years_open":null,"ai_system":["Claude","Codex","GPT-5.6 Sol"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Alpöge 'coaxing Claude' with 'some small amount of direction'; Sussman proposed the program and ran the earlier Codex QED3 experiment; both say they cannot yet tell how reliable the output is","result":"Claimed, not established: a very long Claude-generated argument for the existence of SU(2) Yang–Mills on ℝ⁴ (continuum limit with Osterwalder–Schrader properties and non-triviality), conditional on listed inputs; no mass gap, no uniqueness of the infinite-volume limit.","field":"physics"},{"entry":{"id":"2026-10-13-rational-exponents-conjecture-proof-paper-conlon","url":"https://postcutoff.com/e/2026-10-13-rational-exponents-conjecture-proof-paper-conlon/","date":"2026-10-13","date_precision":"day","short_title":"Eight extremal combinatorialists write up GPT-6 Astra's proof of the Erdős–Simonovits rational exponents conjecture (Conlon, Janzer, Liu et al.)","deck":null,"takeaway":"The rational exponents proof that GPT-6 Astra found during Epoch AI's FrontierMath Erdős runs now has a human-written paper by leading experts in the area, who explain and situate it and call it one of 'the recent AI-driven breakthroughs'.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-10-14","updated":"2026-10-14","orgs":["Caltech","EPFL","KAIST","IBS","Emory University","Epoch AI","OpenAI"]},"problem":"Erdős–Simonovits rational exponents conjecture (Erdős problem #571): every rational r in [1, 2] is the Turán exponent of some graph","short":"Erdős–Simonovits rational exponents conjecture","year_posed":null,"years_open":null,"ai_system":["GPT-6 Astra"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"The proof was found by GPT-6 Astra in Epoch AI's FrontierMath Erdős runs (Sept 2026); eight human experts wrote, explained and situated it in this paper","result":"Human-written exposition of the AI-found proof: for every rational r in [1, 2] there is a graph H with ex(n, H) = Θ(n^r).","field":"mathematics"},{"entry":{"id":"2026-10-09-tao-math-2-0-caltech-lecture","url":"https://postcutoff.com/e/2026-10-09-tao-math-2-0-caltech-lecture/","date":"2026-10-09","date_precision":"day","short_title":"Terence Tao's Caltech lecture 'Math 2.0'","deck":"AI has ended 'proof scarcity', and blind problem-solving is now 'actively harmful' to mathematics","takeaway":"Tao is the mathematician most often cited on AI and maths, and this is his fullest public statement after OpenAI's release.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":7,"official":4,"filed":"2026-10-10","updated":"2026-10-10","orgs":["UCLA","Caltech"]},"problem":"How mathematics should respond to AI-driven 'proof abundance' after OpenAI's 722-manuscript release","short":"How mathematics should respond to AI-driven 'proof abundance' after OpenAI's 722-manuscript release","year_posed":null,"years_open":null,"ai_system":["n/a"],"status":"confirmed","verification":{"key":"none","label":"Unverified claim"},"human_role":"Human-led","result":"Public lecture and slides arguing that problem-solving has become a misaligned proxy and proposing 'Math 2.0' uses of AI (exposition problems, living result networks, formal libraries, open interpretable maths models).","field":"mathematics"},{"entry":{"id":"2026-10-08-dualization-eth-lower-bound-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-08-dualization-eth-lower-bound-gpt-6-astra/","date":"2026-10-08","date_precision":"day","short_title":"GPT-6 Astra finds a 3SAT reduction showing monotone Dualization (minimal hypergraph transversals) has no polynomial algorithm under ETH","deck":null,"takeaway":"Whether monotone Boolean dualization, equivalently enumerating all minimal transversals of a hypergraph, can be done in output-polynomial time has been open since the 1990s. A short preprint says no, assuming the Exponential Time Hypothesis, and its authors say the proof 'was originally discovered by GPT-6 Astra'.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-09","updated":"2026-10-13","orgs":["Hokkaido University","Okayama University","Hosei University","OpenAI"]},"problem":"Is monotone Boolean dualization (hypergraph transversal enumeration) solvable in output-polynomial time?","short":"Dualization lower bound under ETH","year_posed":null,"years_open":null,"ai_system":["GPT-6 Astra"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI found the proof of the main theorem; the authors reconstructed, revised and verified it and rewrote the exposition","result":"No, under the Exponential Time Hypothesis: a 3SAT reduction rules out N^o(√(log N/log log N)) algorithms, leaving a gap only between √(log N) and log N in the exponent.","field":"computer-science"},{"entry":{"id":"2026-10-08-hyperkahler-syz-conjecture-engel-mauri","url":"https://postcutoff.com/e/2026-10-08-hyperkahler-syz-conjecture-engel-mauri/","date":"2026-10-08","date_precision":"day","short_title":"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","deck":null,"takeaway":"The hyperkähler SYZ conjecture (every nef line bundle on a compact hyperkähler manifold is semiample) has a human-written proof by Philip Engel and Mirko Mauri, which also gives finitely many deformation classes in each dimension when b2 ≥ 5. OpenAI's catalogue claimed the same theorem on 6 Oct; the authors say the proof structures 'appear similar'.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-09","updated":"2026-10-09","orgs":["University of Illinois Chicago","Université Paris Cité","OpenAI"]},"problem":"Hyperkähler SYZ conjecture / abundance for nef line bundles on compact hyperkähler manifolds","short":"Hyperkähler SYZ conjecture","year_posed":null,"years_open":null,"ai_system":["ChatGPT Astra (proofreading only)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led; the proof was written by the authors, AI used only to proofread after OpenAI's release","result":"Proved (preprint): every nef line bundle on a compact irreducible hyperkähler manifold is semiample; finitely many deformation classes in each dimension for b2 ≥ 5.","field":"mathematics"},{"entry":{"id":"2026-10-06-artin-davenport-cubic-forms-ten-variables","url":"https://postcutoff.com/e/2026-10-06-artin-davenport-cubic-forms-ten-variables/","date":"2026-10-06","date_precision":"day","short_title":"Artin–Davenport conjecture proved","deck":"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)","takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-07","updated":"2026-10-13","orgs":["TU Graz","ISTA","Imperial College London","Leibniz Universität Hannover","Academia Sinica","OpenAI"]},"problem":"Artin–Davenport conjecture: integral cubic forms in at least 10 variables have nontrivial integer zeros","short":"Artin–Davenport conjecture","year_posed":null,"years_open":null,"ai_system":["ChatGPT 6.0 (GPT-6)","ChatGPT 5.5/5.6/6.1","OpenAI Codex"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led over two years; ChatGPT 6.0 supplied the improved incidence-variety dimension bound (Lemma 3.8) and the Galois-descent approach of Lemma 3.6; GPT models helped combine estimates and proofread; Codex produced a partial Lean auto-formalisation","result":"Proved in full: n ≥ 10 suffices (previous record n ≥ 14, Heath-Brown 2007); 10 is sharp.","field":"mathematics"},{"entry":{"id":"2026-10-06-free-group-factors-isomorphism-openai","url":"https://postcutoff.com/e/2026-10-06-free-group-factors-isomorphism-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the free group factor problem","deck":"L(F₂) ≅ L(F₃), so all free group factors are isomorphic (Lean-formalized)","takeaway":"The problem has been one of the best-known open questions in operator algebras for decades.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":4,"filed":"2026-10-07","updated":"2026-10-19","orgs":["OpenAI"]},"problem":"Free group factor isomorphism problem (is L(F₂) ≅ L(F₃)?)","short":"Free group factor isomorphism problem","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (unreleased)"],"status":"confirmed","verification":{"key":"expert","label":"Expert-checked"},"human_role":"Autonomous by OpenAI's fixed procedure (per README)","result":"Claimed: yes; all interpolated free group factors L(F_r), 1 < r ≤ ∞, are isomorphic, with fundamental group ℝ₊.","field":"mathematics"},{"entry":{"id":"2026-10-06-generalized-lax-conjecture-openai-counterexample-kummer-netzer-generic-case","url":"https://postcutoff.com/e/2026-10-06-generalized-lax-conjecture-openai-counterexample-kummer-netzer-generic-case/","date":"2026-10-06","date_precision":"day","short_title":"Generalized Lax conjecture: OpenAI counterexample claim; AI-found generic case (Kummer–Netzer)","deck":null,"takeaway":"The generalized Lax conjecture says every hyperbolicity cone is spectrahedral, a slice of a positive semidefinite cone. On 21 Sept 2026 Mario Kummer and Tim Netzer posted 'a proof found by an artificial intelligence system' for the generic case, strictly hyperbolic polynomials (arXiv 2609.24542). Family 095 of OpenAI's 6 Oct maths release (manuscripts of 24 Sept and 5 Oct) claims the general conjecture is false. It gives hyperbolicity cones that are not even spectrahedral shadows, which also disproves the projected Lax conjecture, and the main statement is in Lean. Kevin Shu's exposition on the Quasicoherent blog accepts the counterexample. Neither result is refereed. We found both only on 31 Oct, through a citation.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI"]},"problem":"Generalized Lax conjecture (every hyperbolicity cone is spectrahedral) and the projected Lax conjecture (every hyperbolicity cone is a spectrahedral shadow)","short":"Generalized Lax conjecture","year_posed":null,"years_open":null,"ai_system":["unnamed AI system (Kummer–Netzer)","OpenAI internal model"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Kummer–Netzer: AI produced the proof, humans posed the question, suggested approaches, checked and edited. OpenAI: autonomous model output, Lean statement","result":"Claimed: true for strictly hyperbolic polynomials (Kummer–Netzer, AI-found); false in general, even for shadows (OpenAI family 095)","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-bloch-conjecture-surfaces-claim","url":"https://postcutoff.com/e/2026-10-06-openai-bloch-conjecture-surfaces-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Bloch's conjecture for surfaces","deck":null,"takeaway":"Family 040 of OpenAI's 6 Oct 2026 release claims Bloch's conjecture: on every smooth complex projective surface without holomorphic 2-forms (p_g = 0), zero-cycles modulo rational equivalence are completely described by their degree and Albanese image. Bloch, Kas and Lieberman proved this in 1976 for surfaces not of general type; for general type only special families were known. The 45-page proof uses virtual localization on Quot schemes. It is not formalized, and the only public expert comment so far, from Daniel Litt, flags a citation problem he thinks is harmless.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":2,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Bloch's conjecture (1970s): for a complex surface with p_g = 0, the Albanese map on degree-zero zero-cycles is an isomorphism","short":"Bloch's conjecture","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"expert","label":"Expert-checked"},"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","result":"Claimed in full: for p_g = q = 0, CH₀(S) ≅ ℤ via the degree; hence Bloch's conjecture for all surfaces with p_g = 0","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-cannon-conjecture-proof-claim","url":"https://postcutoff.com/e/2026-10-06-openai-cannon-conjecture-proof-claim/","date":"2026-10-06","date_precision":"day","short_title":"Cannon's conjecture claimed (OpenAI, Lean); Bonk's seminar talk","deck":null,"takeaway":"A 31-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims Cannon's conjecture: every word-hyperbolic group whose boundary is a 2-sphere acts properly and cocompactly by isometries on hyperbolic 3-space. For torsion-free groups this makes them fundamental groups of closed hyperbolic 3-manifolds, a group-theoretic counterpart of geometrization. The proof bounds combinatorial modulus and then applies a Bourdon–Kleiner criterion. The Lean statement was accepted in a community comparator run. On 22 Oct Mario Bonk, who with Bruce Kleiner proved a key special case, gave a seminar talk titled 'The AI Proof of Cannon's conjecture'.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Cannon's conjecture: does every word-hyperbolic group whose Gromov boundary is a 2-sphere act geometrically (properly, cocompactly, isometrically) on hyperbolic 3-space?","short":"Cannon's conjecture","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; presented in a seminar talk by Mario Bonk; no written human check yet","result":"Claimed proof for all such groups (finite kernel allowed), via a fixed-scale combinatorial 2-modulus bound and the Bourdon–Kleiner criterion; Lean formalization of the conclusion","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-catalan-constant-irrational","url":"https://postcutoff.com/e/2026-10-06-openai-catalan-constant-irrational/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI: Catalan's constant is irrational (Lean)","deck":null,"takeaway":"Family 005 of OpenAI's 6 Oct 2026 release is a 44-page manuscript, dated 24 Sep, proving that Catalan's constant G = 1 − 1/9 + 1/25 − 1/49 + … is irrational, one of the classical open questions about special values. Its Lean statement is one line on top of Mathlib and passed the community comparator runs. Frank Calegari sees 'a bunch of new ideas, or at least new optimizations', and outside authors have already turned the method into explicit irrationality measures.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Is Catalan's constant G = β(2) irrational?","short":"Is Catalan's constant G","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","result":"Proved irrational (OpenAI manuscript with a Lean proof of the exact statement); follow-ups give explicit irrationality measures","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-closed-geodesics-spheres-infinitely-many","url":"https://postcutoff.com/e/2026-10-06-openai-closed-geodesics-spheres-infinitely-many/","date":"2026-10-06","date_precision":"day","short_title":"Infinitely many closed geodesics on every sphere, claimed and re-proved","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release (family 345, dated 24 Sept) claims that every smooth Riemannian metric on the n-sphere has infinitely many geometrically distinct closed geodesics, for every n ≥ 2. Only the 2-sphere was known (Bangert, Franks, Hingston, early 1990s). It also covers every closed 3-manifold. On 15 Oct Zhiqi Chen, Shaoqiang Deng and Hui Zhang posted a proof that does not use OpenAI's lemmas, written with OpenAI Codex and checked by them, and extended it to the complex, quaternionic and Cayley projective spaces. They credit OpenAI's paper as the source of the strategy.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Nankai University","Guangdong University of Technology","Southeast University"]},"problem":"Does every Riemannian metric on Sⁿ (n ≥ 3) have infinitely many geometrically distinct closed geodesics?","short":"Does every Riemannian metric on Sⁿ have infinitely many geometrically distinct closed geodesics?","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (math release)","OpenAI Codex"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"OpenAI: autonomous output. Chen–Deng–Zhang: human-verified proof developed with Codex, crediting OpenAI's strategy","result":"Claimed for all n ≥ 2 and every metric, plus every closed 3-manifold (OpenAI); independently re-proved and extended to all compact rank-one symmetric spaces' underlying manifolds (Chen–Deng–Zhang)","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-finitely-presented-burnside-periodic-group-claim","url":"https://postcutoff.com/e/2026-10-06-openai-finitely-presented-burnside-periodic-group-claim/","date":"2026-10-06","date_precision":"day","short_title":"Finitely presented Burnside problem answered negatively? (OpenAI, Lean)","deck":null,"takeaway":"Two manuscripts credited to 'OpenAI' claim an infinite group with a finite presentation in which every element has finite order. Whether such a group exists, the finitely presented form of Burnside's 1902 question, had stayed open even after the infinite finitely generated examples of Golod–Shafarevich and Novikov–Adian. The claimed group is the Steinberg group St₁₂(R) over a characteristic-two algebra and also has Kazhdan's property (T). A companion dated 5 Oct makes it residually finite, with all element orders powers of 2. The existence statement is formalized in Lean and was accepted in a community comparator run; no group theorist has published an assessment.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":9,"filed":"2026-10-29","updated":"2026-10-31","orgs":["OpenAI"]},"problem":"Finitely presented Burnside problem: must a finitely presented group in which every element has finite order be finite?","short":"Finitely presented Burnside problem","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; no human exposition or check yet","result":"Claimed infinite finitely presented periodic group St₁₂(R) over an F₂-algebra, with property (T); claimed residually finite version with a finite-index 2-power-torsion subgroup; claimed finitely presented nil algebra that is not nilpotent","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc","url":"https://postcutoff.com/e/2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims the aspherical positive-scalar-curvature conjecture","deck":null,"takeaway":"Two manuscripts in OpenAI's 6 Oct 2026 math release (family 335) claim that a closed manifold with positive scalar curvature is rationally inessential, in every dimension and without spin assumptions. That would prove the Gromov–Lawson (Schoen–Yau) conjecture that closed aspherical manifolds, such as tori and hyperbolic manifolds, admit no metric of positive scalar curvature. It was known in dimensions up to five (Chodosh–Li, Gromov). A second paper proves Gromov's proposed integral bound of simplicial volume by negative scalar curvature. Roman Sauer, whose technique appears in the proofs, wrote 'I am stunned.' No Lean proof; no expert verification yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":2,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Gromov–Lawson / Schoen–Yau aspherical conjecture: a closed aspherical manifold admits no metric of positive scalar curvature; Gromov's integral scalar curvature bound for simplicial volume","short":"Gromov–Lawson / Schoen–Yau aspherical conjecture","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"expert","label":"Expert-checked"},"human_role":"Autonomous AI output released by OpenAI","result":"Claimed in every dimension, without spin hypotheses, via rational inessentiality of positive-scalar-curvature manifolds; plus the integral inequality with a dimensional constant","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-heisenberg-ferromagnet-bose-einstein-condensation","url":"https://postcutoff.com/e/2026-10-06-openai-heisenberg-ferromagnet-bose-einstein-condensation/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims quantum ferromagnet order and interacting Bose condensation","deck":null,"takeaway":"Two families in OpenAI's 6 Oct 2026 math release claim long-open theorems of quantum statistical mechanics. Family 271: the quantum Heisenberg ferromagnet in dimension 3 or more has a spontaneously magnetized equilibrium state at low temperature, plus Bloch's T^3/2 law with its exact coefficient. Family 267: a dilute gas of hard-sphere bosons Bose-condenses in its ground state and at a fixed positive temperature. The magnetization and ground-state condensation statements have Lean proofs that passed an independent Comparator run. OpenAI published a reasoning summary for 271, and Scientific American listed both among the release's most important claims. No expert verification yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Long-range order (spontaneous magnetization) of the quantum Heisenberg ferromagnet in d ≥ 3; Bose–Einstein condensation of an interacting continuum Bose gas at fixed density in the thermodynamic limit","short":"Long-range order of the quantum Heisenberg ferromagnet in d ≥ 3; Bose–Einstein condensation of an interacting continuum Bose gas at fixed density in the thermodynamic limit","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Autonomous AI output released by OpenAI, with Lean formalizations of the two core theorems","result":"Claimed: magnetized KMS states at low temperature for all d ≥ 3 and all spins, Bloch's T^{3/2} law with exact coefficient; condensation in every hard-sphere ground state and at a fixed positive temperature, with the Bogoliubov depletion law","field":"physics"},{"entry":{"id":"2026-10-06-openai-l-equals-rl-equals-bpl-lean-claim","url":"https://postcutoff.com/e/2026-10-06-openai-l-equals-rl-equals-bpl-lean-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims L = RL = BPL with a Lean proof","deck":null,"takeaway":"OpenAI's 6 Oct 2026 math release (family 103, manuscript dated 23 Sept) claims L = RL = BPL. That would mean randomness does not help logarithmic-space computation, a central open problem of complexity theory. The best unconditional bound was Saks and Zhou's O(log^{3/2} n) space, slightly improved by Hoza. A Lean formalization added on 7 Oct proves the equality in an explicit Turing-machine model. The community comparator run of 26 Oct accepted it. A re-check found the proof's definitions identical to the statement file's and the proof built on standard axioms only. It notes that it does not show 'that the machine model is the standard one'. No complexity theorist has assessed it publicly as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Is BPL = L? Can bounded-error randomized logarithmic-space computation be simulated deterministically in logarithmic space (also RL = L)?","short":"Is BPL","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"peer-review","label":"Peer-reviewed"},"human_role":"Autonomous AI output released by OpenAI; community volunteers (with Claude Code) re-checked the Lean statement","result":"Claimed proof of L = RL = BPL, with a Lean formalization in an explicit Turing-machine model whose statement definitions have been community-checked against the proof's","field":"computer-science"},{"entry":{"id":"2026-10-06-openai-modularity-elliptic-curves-imaginary-quadratic","url":"https://postcutoff.com/e/2026-10-06-openai-modularity-elliptic-curves-imaginary-quadratic/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI: elliptic curves over imaginary quadratic fields are modular","deck":null,"takeaway":"Family 030 of OpenAI's 6 Oct 2026 release is a 30-page manuscript proving that every elliptic curve over every imaginary quadratic field is modular. Caraiani and Newton had done this for a positive proportion of fields; the new step handles curves whose mod-3 and mod-5 representations are both reducible. Frank Calegari calls it 'a really nice trick rather than a fundamental breakthrough'. Within two days Bao Le Hung posted a note saying the same idea also proves modularity over all totally real fields, itself a long-open problem. Nothing is formalized or refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Modularity of elliptic curves over imaginary quadratic fields (and, by extension, over totally real and CM fields)","short":"Modularity of elliptic curves over imaginary quadratic fields","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (openai/math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI; follow-up notes by human experts written with AI drafts","result":"Claimed: every elliptic curve over every imaginary quadratic field is modular; Le Hung's note states the method also covers all totally real fields and CM fields without ζ5","field":"mathematics"},{"entry":{"id":"2026-10-06-openai-nearby-lagrangian-conjecture-counterexample","url":"https://postcutoff.com/e/2026-10-06-openai-nearby-lagrangian-conjecture-counterexample/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims a counterexample to the nearby Lagrangian conjecture","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release (family 340, dated 23 Sept) claims to disprove the nearby Lagrangian conjecture, a central conjecture of symplectic topology posed by Arnold, Lalonde and Sikorav. For some large even N, it builds a closed exact Lagrangian in the cotangent bundle of S⁹×S^{N−1} that is diffeomorphic to the base but cannot be moved onto the zero section by a Hamiltonian isotopy. Earlier work had shown such Lagrangians are always simple homotopy equivalent to the base (Abouzaid–Kragh). On 15 Oct Wenyuan Li gave a seminar talk on how the construction works, 'assuming the counterexample is correct'. No Lean proof; no expert verdict.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":1,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},"problem":"Nearby Lagrangian conjecture (Arnold): every closed exact Lagrangian in T*Q is Hamiltonian isotopic to the zero section","short":"Nearby Lagrangian conjecture","year_posed":null,"years_open":null,"ai_system":["OpenAI internal model (math release)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Autonomous AI output released by OpenAI","result":"Claimed counterexample in T*(S⁹ × S^{N−1}) for some large even N: an embedded exact Lagrangian diffeomorphic to the base, not Hamiltonian isotopic to the zero section","field":"mathematics"},{"entry":{"id":"2026-10-02-epsilon-dvoretzky-conjecture-klartag-moshe","url":"https://postcutoff.com/e/2026-10-02-epsilon-dvoretzky-conjecture-klartag-moshe/","date":"2026-10-02","date_precision":"day","short_title":"Klartag and Moshe prove the ε-Dvoretzky conjecture","deck":"The key probabilistic idea came from a ChatGPT discussion","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-05","updated":"2026-10-05","orgs":["Tel Aviv University","Weizmann Institute of Science","OpenAI"]},"problem":"ε-Dvoretzky conjecture: polynomial dependence on 1/ε in Dvoretzky's theorem (V. Milman; Klartag–Novikov)","short":"ε-Dvoretzky conjecture","year_posed":null,"years_open":null,"ai_system":["ChatGPT"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"Human-led. The key idea (the probability distribution on the Grassmannian) came out of a discussion between ChatGPT and Shahar Moshe; ChatGPT also checked and improved proofs.","result":"Polynomial bound n ≥ (C/ε)^((ℓ+1)/2)|log ε| for nearly ellipsoidal ℓ-dimensional sections of arbitrary (not necessarily symmetric) convex bodies, and a Euclidean-ball version.","field":"mathematics"},{"entry":{"id":"2026-09-03-dying-percolation-theta-pc-zero","url":"https://postcutoff.com/e/2026-09-03-dying-percolation-theta-pc-zero/","date":"2026-09-03","date_precision":"day","short_title":"Claude-written Lean proof claims the dying percolation conjecture θ(p_c)=0 in every dimension","deck":null,"takeaway":"If the formal statement matches the intended theorem, a famous problem in mathematical physics is settled by machine-written formal mathematics.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":6,"filed":"2026-09-29","updated":"2026-10-22","orgs":["Anthropic","OpenAI"]},"problem":"Dying percolation conjecture θ(p_c)=0 for Bernoulli bond percolation on Z^d","short":"Dying percolation conjecture","year_posed":null,"years_open":null,"ai_system":["Claude (Anthropic)","Claude Fable 5.1","GPT-5.6 Sol"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"Autonomous formalization: Claude wrote all the Lean code under Justin Leder's direction. The follow-up proofs of stronger conjectures were produced by GPT-5.6 Sol + Claude Fable 5.1 with 'minimal human intervention' from Ahmed Bou-Rabee","result":"Claimed Lean-verified proof that θ(p_c)=0 for all d ≥ 2, via a new additive gluing inequality that settles Kozma–Nitzan Conjecture 3.","field":"mathematics"},{"entry":{"id":"2026-08-28-matrix-spencer-conjecture-proved","url":"https://postcutoff.com/e/2026-08-28-matrix-spencer-conjecture-proved/","date":"2026-08-28","date_precision":"day","short_title":"Matrix Spencer conjecture proved","deck":"Authors credit GPT-5.6 Sol Pro with 'the heavy-lifting' for the key lemma","takeaway":"Matrix Spencer was one of the headline open problems in discrepancy theory.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Emrullah Akbas","Suvrit Sra","OpenAI"]},"problem":"Matrix Spencer conjecture","short":"Matrix Spencer conjecture","year_posed":null,"years_open":null,"ai_system":["GPT-5.6 Sol Pro"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-assisted: humans chose the strategy and verified and polished; the model produced the key lemma's proof and most calculations","result":"O(√n) discrepancy for n symmetric n×n matrices of norm at most 1, achievable efficiently.","field":"mathematics"},{"entry":{"id":"2026-08-26-erdos-rankin-large-prime-gaps-improved","url":"https://postcutoff.com/e/2026-08-26-erdos-rankin-large-prime-gaps-improved/","date":"2026-08-26","date_precision":"day","short_title":"GPT-5.6 improves the Erdős–Rankin / Ford–Green–Konyagin–Maynard–Tao bound for large prime gaps","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":2,"filed":"2026-09-29","updated":"2026-10-15","orgs":["OpenAI"]},"problem":"Large gaps between consecutive primes (Erdős–Rankin; Erdős problem #4)","short":"Large gaps between consecutive primes","year_posed":null,"years_open":null,"ai_system":["GPT-5.6 Pro/Sol","GPT-6 Astra"],"status":"pending","verification":{"key":"lean-unreviewed","label":"Lean proof, not yet reviewed"},"human_role":"AI-assisted: pseudonymous user DottedCalculator prompted the model; Thomas Bloom wrote the exposition","result":"Improved lower bound for infinitely many large prime gaps, saving a log log log n factor over FGKMT 2018.","field":"mathematics"},{"entry":{"id":"2026-08-19-yau-tian-donaldson-cscK-disproof-claim","url":"https://postcutoff.com/e/2026-08-19-yau-tian-donaldson-cscK-disproof-claim/","date":"2026-08-19","date_precision":"day","short_title":"Peking University preprint claims an AI-found disproof of the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics","deck":null,"takeaway":"The Yau–Tian–Donaldson programme is central to modern complex geometry, and the paper is explicit that its main result is AI-generated.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-10-07","orgs":["Peking University","Jihao Liu","Anthropic","OpenAI"]},"problem":"Yau–Tian–Donaldson conjecture for constant scalar curvature Kähler metrics","short":"Yau–Tian–Donaldson conjecture (cscK metrics)","year_posed":null,"years_open":null,"ai_system":["Danus (Rethlas-based agent)","Claude Fable 5 (Claude Code)","GPT-5.6 Sol (Codex)"],"status":"pending","verification":{"key":"preprint","label":"Preprint only"},"human_role":"AI-generated main result with one key human redirection; the author checked and polished the text","result":"Claimed explicit K-polystable polarized smooth projective fivefold without a cscK metric.","field":"mathematics"},{"entry":{"id":"2026-08-10-claude-riemann-zeta-zeros-two-thirds","url":"https://postcutoff.com/e/2026-08-10-claude-riemann-zeta-zeros-two-thirds/","date":"2026-08-10","date_precision":"day","short_title":"Claude proves more than two-thirds of Riemann zeta zeros are simple and on the critical line (up from 41.6%)","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":10,"official":9,"filed":"2026-09-29","updated":"2026-10-23","orgs":["Anthropic"]},"problem":"Proportion of nontrivial zeros of ζ(s) on the critical line (toward the Riemann hypothesis)","short":"Zeros of ζ(s) on the critical line","year_posed":null,"years_open":null,"ai_system":["Claude (unreleased research model) in Claude Code"],"status":"confirmed","verification":{"key":"lean","label":"Lean-verified"},"human_role":"Near-autonomous: non-mathematician prompted; experts verified","result":"Unconditional proof that more than 67.2% of zeta zeros are simple and lie on the critical line.","field":"mathematics"}],"all":[{"id":"2026-11-13-ai-assisted-results-november-13-listing","url":"https://postcutoff.com/e/2026-11-13-ai-assisted-results-november-13-listing/","date":"2026-11-13","date_precision":"day","short_title":"AI-assisted results, 13 Nov","deck":null,"takeaway":"The arXiv listings of 12–13 Nov carried about 190 maths papers with an AI statement. Most disclose editing or literature search; about forty credit models with proofs, constructions or counterexamples. Seven got their own entries; the rest are catalogued here with the disclosure quoted. All are unrefereed preprints.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":27,"official":27,"filed":"2026-11-13","updated":"2026-11-13","orgs":["OpenAI","Anthropic","Google DeepMind","Axiom Math","various researchers"]},{"id":"2026-11-12-binary-search-tree-unified-bound-ai","url":"https://postcutoff.com/e/2026-11-12-binary-search-tree-unified-bound-ai/","date":"2026-11-12","date_precision":"day","short_title":"A BST meets Iacono's 2001 unified bound, with Claude and GPT","deck":null,"takeaway":"Since 2001 it was open whether any binary search tree, even offline, achieves Iacono's unified bound, which combines the working-set and dynamic-finger bounds. A new preprint gives an online deterministic one, so dynamic optimality would imply Iacono's conjecture for splay trees. Five models from Anthropic and OpenAI helped design and analyse it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-13","updated":"2026-11-13","orgs":["Anthropic","OpenAI","Pennsylvania State University","Jilin University","Tsinghua University"]},{"id":"2026-11-12-fixed-profile-soliton-resolution-counterexamples-gpt-6-astra","url":"https://postcutoff.com/e/2026-11-12-fixed-profile-soliton-resolution-counterexamples-gpt-6-astra/","date":"2026-11-12","date_precision":"day","short_title":"Fixed-profile soliton resolution fails (3D, 5D wave equation), with GPT-6 Astra","deck":null,"takeaway":"Soliton resolution predicts that solutions of the focusing energy-critical wave equation split into finitely many fixed solitons plus radiation. Two preprints build bounded global solutions whose core keeps changing shape, contradicting the fixed-profile formulation of Duyckaerts, Kenig and Merle. The authors had the mechanism; GPT-6 Astra did the estimates.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-13","updated":"2026-11-13","orgs":["OpenAI","University of Rochester","Brown University","various researchers"]},{"id":"2026-11-12-grid-minor-sixth-power-hajiaghayi-astra","url":"https://postcutoff.com/e/2026-11-12-grid-minor-sixth-power-hajiaghayi-astra/","date":"2026-11-12","date_precision":"day","short_title":"Grid-minor exponent 6 in polynomial time (Hajiaghayi with Astra)","deck":null,"takeaway":"Robertson and Seymour's grid-minor theorem is central to graph minors and bidimensionality; Chuzhoy and Tan's g⁹ bound (2021) was the best. Two AI-assisted claims now go below it: Édouard Bonnet's Lean-checked g⁸·polylog with Codex, and Hajiaghayi's g⁶·polylog with a polynomial-time algorithm, developed with OpenAI Astra.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-11-13","updated":"2026-11-13","orgs":["OpenAI","University of Maryland","various researchers"]},{"id":"2026-11-12-test-free-pdl-no-craig-interpolation-claude-fable","url":"https://postcutoff.com/e/2026-11-12-test-free-pdl-no-craig-interpolation-claude-fable/","date":"2026-11-12","date_precision":"day","short_title":"Test-free PDL lacks interpolation; counterexample by Claude Fable in Lean","deck":null,"takeaway":"After Borzechowski et al. proved Craig interpolation for PDL (2026), the test-free fragment remained open. Malvin Gattinger asked Claude Fable to turn failed proof attempts into a counterexample search; it returned one with a Lean formalization in minutes. The paper, written by Gattinger, publishes all prompts.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-11-13","updated":"2026-11-13","orgs":["Anthropic","Harmonic","University of Amsterdam"]},{"id":"2026-11-12-ai-assisted-results-november-12-listing","url":"https://postcutoff.com/e/2026-11-12-ai-assisted-results-november-12-listing/","date":"2026-11-12","date_precision":"day","short_title":"AI-assisted results, 12 Nov: Hexagon","deck":null,"takeaway":"The largest Hexagon day yet: 94 uncited deposits on 12 Nov, a quarter of them Tejasvi Singh Tomar's Lean-checked bounds on classical constants. Several claim named conjectures (Petty, Marcus–Minc, Kim–Roush, Humphreys, quantum Kazhdan–Lusztig) with little or no human checking; read each as a claim.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":102,"official":102,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","Anthropic","various researchers"]},{"id":"2026-11-11-band-matrix-que-critical-crossover-axiomprover-lean","url":"https://postcutoff.com/e/2026-11-11-band-matrix-que-critical-crossover-axiomprover-lean/","date":"2026-11-11","date_precision":"day","short_title":"Band-matrix QUE and critical crossover, 1.2M lines of Lean by AxiomProver","deck":null,"takeaway":"Random band matrices are the standard model for the Anderson localization transition. A Chicago–Axiom Math preprint proves delocalization (QUE) right down to W ≫ √N and the predicted critical law at W² ≍ N, and says AxiomProver formalized all of it in Lean, including the published results it relies on.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-13","updated":"2026-11-13","orgs":["Axiom Math","University of Chicago"]},{"id":"2026-11-11-benson-etingof-ostrik-conjecture-gpt-6-astra","url":"https://postcutoff.com/e/2026-11-11-benson-etingof-ostrik-conjecture-gpt-6-astra/","date":"2026-11-11","date_precision":"day","short_title":"Benson–Etingof–Ostrik conjecture: full proof generated by GPT-6 Astra","deck":null,"takeaway":"A month after the finite case, the full 2023 conjecture on symmetric tensor categories in positive characteristic is claimed in a paper 'generated by GPT-6 Astra', posted by a team that includes Etingof and Ostrik, who posed it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","MIT","University of Sydney","University of Bonn","University of Oregon"]},{"id":"2026-11-11-hyperelliptic-schottky-problem-huang-jiang-ai","url":"https://postcutoff.com/e/2026-11-11-hyperelliptic-schottky-problem-huang-jiang-ai/","date":"2026-11-11","date_precision":"day","short_title":"Hyperelliptic Schottky problem solved; AI suggested the final step","deck":null,"takeaway":"Which abelian varieties are Jacobians of hyperelliptic curves? Huang and Jiang say: exactly those whose theta divisor has singular locus of codimension three. Their acknowledgement says AI tools suggested completing the proof of the main theorem with a known result and constructed the paper's sharpness example.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-13","updated":"2026-11-13","orgs":["Fudan University","various researchers"]},{"id":"2026-11-11-smale-mean-value-constant-equals-4-goel-lean","url":"https://postcutoff.com/e/2026-11-11-smale-mean-value-constant-equals-4-goel-lean/","date":"2026-11-11","date_precision":"day","short_title":"Smale's mean value constant = 4 (claimed, Lean-checked)","deck":null,"takeaway":"Two months after GPT-6 Astra refuted the constant-1 form, a Hexagon deposit says the best constant in Smale's mean value theorem is exactly 4, so Smale's 1981 bound cannot be improved at all; a Lean proof accompanies it, experts have not yet commented.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":5,"filed":"2026-11-12","updated":"2026-11-12","orgs":["Anthropic","OpenAI"]},{"id":"2026-11-11-talagrand-ultimate-matching-conjecture-gpt-6","url":"https://postcutoff.com/e/2026-11-11-talagrand-ultimate-matching-conjecture-gpt-6/","date":"2026-11-11","date_precision":"day","short_title":"Talagrand's Ultimate Matching Conjecture: GPT-6 proof (Hexagon)","deck":null,"takeaway":"Three specialists in random matching say GPT-6 found the critical-exponent proof of a research problem from Talagrand's book and then the endpoint answer; the text is mostly model-written and the authors promise a digested version.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","CNRS","University of Pisa"]},{"id":"2026-11-11-alphagenome-2","url":"https://postcutoff.com/e/2026-11-11-alphagenome-2/","date":"2026-11-11","date_precision":"day","short_title":"Google DeepMind launches AlphaGenome 2, predicting how DNA variants act in specific cell types","deck":null,"takeaway":"Most disease-linked variants sit in non-coding DNA and act only in some cells; a model that predicts cell-type-specific effects can help narrow down causes of rare and common diseases.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":2,"filed":"2026-11-12","updated":"2026-11-13","orgs":["Google DeepMind"]},{"id":"2026-11-11-triangles-positive-density-plane-gpt-6-astra-pro","url":"https://postcutoff.com/e/2026-11-11-triangles-positive-density-plane-gpt-6-astra-pro/","date":"2026-11-11","date_precision":"day","short_title":"Large triangles in planar sets of positive density: GPT-6 Astra Pro proof","deck":null,"takeaway":"The planar case of Bourgain's simplex theorem, previously known only for density above 1/2, is claimed in a 9-page AI-written note steered by a harmonic analyst who works on trilinear estimates; Claude Opus 5.5 wrote a Lean formalization.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","Anthropic","Chapman University"]},{"id":"2026-11-11-ai-assisted-results-november-11-listing","url":"https://postcutoff.com/e/2026-11-11-ai-assisted-results-november-11-listing/","date":"2026-11-11","date_precision":"day","short_title":"AI-assisted results, 11 Nov: arXiv and Hexagon","deck":null,"takeaway":"Own entries today: Smale's constant = 4 (Goel, Lean), Talagrand's Ultimate Matching Conjecture (GPT-6), the Benson–Etingof–Ostrik conjecture (GPT-6 Astra), large triangles in planar dense sets (GPT-6 Astra Pro) and explicit RIP matrices (GPT-6 Astra). Below: the rest, quoted from Hexagon provenance and arXiv PDFs; nothing here is refereed.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":113,"official":113,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","Anthropic","various researchers"]},{"id":"2026-11-10-explicit-rip-matrices-two-thirds-gpt-6-astra","url":"https://postcutoff.com/e/2026-11-10-explicit-rip-matrices-two-thirds-gpt-6-astra/","date":"2026-11-10","date_precision":"day","short_title":"Explicit RIP matrices reach k = m^(2/3): main theorem by GPT-6 Astra","deck":null,"takeaway":"Since 2011 explicit compressed-sensing matrices had only beaten the square-root bottleneck by a tiny ε. A GPT-6 Astra proof, digested and written up by two specialists and formalized in Lean, reaches exponent 2/3.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":1,"filed":"2026-11-12","updated":"2026-11-12","orgs":["OpenAI","University of Central Florida","Ohio State University"]},{"id":"2026-11-10-ai-agents-maya-site-claim-rediscovered","url":"https://postcutoff.com/e/2026-11-10-ai-agents-maya-site-claim-rediscovered/","date":"2026-11-10","date_precision":"day","short_title":"Viral claim that AI agents found an unmapped Maya site turns out to be a rediscovery of a site published in 2025","deck":null,"takeaway":"A typical viral \"AI discovers X\" story: a real capability (fast triage of lidar data) presented as a discovery before a literature check.","category":"science","category_label":"Science & math","importance":2,"confidence":"high","status":{"key":"disputed","labels":["Disputed"]},"sources":3,"official":0,"filed":"2026-11-12","updated":"2026-11-12","orgs":[]},{"id":"2026-11-10-ai-assisted-results-november-10-listing","url":"https://postcutoff.com/e/2026-11-10-ai-assisted-results-november-10-listing/","date":"2026-11-10","date_precision":"day","short_title":"AI-assisted results, 10 Nov: arXiv and Hexagon","deck":null,"takeaway":"Own entries today: X_ns+(121) (two AI proofs) and the Benjamini–Schramm p_c < p_u claims. Below: a catalogue of the rest, quoted from Hexagon provenance and arXiv PDFs; nothing here is refereed.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":82,"official":82,"filed":"2026-11-10","updated":"2026-11-10","orgs":["OpenAI","Anthropic","various researchers"]},{"id":"2026-11-09-non-split-cartan-121-two-ai-proofs","url":"https://postcutoff.com/e/2026-11-09-non-split-cartan-121-two-ai-proofs/","date":"2026-11-09","date_precision":"day","short_title":"X_ns+(121): two AI-generated proofs in two days","deck":null,"takeaway":"Arango-Piñeros, Najman and Zureick-Brown (Claude Opus 5.5, GPT-6.1 Sol, GPT-6 Astra) and, separately, Manvir Jaswal (an agent system on Claude Opus 5.5) posted proofs on Hexagon that X_ns+(121) has exactly 7 rational points, all CM. Neither is fully human-checked.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-11-10","updated":"2026-11-10","orgs":["Anthropic","OpenAI","University of Zagreb","UMass Amherst","Amherst College"]},{"id":"2026-11-09-nill-reflexive-polytopes-conjecture-disproved-gpt-6-1-sol","url":"https://postcutoff.com/e/2026-11-09-nill-reflexive-polytopes-conjecture-disproved-gpt-6-1-sol/","date":"2026-11-09","date_precision":"day","short_title":"Reflexive polytopes conjecture disproved, GPT-6.1 Sol","deck":null,"takeaway":"The conjecture's own author confirms the disproof, and the low-dimensional counterexamples are explicit, which makes the claim easy to check.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-09","updated":"2026-11-09","orgs":["OpenAI","various researchers"]},{"id":"2026-11-09-ai-assisted-results-november-9-listing","url":"https://postcutoff.com/e/2026-11-09-ai-assisted-results-november-9-listing/","date":"2026-11-09","date_precision":"day","short_title":"AI-assisted results, 9 Nov: arXiv and Hexagon","deck":null,"takeaway":"The 9 Nov sweep listed 175 new arXiv papers mentioning AI. Own entries: the matroid secretary conjecture (strong form, GPT Astra) and Nill's reflexive polytope conjecture disproved with GPT-6.1 Sol.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":44,"official":44,"filed":"2026-11-09","updated":"2026-11-09","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]},{"id":"2026-11-08-ai-assisted-results-november-8-listing","url":"https://postcutoff.com/e/2026-11-08-ai-assisted-results-november-8-listing/","date":"2026-11-08","date_precision":"day","short_title":"AI-assisted results, 8 Nov: Hexagon of 7 Nov and arXiv","deck":null,"takeaway":"The 8 Nov sweep listed 22 uncited Hexagon works of 7 Nov and 352 arXiv papers mentioning AI. Own entries: 2×2 matrix mortality undecidable, the quasiplanar conjecture proved by ChatGPT-6, D-equivalence for generalized Kummer type, and Anthropic's two Claude physics papers.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":61,"official":61,"filed":"2026-11-08","updated":"2026-11-08","orgs":["OpenAI","Anthropic","Google","various researchers"]},{"id":"2026-11-07-anthropic-claude-hawking-page-modular-bootstrap-hexagon","url":"https://postcutoff.com/e/2026-11-07-anthropic-claude-hawking-page-modular-bootstrap-hexagon/","date":"2026-11-07","date_precision":"day","short_title":"Anthropic's Claude physics papers: Hawking–Page and modular bootstrap","deck":null,"takeaway":"Anthropic's own Hexagon account posted two 'primarily AI-generated' hep-th papers on 7 Nov: an internal Claude research model computed a three-loop coefficient unknown for two decades (b = −1.48×10⁻³ λ²), and found the optimal slope of the spinless modular bootstrap, later reproduced by public Claude Opus 5.5. A physicist checked the results.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-11-08","updated":"2026-11-09","orgs":["Anthropic"]},{"id":"2026-11-07-ai-assisted-results-november-7-listing","url":"https://postcutoff.com/e/2026-11-07-ai-assisted-results-november-7-listing/","date":"2026-11-07","date_precision":"day","short_title":"AI-assisted results, 7 Nov: Hexagon of 6 Nov and arXiv","deck":null,"takeaway":"The 7 Nov sweep listed 35 uncited Hexagon works of 6 Nov and 200 arXiv papers of the latest announcement mentioning AI. Own entries: Axiom's Lean proofs of the Weil conjectures and the Stacks Project, and the unforced Navier–Stokes non-uniqueness deposit.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":65,"official":64,"filed":"2026-11-07","updated":"2026-11-07","orgs":["OpenAI","Anthropic","Google","various researchers"]},{"id":"2026-11-06-psi-aaronson-ambainis-conjecture-proof","url":"https://postcutoff.com/e/2026-11-06-psi-aaronson-ambainis-conjecture-proof/","date":"2026-11-06","date_precision":"day","short_title":"PSI claims Lean-checked proof of the Aaronson–Ambainis conjecture","deck":null,"takeaway":"The conjecture is the main missing step for showing that quantum computers can't get more than a polynomial speedup on unstructured problems for most inputs.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":4,"filed":"2026-11-07","updated":"2026-11-12","orgs":["Physical Superintelligence","Anthropic"]},{"id":"2026-11-06-axiom-weil-conjectures-lean-formalization","url":"https://postcutoff.com/e/2026-11-06-axiom-weil-conjectures-lean-formalization/","date":"2026-11-06","date_precision":"day","short_title":"Axiom: the Weil conjectures proved in Lean","deck":null,"takeaway":"On 6 Nov 2026 Axiom Math published AxiomMath/Weil, a kernel-checked Lean 4 proof of rationality, the functional equation, the Riemann hypothesis (Deligne) and the Betti-number statement, following Freitag–Kiehl. About 980 modules were written by AI agents; the code has not been reviewed by humans.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-11-07","updated":"2026-11-08","orgs":["Axiom Math"]},{"id":"2026-11-06-navier-stokes-unforced-leray-hopf-nonuniqueness-buckmaster-coiculescu-li","url":"https://postcutoff.com/e/2026-11-06-navier-stokes-unforced-leray-hopf-nonuniqueness-buckmaster-coiculescu-li/","date":"2026-11-06","date_precision":"day","short_title":"Navier–Stokes: unforced Leray–Hopf non-uniqueness, AI-written","deck":null,"takeaway":"On 6 Nov 2026 three NYU mathematicians deposited on Hexagon an 80-page proof, written mainly by Claude Opus and Codex, that the unforced 3D Navier–Stokes equations have a continuum of Leray–Hopf solutions with the same initial data, following the Jia–Šverák programme. The authors say they understand the broad steps; a Lean formalization is announced but not yet public.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-11-07","updated":"2026-11-08","orgs":["New York University","Anthropic","OpenAI"]},{"id":"2026-11-06-tusnady-problem-settled-two-ai-assisted-preprints","url":"https://postcutoff.com/e/2026-11-06-tusnady-problem-settled-two-ai-assisted-preprints/","date":"2026-11-06","date_precision":"day","short_title":"Tusnády's problem claimed settled twice, with AI help","deck":null,"takeaway":"Tusnády's problem is a standard open problem of geometric discrepancy theory, and the gap in dimensions d ≥ 3 had been open for decades.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-06","updated":"2026-11-06","orgs":["Anthropic","OpenAI"]},{"id":"2026-11-06-ai-assisted-results-november-6-listing","url":"https://postcutoff.com/e/2026-11-06-ai-assisted-results-november-6-listing/","date":"2026-11-06","date_precision":"day","short_title":"AI-assisted results, 6 Nov: arXiv of 6 Nov and Hexagon of 5 Nov","deck":null,"takeaway":"The 6 Nov sweep listed 183 new arXiv papers mentioning AI and 26 uncited Hexagon works. Own entries: Tusnády's problem (two preprints), Singmaster's bound by Claude Opus 5.5 and the Hopfield capacity bracket.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":48,"official":48,"filed":"2026-11-06","updated":"2026-11-06","orgs":["OpenAI","Anthropic","various researchers"]},{"id":"2026-11-05-z-diophantine-in-q-glazer-burungale","url":"https://postcutoff.com/e/2026-11-05-z-diophantine-in-q-glazer-burungale/","date":"2026-11-05","date_precision":"day","short_title":"ℤ is Diophantine in ℚ: Glazer (Astra, Claude) and Burungale claim Mazur's conjecture fails","deck":null,"takeaway":"OpenAI's October preprint claimed Hilbert's tenth problem over ℚ is undecidable without defining ℤ in ℚ. A month later two independent manuscripts claim the stronger statement: one polynomial equation picks out exactly the integers among the rationals. Glazer's was produced with Astra 6, Sol 6.1 and Opus 5.5; Burungale says ChatGPT assisted two sections. Neither is refereed.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":4,"filed":"2026-11-13","updated":"2026-11-13","orgs":["OpenAI","Anthropic","Principia Labs","University of Texas at Austin"]},{"id":"2026-11-05-3-uniform-hyperclique-hypothesis-false-chatgpt-sol","url":"https://postcutoff.com/e/2026-11-05-3-uniform-hyperclique-hypothesis-false-chatgpt-sol/","date":"2026-11-05","date_precision":"day","short_title":"3-uniform hyperclique hypothesis refuted with ChatGPT Sol 6.1","deck":null,"takeaway":"The h-uniform hyperclique hypothesis was a standard assumption for conditional lower bounds.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-05","updated":"2026-11-05","orgs":["OpenAI","University of Illinois Urbana-Champaign"]},{"id":"2026-11-05-hopfield-critical-capacity-claude-opus-5-5","url":"https://postcutoff.com/e/2026-11-05-hopfield-critical-capacity-claude-opus-5-5/","date":"2026-11-05","date_precision":"day","short_title":"Hopfield critical capacity bracketed, by Claude Opus 5.5","deck":null,"takeaway":"Turning the replica prediction for the Hopfield model into a theorem has been a long-standing goal in the mathematics of spin glasses.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-06","updated":"2026-11-06","orgs":["Anthropic","OpenAI"]},{"id":"2026-11-05-quasiplanar-graphs-linear-edges-chatgpt-6-pach-tardos-zeng","url":"https://postcutoff.com/e/2026-11-05-quasiplanar-graphs-linear-edges-chatgpt-6-pach-tardos-zeng/","date":"2026-11-05","date_precision":"day","short_title":"Quasiplanar graph conjecture proved by ChatGPT-6 (Pach, Tardos, Zeng)","deck":null,"takeaway":"János Pach, one of the conjecture's authors, with Gábor Tardos and Ji Zeng, writes that ChatGPT-6 proved that graphs drawn with no k pairwise crossing edges have linearly many edges, a conjecture known only for k ≤ 4. Their note gives a self-contained combinatorial version; an independent human preprint by Ankan Sadhu (1 Nov) claims a similar proof.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-11-08","updated":"2026-11-08","orgs":["OpenAI","Rényi Institute","ISTA"]},{"id":"2026-11-05-singmaster-bound-claude-opus-5-5-whitmore","url":"https://postcutoff.com/e/2026-11-05-singmaster-bound-claude-opus-5-5-whitmore/","date":"2026-11-05","date_precision":"day","short_title":"Singmaster's problem: new bound by Claude Opus 5.5","deck":null,"takeaway":"Singmaster's problem is a well-known open problem in elementary number theory, and the claimed exponent 1/2 improves even on what was previously known under Cramér's conjecture.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-06","updated":"2026-11-06","orgs":["Anthropic"]},{"id":"2026-11-05-ai-assisted-results-november-5-listing","url":"https://postcutoff.com/e/2026-11-05-ai-assisted-results-november-5-listing/","date":"2026-11-05","date_precision":"day","short_title":"AI-assisted results, 5 Nov: arXiv of 5 Nov and Hexagon of 4 Nov","deck":null,"takeaway":"The 5 Nov sweep listed 424 arXiv papers mentioning AI (about 240 new) and 44 uncited Hexagon works. Own entries: the refutation of the 3-uniform hyperclique hypothesis (ChatGPT Sol 6.1), Najman's elliptic-curve torsion in degrees 6–9 (GPT-6.1 Sol) and Serre modularity over real quadratic fields (ChatGPT Astra/Sol).","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":67,"official":67,"filed":"2026-11-05","updated":"2026-11-05","orgs":["OpenAI","Anthropic","various researchers"]},{"id":"2026-11-04-axiom-stacks-project-lean-formalization","url":"https://postcutoff.com/e/2026-11-04-axiom-stacks-project-lean-formalization/","date":"2026-11-04","date_precision":"day","short_title":"Axiom: the Stacks Project formalized in Lean","deck":null,"takeaway":"On 4 Nov 2026 Axiom Math released AxiomMath/Stacks, a Lean 4 formalization of 'every tag of the Stacks Project with formalizable content' (about 63,000 modules), with an agent-written StacksFrontier tier and a human-reviewed StacksCore tier. Axiom's X post had 6.9M views.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-11-07","updated":"2026-11-09","orgs":["Axiom Math"]},{"id":"2026-11-04-elliptic-curve-torsion-degrees-6-9-gpt-6-1-sol-najman","url":"https://postcutoff.com/e/2026-11-04-elliptic-curve-torsion-degrees-6-9-gpt-6-1-sol-najman/","date":"2026-11-04","date_precision":"day","short_title":"Elliptic-curve torsion in degrees 6–9, mostly by GPT-6.1 Sol","deck":null,"takeaway":"Filip Najman deposited on Hexagon a paper that determines every group occurring as the torsion of an elliptic curve over a number field of degree 6, 7, 8 or 9, including three groups not previously known to occur.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-05","updated":"2026-11-06","orgs":["OpenAI","Anthropic","University of Zagreb"]},{"id":"2026-11-04-glauber-dynamics-colorings-1-plus-1-over-e-gpt-6-astra-pro","url":"https://postcutoff.com/e/2026-11-04-glauber-dynamics-colorings-1-plus-1-over-e-gpt-6-astra-pro/","date":"2026-11-04","date_precision":"day","short_title":"Colourings Glauber dynamics above (1+1/e)Δ (GPT-6 Astra Pro)","deck":null,"takeaway":"If it holds, this is a major step towards the Δ+2 conjecture, and another case of a frontier model supplying the central idea of a theoretical computer science result.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-04","updated":"2026-11-04","orgs":["OpenAI"]},{"id":"2026-11-04-huffman-unequal-letter-costs-np-complete-codex","url":"https://postcutoff.com/e/2026-11-04-huffman-unequal-letter-costs-np-complete-codex/","date":"2026-11-04","date_precision":"day","short_title":"Huffman coding with unequal letter costs is NP-complete (Codex)","deck":null,"takeaway":"A classical problem in information theory, listed as open in surveys of prefix coding for decades, appears settled with the key reduction attributed to a coding agent.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-04","updated":"2026-11-04","orgs":["OpenAI","Indian Institute of Science"]},{"id":"2026-11-04-matrix-mortality-2x2-undecidable-codex-claude-lean","url":"https://postcutoff.com/e/2026-11-04-matrix-mortality-2x2-undecidable-codex-claude-lean/","date":"2026-11-04","date_precision":"day","short_title":"2×2 matrix mortality claimed undecidable (AI-assisted, Lean)","deck":null,"takeaway":"Paterson showed in 1970 that mortality is undecidable for 3×3 integer matrices; the 2×2 case was the one dimension left open. A Waterloo preprint claims it is undecidable for families of 5,799 2×2 integer matrices, with AI used for research, proofs and a Lean formalization of the reduction.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-11-08","updated":"2026-11-08","orgs":["University of Waterloo","OpenAI","Anthropic"]},{"id":"2026-11-04-amplituhedron-magic-number-conjecture-counterexample-chatgpt","url":"https://postcutoff.com/e/2026-11-04-amplituhedron-magic-number-conjecture-counterexample-chatgpt/","date":"2026-11-04","date_precision":"day","short_title":"Amplituhedron magic number conjecture: counterexample found by ChatGPT","deck":null,"takeaway":"A senior algebraic combinatorialist (Lam) reports an AI-found counterexample to a conjecture in a field linked to physics.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-04","updated":"2026-11-04","orgs":["OpenAI","University of Michigan"]},{"id":"2026-11-04-anthropic-claude-bacterial-vault-particle-structure","url":"https://postcutoff.com/e/2026-11-04-anthropic-claude-bacterial-vault-particle-structure/","date":"2026-11-04","date_precision":"day","short_title":"Anthropic's biology team, working with Claude, reports the first 3D structure of a vault particle from bacterial life","deck":null,"takeaway":"It is a first-hand example of a frontier lab using its own model as a research assistant across a full wet-lab project, with human scientists directing.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-05","updated":"2026-11-05","orgs":["Anthropic"]},{"id":"2026-11-04-nature-ai-native-autonomous-biomolecular-lab","url":"https://postcutoff.com/e/2026-11-04-nature-ai-native-autonomous-biomolecular-lab/","date":"2026-11-04","date_precision":"day","short_title":"Nature: AI agents run an automated biology lab","deck":null,"takeaway":"Nature's peer review of an agent-run wet lab moves \"self-driving labs\" from demonstrations toward published, reviewed practice.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-11-07","updated":"2026-11-07","orgs":[]},{"id":"2026-11-04-serre-modularity-real-quadratic-fields-chatgpt-astra","url":"https://postcutoff.com/e/2026-11-04-serre-modularity-real-quadratic-fields-chatgpt-astra/","date":"2026-11-04","date_precision":"day","short_title":"Serre modularity over real quadratic fields, primarily AI-generated","deck":null,"takeaway":"Dieulefait posted two Claude-assisted Fontaine–Mazur modularity papers the same day.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-11-05","updated":"2026-11-05","orgs":["OpenAI","Anthropic"]},{"id":"2026-11-04-ai-assisted-results-november-4-listing","url":"https://postcutoff.com/e/2026-11-04-ai-assisted-results-november-4-listing/","date":"2026-11-04","date_precision":"day","short_title":"AI-assisted results, 4 Nov: arXiv of 4 Nov and Hexagon of 2–3 Nov","deck":null,"takeaway":"The 4 Nov sweep listed 374 arXiv papers that mention AI (193 new), 104 conjecture papers and 33 uncited Hexagon works. Three got own entries: Chandran–Gajjala's NP-completeness of Huffman coding with unequal letter costs (Codex), Yang's (1+1/e)Δ colouring mixing claim (GPT-6 Astra Pro) and Akhmedova–Lam–Yost-Wolff's ChatGPT-found counterexample to the amplituhedron magic number conjecture.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":60,"official":60,"filed":"2026-11-04","updated":"2026-11-04","orgs":["OpenAI","Anthropic","various researchers"]},{"id":"2026-11-03-alon-seymour-thomas-separator-conjecture-gpt-6-astra-two-proofs","url":"https://postcutoff.com/e/2026-11-03-alon-seymour-thomas-separator-conjecture-gpt-6-astra-two-proofs/","date":"2026-11-03","date_precision":"day","short_title":"K_h-minor-free separators O(h√n): two AI-assisted proofs","deck":null,"takeaway":"The separator theorem underlies many algorithms on minor-closed graph classes.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-03","updated":"2026-11-03","orgs":["OpenAI","Anthropic"]},{"id":"2026-11-03-kronheimer-mrowka-dodecahedron-web-instanton-homology-64-chatgpt","url":"https://postcutoff.com/e/2026-11-03-kronheimer-mrowka-dodecahedron-web-instanton-homology-64-chatgpt/","date":"2026-11-03","date_precision":"day","short_title":"Kronheimer–Mrowka: dodecahedron instanton homology is 64 (ChatGPT Astra 6)","deck":null,"takeaway":"Kronheimer and Mrowka built J♯ in 2015 as a possible route to the four-colour theorem, asking whether its dimension counts Tait colourings for planar webs.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-03","updated":"2026-11-03","orgs":["OpenAI","Anthropic","Harvard University","MIT"]},{"id":"2026-11-03-mixed-volume-fpras-gurvits-conjecture-disproved-gpt-6-astra","url":"https://postcutoff.com/e/2026-11-03-mixed-volume-fpras-gurvits-conjecture-disproved-gpt-6-astra/","date":"2026-11-03","date_precision":"day","short_title":"FPRAS for mixed volumes, ideas from GPT-6 Astra","deck":null,"takeaway":"Mixed volumes generalize permanents and volumes; Gurvits had conjectured that no such scheme exists.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-03","updated":"2026-11-03","orgs":["OpenAI"]},{"id":"2026-11-03-d-equivalence-generalized-kummer-chatgpt-6","url":"https://postcutoff.com/e/2026-11-03-d-equivalence-generalized-kummer-chatgpt-6/","date":"2026-11-03","date_precision":"day","short_title":"D-equivalence for generalized Kummer hyperkählers (ChatGPT-6 assisted)","deck":null,"takeaway":"Donovan, Wang and Zhang claim that birational hyperkähler varieties of generalized Kummer type are derived equivalent; with earlier work this would settle the D-equivalence conjecture for every known deformation type. ChatGPT 6.0 helped realise the key kernel construction.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-08","updated":"2026-11-08","orgs":["Tsinghua University","OpenAI"]},{"id":"2026-11-03-reed-muller-rm19-covering-radius-242-three-proofs","url":"https://postcutoff.com/e/2026-11-03-reed-muller-rm19-covering-radius-242-three-proofs/","date":"2026-11-03","date_precision":"day","short_title":"RM(1,9) covering radius = 242, three proofs","deck":null,"takeaway":"Nonlinearity bounds matter for cipher design; agreement between three independent computer-assisted proofs makes the value 242 more credible.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-11-03","updated":"2026-11-03","orgs":["OpenAI","University of South Florida"]},{"id":"2026-11-03-ai-assisted-results-november-3-listing","url":"https://postcutoff.com/e/2026-11-03-ai-assisted-results-november-3-listing/","date":"2026-11-03","date_precision":"day","short_title":"AI-assisted results, 3 Nov: arXiv of 3 Nov and Hexagon of 2–3 Nov","deck":null,"takeaway":"The 3 Nov sweep listed 376 arXiv papers that mention AI, 124 conjecture papers and 69 uncited Hexagon works. Four results got own entries: Kronheimer–Mrowka's primarily AI-generated proposed proof that the dodecahedron web has instanton homology of dimension 64, a second AI-derived proof of the Alon–Seymour–Thomas separator conjecture (with Lean), Chen–Liu's FPRAS for mixed volumes (ideas 'generated entirely by GPT 6 Astra'), and the convergence of three proofs that RM(1,9) has covering radius 242.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":13,"filed":"2026-11-03","updated":"2026-11-03","orgs":["OpenAI","Anthropic","Google","various researchers"]},{"id":"2026-11-02-ai-assisted-results-november-2-listing","url":"https://postcutoff.com/e/2026-11-02-ai-assisted-results-november-2-listing/","date":"2026-11-02","date_precision":"day","short_title":"AI-assisted results, 2 Nov: arXiv of 2 Nov and Hexagon of 1–2 Nov","deck":null,"takeaway":"The 2 Nov sweep listed 319 arXiv papers that mention AI, 75 conjecture papers without an AI statement and 52 uncited Hexagon works. Three results got own entries: Colbrook's proof of the 1974 Kung–Traub conjecture (Lean, with ChatGPT 6), Yu–Yu's GPT-6 Astra counterexample to the Alon–Krivelevich conjecture, and Barra–Durante's AI-generated proof of the Hopfield thermodynamic limit. Listed here: Okayasu's answer to Haagerup's 1985 problem with Claude Opus 5.5, Suvrit Sra's proof of Petz's 1994 conjecture with GPT Pro, Qiyue Tang's claimed proof of Kamnitzer's ICM 2022 conjecture (after an AI reviewer found a fatal error that was then repaired), and Niels Feld's counterexamples to Morel's purity conjecture.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":50,"official":50,"filed":"2026-11-02","updated":"2026-11-02","orgs":["OpenAI","Anthropic","various researchers"]},{"id":"2026-11-01-ai-assisted-results-november-1-listing","url":"https://postcutoff.com/e/2026-11-01-ai-assisted-results-november-1-listing/","date":"2026-11-01","date_precision":"day","short_title":"AI-assisted results, 1 Nov: arXiv leftovers and result repos","deck":null,"takeaway":"The 1 Nov sweep listed 325 arXiv papers that mention AI (mostly already-skimmed submissions of 26–29 Oct) and 66 conjecture papers without an AI statement. No new result reached own-entry level. Listed here: Chao Liu's f(k) = k²/2 + O(k log k) for subset sums of zero-sum-free sets ('AI tools developed by OpenAI were used'), Bednorz–Łochowski's proof of the Das–Łochowski–Matsuda–Perkowski fBm conjecture with ChatGPT proof strategies, Gao–Zhao on Duke–Erdős–Rödl, and a Lean proof (drafted with Claude Code) of Long's ChatGPT-assisted Poisson conjecture counterexample.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":13,"filed":"2026-11-01","updated":"2026-11-01","orgs":["OpenAI","Anthropic","Google","various researchers"]},{"id":"2026-10-31-bhargava-cyclic-reduction-elliptic-curves-positive-proportion-gpt-6","url":"https://postcutoff.com/e/2026-10-31-bhargava-cyclic-reduction-elliptic-curves-positive-proportion-gpt-6/","date":"2026-10-31","date_precision":"day","short_title":"Bhargava: elliptic curves have cyclic reduction at a positive proportion of primes (claim)","deck":null,"takeaway":"Serre showed under the generalized Riemann hypothesis that an elliptic curve over ℚ without full rational 2-torsion has cyclic reduction E(F_p) at a positive proportion of primes; Murty removed GRH for CM curves, and Gupta–Murty proved infinitely many such primes in general. On 31 Oct Manjul Bhargava (Princeton) posted on Hexagon an unconditional positive lower proportion for every such curve. The argument uses Cureton–Thorner's zero-free half-plane and 'the marked Type II estimates of OpenAI'. Per the provenance statement, handwritten notes were turned into LaTeX 'with additional writing and details added by both the author and GPT-6'. GPT-6 also helped formalise the deduction in Lean, with the analytic inputs left as hypotheses. Only some parts have been checked by a human, and it is not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Princeton University"]},{"id":"2026-10-31-onsager-hard-rods-first-order-transition-gpt-6-astra-unverified","url":"https://postcutoff.com/e/2026-10-31-onsager-hard-rods-first-order-transition-gpt-6-astra-unverified/","date":"2026-10-31","date_precision":"day","short_title":"Onsager hard rods: AI-written first-order transition claim, not human-verified","deck":null,"takeaway":"Onsager predicted in 1949 that elongated hard rods undergo a first-order isotropic–nematic transition, but a rigorous proof for rods with continuous orientations in three dimensions has been missing. On 31 Oct Qidong He (Northeastern University) deposited Hexagon 2610.00667, which claims such a transition for small diameter-to-length ratio. The provenance statement says GPT-6 Astra produced the candidate proof 'in an essentially zero-shot run lasting about six hours'. It says the argument follows Lebowitz–Mazel–Presutti, and that no human has checked any of it: '***THE RESULT IS NOT HUMAN-VERIFIED***'.","category":"science","category_label":"Science & math","importance":3,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Anthropic"]},{"id":"2026-10-31-ai-assisted-results-october-31-listing","url":"https://postcutoff.com/e/2026-10-31-ai-assisted-results-october-31-listing/","date":"2026-10-31","date_precision":"day","short_title":"AI-assisted results, 31 Oct: arXiv of 30 Oct and Hexagon of 29–31 Oct","deck":null,"takeaway":"The 31 Oct sweep listed 344 arXiv papers that mention AI, 75 conjecture papers with no AI statement and 48 uncited Hexagon works. We skimmed every AI statement and read about 70 PDFs or Hexagon records. Seven results got their own entries. Two come from senior authors: Panchenko's Mézard–Parisi formula ('found by ChatGPT Astra') and Bhargava's positive-density cyclicity of elliptic curves, written with GPT-6. The other five are Tianchi Yang's count of C₄-free graphs, Kolpakov's sharp dimension bounds for right-angled hyperbolic polytopes, the generalized Lax conjecture (Kummer–Netzer's AI-found generic case and OpenAI's counterexample), Diening–Storn's AFEM optimality for every bulk parameter, and an unverified GPT-6 Astra claim on Onsager's hard rods. The rest are listed here, among them Pongrácz's R(5,6) ≤ 84 certificate with GPT-6 Astra and Bowler–Pitz's negative answers to Halin-type questions, where Claude found key lemmas.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":44,"official":42,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Anthropic","Google","various researchers"]},{"id":"2026-10-30-alon-krivelevich-conjecture-disproved-gpt-6-astra-xean","url":"https://postcutoff.com/e/2026-10-30-alon-krivelevich-conjecture-disproved-gpt-6-astra-xean/","date":"2026-10-30","date_precision":"day","short_title":"Alon–Krivelevich conjecture disproved with GPT-6 Astra","deck":null,"takeaway":"The conjecture, from Noga Alon and Michael Krivelevich's 1998 paper, was a central open question on list colouring of bipartite graphs (the best known upper bound is Δ/log Δ-type).","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-11-02","updated":"2026-11-02","orgs":["OpenAI","Nanjing Normal University","William & Mary"]},{"id":"2026-10-30-mezard-parisi-formula-panchenko-proof-found-by-chatgpt-astra","url":"https://postcutoff.com/e/2026-10-30-mezard-parisi-formula-panchenko-proof-found-by-chatgpt-astra/","date":"2026-10-30","date_precision":"day","short_title":"Mézard–Parisi formula: Panchenko's proof 'found by ChatGPT Astra'","deck":null,"takeaway":"Dmitry Panchenko (Toronto), a leading probabilist of spin glasses, posted a proof of the Mézard–Parisi formula on 29–30 Oct 2026. The formula gives the free energy of random K-SAT and diluted p-spin models with even p. Franz and Leone had proved the upper bound; Panchenko's paper claims the matching lower bound. His AI statement says: 'The proof in this paper was found by ChatGPT Astra.' A companion note was 'written by ChatGPT Astra'. He adds that OpenAI released another proof a few days later, which 'appears to follow a different approach'. OpenAI's version is family 221 of its maths repository. Both are unrefereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","University of Toronto"]},{"id":"2026-10-30-number-of-c4-free-graphs-erdos-question-tianchi-yang-gpt-6-pro","url":"https://postcutoff.com/e/2026-10-30-number-of-c4-free-graphs-erdos-question-tianchi-yang-gpt-6-pro/","date":"2026-10-30","date_precision":"day","short_title":"Number of C₄-free graphs is 2^{(1+o(1))ex(n,C₄)} (claim, GPT-6 Pro)","deck":null,"takeaway":"Tianchi Yang's preprint (arXiv 2610.39595, 29 Oct 2026) claims that the number of graphs on n vertices with no 4-cycle is 2^{(1+o(1))ex(n,C₄)}. That answers a question of Erdős. The best earlier upper bound was Kleitman and Winston's 2^{(1.0819…+o(1))n^{3/2}} from 1982, and Morris and Saxton showed the same statement fails for C₆. The AI statement says the author supplied the main ideas, a modified hypergraph container method, and 'GPT-6 Pro assisted in completing the proofs.' Single author, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI","Anthropic"]},{"id":"2026-10-30-afem-optimal-convergence-any-bulk-parameter-diening-storn-chatgpt","url":"https://postcutoff.com/e/2026-10-30-afem-optimal-convergence-any-bulk-parameter-diening-storn-chatgpt/","date":"2026-10-30","date_precision":"day","short_title":"AFEM rate-optimal for any bulk parameter θ < 1 (ChatGPT framework)","deck":null,"takeaway":"Rate-optimality of the standard adaptive finite element method (Stevenson 2007; Cascón–Kreuzer–Nochetto–Siebert 2008) has always required the Dörfler marking parameter θ to be small. Lars Diening and Johannes Storn claim it for every θ in (0,1), for conforming Lagrange elements and the Poisson problem (arXiv 2610.39719). They say an extended discussion between Storn and ChatGPT 'led to an initial framework for the proof', and that they do not try to separate the human and AI contributions. The authors then verified and rewrote it. Not refereed.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI"]},{"id":"2026-10-30-hopfield-model-thermodynamic-limit-barra-durante-claude-chatgpt","url":"https://postcutoff.com/e/2026-10-30-hopfield-model-thermodynamic-limit-barra-durante-claude-chatgpt/","date":"2026-10-30","date_precision":"day","short_title":"Hopfield model thermodynamic limit, AI-generated proofs","deck":null,"takeaway":"It follows Panchenko's Mézard–Parisi proof found by ChatGPT Astra in the same week, another rigorous statistical-mechanics result whose authors attribute the derivation to models.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-11-02","updated":"2026-11-02","orgs":["Anthropic","OpenAI"]},{"id":"2026-10-30-right-angled-hyperbolic-polytopes-dimension-bounds-kolpakov-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-30-right-angled-hyperbolic-polytopes-dimension-bounds-kolpakov-gpt-6-astra/","date":"2026-10-30","date_precision":"day","short_title":"Right-angled hyperbolic polytopes exist only up to dimension 8 (claim)","deck":null,"takeaway":"Finite-volume right-angled hyperbolic polytopes were known in dimensions up to 8 (Potyagailo–Vinberg), and Dufour had shown they cannot exist above 12. For polytopes with all vertices at infinity the gap was 4 to 6. Alexander Kolpakov's preprint (arXiv 2610.38900, 29 Oct 2026) claims both gaps are closed: at most 8 in general and at most 4 when all vertices are ideal. He used GPT-6 Astra and GPT-6.1 Sol 'to investigate proof ideas, help strengthen the results', and Lean modules check the internal deductions. Not refereed.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI"]},{"id":"2026-10-30-ai-assisted-results-october-30-listing","url":"https://postcutoff.com/e/2026-10-30-ai-assisted-results-october-30-listing/","date":"2026-10-30","date_precision":"day","short_title":"AI-assisted results, 30 Oct: arXiv announcement of 29–30 Oct","deck":null,"takeaway":"The 30 Oct sweep listed 374 arXiv papers that mention AI. We skimmed all their AI statements and read about 35 PDFs. Six results got their own entries: Das–Fox's Murty–Simon proof (GPT-5.6 Sol found the bound), Colbrook–Stepaniants's spectral isometry proof (found by GPT-6 Astra, Lean), the Θ(log k log log k) ratio of greedy for online Steiner forest (GPT Astra, Lean), Gáspár's Laczkovich solution (key proposition by ChatGPT), Ren's pure-semisimplicity counterexample (GPT-6 Astra, Lean) and Petty's conjecture in dimension 5 (Tencent's hyprover agent). PNNL's OpenConjecture study also has one. This listing has the rest, among them a single-author ChatGPT 6 Pro proof of the Alon–Seymour–Thomas separator conjecture, b₄ ≤ 7 for Borsuk's problem and Fiedorowicz's conjecture from a Claude idea.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":42,"official":42,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Anthropic","Google","Tencent","various researchers"]},{"id":"2026-10-29-berry-esseen-constant-esseen-sharp-certificate-claim","url":"https://postcutoff.com/e/2026-10-29-berry-esseen-constant-esseen-sharp-certificate-claim/","date":"2026-10-29","date_precision":"day","short_title":"Sharp Berry–Esseen constant claimed: C = Esseen's 0.4097","deck":null,"takeaway":"The Berry–Esseen inequality bounds how far a normalised sum of i.i.d. variables is from the normal law. Its best constant has been open since Esseen showed in 1956 that it cannot be below (3+√10)/(6√(2π)) ≈ 0.4097; the best published upper bound is Shevtsova's 0.4690. On 29 Oct Bartosz Kołodziejek and Kacper Rodziewicz (Warsaw) posted on Hexagon a claimed proof that Esseen's value is the exact constant for every sample size n. The proof combines analytic reductions with exact finite certificates in 13 index ranges, checked end to end by their own programs, with a one-command Docker replay. They used ChatGPT, Codex and Gemini/Antigravity 'extensively'. It is not a Lean proof, and no specialist had checked it by 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Warsaw University of Technology","University of Warsaw","OpenAI","Google"]},{"id":"2026-10-29-deterministic-factoring-n-o1-lean-claude-rudich-rousseau","url":"https://postcutoff.com/e/2026-10-29-deterministic-factoring-n-o1-lean-claude-rudich-rousseau/","date":"2026-10-29","date_precision":"day","short_title":"Deterministic factoring in N^o(1) steps, Lean-checked, ideas by Claude","deck":null,"takeaway":"The best proven deterministic factoring algorithms ran in time N^(1/5+o(1)) (Harvey; Harvey–Hittmeir). On 29 Oct Isaac Rudich and Louis-Martin Rousseau (Polytechnique Montréal) posted on Hexagon a Lean 4 proof that one deterministic Turing machine finds the smallest prime factor of every N within C_k·2^(n/k) steps for every k, i.e. N^o(1). The 13-line statement uses only Mathlib and is registered in Palomar; the proof is about 36,000 lines. The authors say every idea and all Lean code came from Claude agents, which they asked to apply the ETH refutation to factoring; they audited the statement, not the proof. The algorithm is 'galactic' and slower than the randomised and heuristic methods used in practice; what is new is a proven deterministic N^o(1) bound.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-29","updated":"2026-10-31","orgs":["Anthropic","Polytechnique Montréal"]},{"id":"2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev","url":"https://postcutoff.com/e/2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev/","date":"2026-10-29","date_precision":"day","short_title":"Kervaire's conjecture: OpenAI proof confirmed in the Kourovka Notebook","deck":null,"takeaway":"Kervaire's conjecture says that adding one generator and one relation to a nontrivial group never gives the trivial group. It was known for torsion-free groups (Klyachko 1993) and residually finite groups (Gerstenhaber–Rothaus 1962). OpenAI's 23–24 Sept release (family 256) claimed it in general, and also the Kervaire–Laudenbach conjecture on independent systems of equations (Howie's form). The Kourovka Notebook's update file now marks problems 17.94 and 18.87(a) solved: 'AI solution confirmed by N. V. Bogachev'. The same file lists 48 AI solutions confirmed by named mathematicians and 35 still unverified, including OpenAI's claim of infinite finitely presented periodic groups (8.52).","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":6,"official":6,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Kourovka Notebook"]},{"id":"2026-10-29-online-steiner-forest-greedy-log-k-loglog-k-gpt-astra-lean","url":"https://postcutoff.com/e/2026-10-29-online-steiner-forest-greedy-log-k-loglog-k-gpt-astra-lean/","date":"2026-10-29","date_precision":"day","short_title":"Online Steiner forest: greedy claimed Θ(log k log log k), with GPT Astra","deck":null,"takeaway":"In online Steiner forest, terminal pairs arrive one by one and must be connected at once. Awerbuch, Azar and Bartal proved in 1996 that the greedy algorithm is O(log² k)-competitive and conjectured O(log k). On 29 Oct a University of Maryland group led by MohammadTaghi Hajiaghayi posted matching bounds of Θ(log k log log k). If correct, that refutes the conjecture and separates greedy from the best online algorithm by a log log k factor. Some proofs 'were developed with the help of OpenAI's GPT Astra', with Gemini Argon assisting, and Lean files for the main theorem are on GitHub. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Google DeepMind","University of Maryland"]},{"id":"2026-10-29-petty-equilateral-set-conjecture-dimension-five-tencent-hyprover","url":"https://postcutoff.com/e/2026-10-29-petty-equilateral-set-conjecture-dimension-five-tencent-hyprover/","date":"2026-10-29","date_precision":"day","short_title":"Petty's conjecture in dimension 5: proof by Tencent's hyprover agent","deck":null,"takeaway":"Petty conjectured that every n-dimensional normed space contains n + 1 points at equal distances. Petty proved dimension 3 and Makeev dimension 4. On 29 Oct Kaiwen Yang (Tongji University and Tencent Hunyuan) posted a proof for dimension 5. Its AI declaration says the proof 'was generated by Tencent Hunyuan's AI agent hyprover on September 29, 2026', and that the author organised, verified and rewrote it. The paper prints the full prompt, which points the agent to Lu–Yang's solution of Banach's isometric conjecture. Few research results in our dataset credit a Chinese lab's agent. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-30","updated":"2026-10-30","orgs":["Tencent","Tongji University"]},{"id":"2026-10-29-pure-semisimplicity-conjecture-counterexample-gpt-6-astra-lean","url":"https://postcutoff.com/e/2026-10-29-pure-semisimplicity-conjecture-counterexample-gpt-6-astra-lean/","date":"2026-10-29","date_precision":"day","short_title":"Pure semisimplicity conjecture: claimed counterexample, GPT-6 Astra, Lean","deck":null,"takeaway":"The pure semisimplicity conjecture says a ring over which every module is a direct sum of finitely generated ones must have finite representation type. It holds for Artin algebras (Auslander) and PI rings (Simson, Herzog). Simson reduced a counterexample to a 'generalized Artin problem' about division ring extensions. On 29 Oct Junpeng Ren posted a division ring extension that solves it. The AI declaration says GPT-6 Astra helped choose the criterion and develop the proof, 'in particular the construction of the division rings'. The main theorem is fully formalised in Lean 4 in a public repository. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Anthropic"]},{"id":"2026-10-29-q-colorability-threshold-existence-aronow-lopatto-ai","url":"https://postcutoff.com/e/2026-10-29-q-colorability-threshold-existence-aronow-lopatto-ai/","date":"2026-10-29","date_precision":"day","short_title":"q-colourability threshold of G(n, d/n) exists, says a machine-written paper","deck":null,"takeaway":"Achlioptas and Friedgut proved in 1999 that q-colourability of the random graph G(n, d/n) has a sharp threshold, but only along a sequence that might depend on n. Whether that sequence converges to a constant d_q was recorded as an open problem and is widely called well known. On 29 Oct P. M. Aronow (Yale) and Patrick Lopatto (UNC Chapel Hill) posted on Hexagon a proof that it converges for every q ≥ 3, with a window of O(n(log n)^(−3/2)) steps for the hitting time. The authors' note says the whole document except that note is machine-written. The provenance names Opus 5.5, GPT 6 Astra, GPT Luna 6 and GPT 6.1 Sol, and says the results were formalised in Lean, but the paper links no Lean repository. Human understanding: 'some parts'. Two companion papers on the dynamics of the spherical Sherrington–Kirkpatrick model came the same day.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","OpenAI","Yale University","University of North Carolina at Chapel Hill"]},{"id":"2026-10-29-spectral-isometry-conjecture-c-star-algebras-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-29-spectral-isometry-conjecture-c-star-algebras-gpt-6-astra/","date":"2026-10-29","date_precision":"day","short_title":"Spectral isometry conjecture for C*-algebras: GPT-6 Astra proof, Lean","deck":null,"takeaway":"Mathieu's spectral isometry conjecture asks whether every unital surjective linear map between unital C*-algebras that preserves the spectral radius is a Jordan isomorphism. Only special classes were known. On 29 Oct Matthew Colbrook and George Stepaniants (Cambridge) posted what they call 'a human digestion of a proof … found by GPT-6 Astra'. The theorem is registered in Lean on Palomar. Their AI declaration says the proof came 'after a sequence of simple prompts'. This is the second such digest from Colbrook in a week, after Smale's mean value conjecture. It is a preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","University of Cambridge"]},{"id":"2026-10-29-euclidean-algorithm-not-optimal-moschovakis-lean","url":"https://postcutoff.com/e/2026-10-29-euclidean-algorithm-not-optimal-moschovakis-lean/","date":"2026-10-29","date_precision":"day","short_title":"Gcd in O(log a / log log a) remainder calls refutes Moschovakis's uniform-process conjecture","deck":null,"takeaway":"Van den Dries and Moschovakis asked whether Euclid's algorithm is optimal among algorithms that compute the gcd from the remainder function. On 29 Oct Jason Davies posted on Hexagon a proof that every pair a ≥ b has a derivation of gcd(a, b) using O(log a / log log a) remainder calls, while Euclid needs order log a in the worst case. That refutes Moschovakis's conjectured logarithmic lower bounds for uniform processes and answers a closure-depth question of van den Dries and Moschovakis; Moschovakis's Main Conjecture for actual algorithms stays open. GPT-6 Astra found the principal result in Codex; GPT-6 Astra and Claude Opus 5.5 wrote the Lean proofs and the paper, and Palomar's comparator confirms the 22 compared statements. Davies says he wrote none of the text or code.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic"]},{"id":"2026-10-29-openconjecture-random-conjectures-gpt-models-pnnl","url":"https://postcutoff.com/e/2026-10-29-openconjecture-random-conjectures-gpt-models-pnnl/","date":"2026-10-29","date_precision":"day","short_title":"OpenConjecture: GPT models on 200 random arXiv conjectures","deck":null,"takeaway":"Reported AI-maths breakthroughs are hand-picked. Researchers at Pacific Northwest National Laboratory sampled 200 conjectures at random from arXiv maths papers posted since February 2026. They ran GPT-5.6 Sol, then GPT-5.6 Sol Pro, then GPT-6 Astra Pro on each, with no problem-specific guidance. The models proposed 70 counterexamples and 55 proofs, and 84% of these passed review by a fresh instance of the same model. No experts checked them. About 60% of the reviewed counterexamples were 'off-the-shelf' and the rest 'standard constructions'. Some exploited missing hypotheses, such as offering the trivial group, and a number of proofs followed closely from known results.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-30","updated":"2026-10-30","orgs":["Pacific Northwest National Laboratory","OpenAI"]},{"id":"2026-10-29-ai-assisted-results-october-29-listing-2","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing-2/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results, 29 Oct 2026 (part 2)","deck":"Claude 'largely carried out' Lipshitz's Khovanov research, ChatGPT found Soundararajan's proof and wrote Etingof's and Mathew's papers, George Elliott's free-product classification 'carried out using ChatGPT', and a first read of Hexagon's backlog","takeaway":"A second pass over the 28 Oct arXiv announcement and a first full read of Hexagon (hexagonmath.org), which the sweep had never covered, found results with large AI roles signed by senior mathematicians. Robert Lipshitz writes that his Khovanov paper's research 'was largely carried out by various Claude models'. Kannan Soundararajan's proof of a problem from Montgomery's CBMS book 'was found by ChatGPT 6'. ChatGPT 6 Pro wrote Pavel Etingof's answer to a 1999 question and produced Akhil Mathew's answer to a question of Drinfeld. George Elliott's classification of reduced free products of nuclear C*-algebras 'was carried out using ChatGPT'. Eight items have their own entries: the unforced 2D Euler non-uniqueness claims, the Erdős–Lovász covering-number bound, UPP^cc under intersection, Kisin's Serre conjecture, Linnik's constant ≤ 3.99, 3D Yang–Mills on a torus, random k-SAT and Chenglong Ma's Hovey–Strickland provenance claim.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":30,"official":29,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]},{"id":"2026-10-29-ai-assisted-results-october-29-listing-3","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing-3/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results, 29 Oct (part 3): Hexagon's evening batch","deck":null,"takeaway":"Hexagon published 58 new works on 28–29 Oct that we had not cited. Four have their own entries: the claimed sharp Berry–Esseen constant (Kołodziejek–Rodziewicz), the existence of the q-colourability threshold (Aronow–Lopatto, machine-written), deterministic factoring in N^o(1) (Rudich–Rousseau, Lean, ideas by Claude) and the refutation of Moschovakis's uniform-process lower bounds for gcd (Davies, Lean). Others claim the quantum Jellium = uniform-electron-gas energy (Csirik), C₂-cofiniteness of all finite-group orbifolds (Riesen), analytic floating bodies of equilibrium in every dimension ≥ 3 (Price–de Groot, GPT-6 Pro) and Cowling's conjecture for almost connected groups (Arhancet). From the backlog: S(13) = 45 comparators for sorting 13 inputs and an area law for 3D U(1) Wilson lattice gauge theory (Tomar, Lean), Enomoto's settlement of the third Brauer–Thrall conjecture over algebraically closed fields (GPT-6 Astra) and Soberón's counterexample to Matschke's table conjecture.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google","various researchers"]},{"id":"2026-10-29-ai-assisted-results-october-29-listing","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results in the 29 Oct 2026 sweep","deck":"Counterexamples to Wilking's 2007 question found by GPT-6 Astra, an LLM-written Plateau paper by De Lellis and Glaudo, explicit Reed–Solomon codes with optimal list size (Guruswami et al.), rank-four quiver mutation decided by GPT Astra, and a dozen named conjectures","takeaway":"The 29 Oct sweep covered the arXiv announcement of 28 Oct, which the previous run had read only through keyword queries. This time 43 papers with an AI statement surfaced, and a PDF scan of a further 405 new and cross-listed submissions in math, cs.CC, cs.DS and cs.DM found about 180 more with an AI statement, most of them describing more than editing help. Four results have their own entries: OpenAI's abundance claim and Jihao Liu's priority remark, OpenAI's Lean-checked L = RL = BPL, R(3,3,3,3) ≤ 61 and the sharp Banach–Mazur exponent. This page lists the rest. GPT-6 Astra found the counterexamples to Wilking's 2007 question 'with minimal human input' and discovered the main results of a rank-four quiver paper. Claude found examples in opinion dynamics and proposed a minimal-surface construction. An LLM supplied 'the main idea' of a paper by Camillo De Lellis, a member of the IAS advisory group on AI and mathematics, and wrote it.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","various researchers"]},{"id":"2026-10-28-kisin-serre-artin-character-conjecture-gpt-astra","url":"https://postcutoff.com/e/2026-10-28-kisin-serre-artin-character-conjecture-gpt-astra/","date":"2026-10-28","date_precision":"day","short_title":"Kisin proves Serre's 1960 Artin-character conjecture with GPT Astra","deck":null,"takeaway":"In 1960 Serre conjectured that a function built from fixed-point multiplicities of a finite group acting on a regular local ring is the character of an ℓ-adic representation. It is the higher-dimensional analogue of the Artin character of local fields. Tomoyuki Abe proved the positive-characteristic case (Inventiones, 2025). On 28 Oct Harvard's Mark Kisin posted the mixed-characteristic case on Hexagon, which finishes the conjecture. He writes that 'extensive use of GPT Astra was made both to arrive at the proofs … and to write a first draft', and that 'significant human effort' went into checking them. Part of the work was done at 'The Overhang', a San Francisco meeting on 9–10 Sept 2026.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Harvard University","OpenAI"]},{"id":"2026-10-28-laczkovich-lipschitz-mapping-problem-chatgpt-gaspar","url":"https://postcutoff.com/e/2026-10-28-laczkovich-lipschitz-mapping-problem-chatgpt-gaspar/","date":"2026-10-28","date_precision":"day","short_title":"Laczkovich's Lipschitz mapping problem: claimed answer, ChatGPT key step","deck":null,"takeaway":"In 1991 Miklós Laczkovich asked whether every measurable set of positive measure in ℝ^d can be mapped onto a cube by a Lipschitz map. Preiss and Matoušek answered it in the plane; dimension 3 and up stayed open. On 28 Oct Attila Gáspár posted a positive answer in every dimension. His disclosure says the proof of Theorem 1.8 'was obtained using ChatGPT on 26 October', and Proposition 2.8 with Lemma 2.7 is 'entirely due to ChatGPT'. He had known other parts and the lower bound since early 2026. It is a short preprint, not refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI"]},{"id":"2026-10-28-murty-simon-conjecture-proof-gpt-5-6-sol-das-fox","url":"https://postcutoff.com/e/2026-10-28-murty-simon-conjecture-proof-gpt-5-6-sol-das-fox/","date":"2026-10-28","date_precision":"day","short_title":"Murty–Simon conjecture: proof for all n, bound found by GPT-5.6 Sol","deck":null,"takeaway":"The Murty–Simon conjecture (Plesník 1975, Murty and Simon) says a diameter-2-critical graph on n vertices has at most ⌊n²/4⌋ edges, with equality only for the balanced complete bipartite graph. Füredi proved it in 1992 only for n above a tower of 2s of height about 10¹⁴. On 28 Oct Sanjana Das and Jacob Fox (Stanford) posted a proof for every n. Their AI statement says the proof of the n²/4 bound 'was mostly found by AI, specifically GPT-5.6 Sol on August 11, 2026'. The authors simplified it and extended it to the equality case for odd n by hand. It is a preprint and has not been refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Stanford University"]},{"id":"2026-10-28-ai-assisted-results-october-28-listing","url":"https://postcutoff.com/e/2026-10-28-ai-assisted-results-october-28-listing/","date":"2026-10-28","date_precision":"day","short_title":"AI-assisted results in the 28 Oct 2026 sweep","deck":"Bialostocki–Lotspeich (1992), part of Gross–Prasad in geometric Langlands, Fuglede in 𝔽₂⁸, a TEMPO structure found by Claude, agent swarms on fractional Chern insulators, and a human digest of GPT-6 Astra's Smale counterexample","takeaway":"The 28 Oct sweep covered the arXiv announcement of 26–27 Oct submissions and late papers from 23–26 Oct. Five results have their own entries: Axiom's Lean proof of the Classification of Finite Simple Groups, OpenAI's spin-one Haldane gap proof (found through Hal Tasaki's follow-up), OpenAI's De Giorgi dimension-eight claim, Morgan–Mullen with Claude and Lean, and formula minimization with GPT-Sol-6.1. This page lists the rest. Claude, GPT-6 Astra and GPT-6 Pro supplied ideas, counterexamples or whole arguments in papers on zero-sum sequences, geometric Langlands, quantum dynamics, fire-blocking and spectral bounds.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":21,"official":19,"filed":"2026-10-28","updated":"2026-10-28","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]},{"id":"2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential","url":"https://postcutoff.com/e/2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential/","date":"2026-10-27","date_precision":"day","short_title":"Anthropic's Claude also refutes ETH","deck":null,"takeaway":"Hours after OpenAI's ETH announcement, Anthropic posted 'A 2^o(n) Time Algorithm for 3-SAT', credited to Claude. It is a deterministic #k-SAT algorithm built on tensor-network contraction, with a different approach from OpenAI's paper, a near-identical bound and a Lean 4 proof that uses only the standard axioms. The paper says the core algorithm was found on 16 Oct with 14M output tokens and no human input. OpenAI reports its model found its disproof that same day. Neither proof has had expert review.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":10,"official":8,"filed":"2026-10-27","updated":"2026-11-06","orgs":["Anthropic"]},{"id":"2026-10-27-axiom-cfsg-classification-finite-simple-groups-lean","url":"https://postcutoff.com/e/2026-10-27-axiom-cfsg-classification-finite-simple-groups-lean/","date":"2026-10-27","date_precision":"day","short_title":"Axiom: Classification of Finite Simple Groups formalized in Lean","deck":null,"takeaway":"On 27 Oct 2026 Axiom Math published a Lean 4 formalization of the Classification of Finite Simple Groups (CFSG). The proof was originally spread over hundreds of papers and an estimated 10,000–15,000 pages. Axiom says it is sorry-free, follows the first-generation proof, and was built by AI agents working with about 30 mathematicians and engineers. Press reports put the build time at 24 days. Along the way the team found a false step in one published route to the Unbalanced Group Theorem and wrote a replacement argument. The project is self-reviewed: outside experts have not yet re-run the build or checked that the Lean statement matches CFSG.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":8,"filed":"2026-10-28","updated":"2026-11-03","orgs":["Axiom Math"]},{"id":"2026-10-27-erdos-lovasz-intersecting-covering-number-frankl-ai-construction","url":"https://postcutoff.com/e/2026-10-27-erdos-lovasz-intersecting-covering-number-frankl-ai-construction/","date":"2026-10-27","date_precision":"day","short_title":"Erdős–Lovász covering-number problem: AI-found construction disproves Frankl's 1996 conjecture","deck":null,"takeaway":"Erdős and Lovász showed in 1975 that an intersecting family of k-sets that needs k points to cover it has at most k^k members. The best constructions stayed near (k/2)^(k−1) for 30 years. In 1996 Frankl, Ota and Tokushige conjectured that m(k) < (ck)^k for some c < 1. A 27 Oct preprint by Peter Frankl, Jian Wang and two Sichuan University students builds families of size k^k·exp(−50√(k log k)). With Zakharov's 2024 upper bound this pins m(k) down to k^k·exp(−k^(1/2+o(1))) and refutes the 1996 conjecture. The authors write that 'the construction was found using generative AI tools'. They do not name the models.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Rényi Institute","Sichuan University"]},{"id":"2026-10-27-banach-mazur-distance-cube-sharp-exponent-gpt-6","url":"https://postcutoff.com/e/2026-10-27-banach-mazur-distance-cube-sharp-exponent-gpt-6/","date":"2026-10-27","date_precision":"day","short_title":"Sharp exponent 2/3 for the Banach–Mazur distance to the cube","deck":null,"takeaway":"Omer Friedland (Sorbonne Université) proves that some n-dimensional normed spaces are at Banach–Mazur distance at least c·n^{2/3}/(log 2n)^10 from the cube ℓ∞ⁿ. With the n^{2/3} upper bound of Friedland, Naor and Youssef from August 2026, the maximal distance is n^{2/3+o(1)}. That settles the exponent in a problem going back to Grünbaum (1960), after a run of improvements from both sides: 5/9, 4/7 and 5/8 below, and 7/8, 5/6 and 2/3 above. The examples are polars of random Gaussian polytopes. Friedland says the strategy is his, and that 'at the author's request, OpenAI's ChatGPT (GPT-6) generated a proof' of a key proposition. Unrefereed preprint.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Sorbonne Université","OpenAI"]},{"id":"2026-10-27-morgan-mullen-conjecture-claude-lean","url":"https://postcutoff.com/e/2026-10-27-morgan-mullen-conjecture-claude-lean/","date":"2026-10-27","date_precision":"day","short_title":"Morgan–Mullen conjecture confirmed with Claude, Lean-checked","deck":null,"takeaway":"Florian Hellwig (University of Zurich) posted a proof on 27 Oct 2026 that every F_{q^n} contains a primitive element that is completely normal over F_q, the 1996 Morgan–Mullen conjecture. The paper also proves Hachenberger's conjectured lower bound for the number of completely normal elements. The paper says the arguments and the text were generated by Claude Fable 5.1 and Claude Opus 5.5 under the author's direction, and the Lean formalization was 'generated entirely by AI'. The author checked the final proof by hand and checked the formal statements.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-28","updated":"2026-10-28","orgs":["University of Zurich","Anthropic"]},{"id":"2026-10-27-ramsey-r3333-upper-bound-61-ai-lean-sat","url":"https://postcutoff.com/e/2026-10-27-ramsey-r3333-upper-bound-61-ai-lean-sat/","date":"2026-10-27","date_precision":"day","short_title":"R(3,3,3,3) ≤ 61 with AI, Lean and SAT","deck":null,"takeaway":"Milos Tatarevic proves that every 4-colouring of the edges of K₆₁ has a monochromatic triangle, so R(3,3,3,3) ≤ 61. The bound had been 62 since Fettes, Kramer and Radziszowski in 2004, and the best lower bound is 51. The proof combines a double-counting argument, classifications of small neighbourhoods and 56,723 SAT formulas, each refuted by both Kissat and CaDiCaL. A Lean 4 formalization checks the reductions but not the SAT part. The paper says AI tools 'assisted extensively': GPT-6 Astra in Codex contributed one of the reductions, and GPT-5.6 in Codex and Fable 5 in Claude Code the others. It is an unrefereed preprint, and no expert comment was found as of 29 Oct.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Impossible Research","OpenAI","Anthropic"]},{"id":"2026-10-27-uppcc-not-closed-under-intersection-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-27-uppcc-not-closed-under-intersection-gpt-6-astra/","date":"2026-10-27","date_precision":"day","short_title":"Unbounded-error communication is not closed under AND, found with GPT-6 Astra","deck":null,"takeaway":"Unbounded-error communication complexity (UPP^cc) is the communication analogue of PP. Beigel, Reingold and Spielman proved in 1995 that PP is closed under intersection. Whether UPP^cc is closed under intersection had been asked since Halstenberg and Reischuk (1988). In a 27 Oct preprint, Mark Bun and Mandar Juvekar (Boston University) give explicit matrices of sign rank Θ(d) whose AND has sign rank 2^Θ(d). The previous gap, by Bun, Mande and Thaler, was only log versus log². The paper began when they asked GPT-6 Astra how to beat that gap. 'To our (in retrospect, naïve) astonishment,' they write, the model 'discovered what appeared to be complete proofs'.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Boston University"]},{"id":"2026-10-27-ai-assisted-results-october-27-listing","url":"https://postcutoff.com/e/2026-10-27-ai-assisted-results-october-27-listing/","date":"2026-10-27","date_precision":"day","short_title":"AI-assisted results in the 27 Oct 2026 sweep","deck":"Muckenhoupt's C_p conjecture, Graham's rearrangement conjecture in abelian groups, a quasi-polynomial χ-bound for P₆-free graphs, Haagerup's 1975 problem, Langberg–Médard, and proofs whose main idea came from Claude or GPT","takeaway":"Tuesday's arXiv announcement (papers submitted 23–26 Oct) was the largest so far for AI-assisted mathematics. Four results have their own entries: Golomb–Welch, 3-coloring with O(n^0.1537) colors, Balko's exposition of OpenAI's Harary–Hill proof, and Ando–Umehara on Alpöge's smooth Carathéodory counterexamples. This page lists the next tier, from 46-year-old conjectures where ChatGPT supplied an alternative argument to short papers where Claude or GPT proposed the main idea.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-27","updated":"2026-10-27","orgs":["OpenAI","Anthropic","Google DeepMind","various researchers"]},{"id":"2026-10-26-openai-eth-false-subexponential-sat-claim","url":"https://postcutoff.com/e/2026-10-26-openai-eth-false-subexponential-sat-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims ETH is false","deck":null,"takeaway":"Anthropic independently posted a different Lean-checked refutation the next day (separate entry). OpenAI's model output claims a deterministic algorithm that counts the satisfying assignments of any k-CNF in time 2^(n/(log n)^(1/2+o(1))). If correct, it refutes ETH, SETH, Gap-ETH and NSETH, the assumptions behind hundreds of fine-grained hardness results. OpenAI says the main theorem is formalized in Lean. A working group led by ETH's proposers, Russell Impagliazzo and Ramamohan Paturi, with Ryan Williams, will examine the proof. No expert has assessed it in public yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":35,"official":11,"filed":"2026-10-26","updated":"2026-11-08","orgs":["OpenAI"]},{"id":"2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim","url":"https://postcutoff.com/e/2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims local smoothing and Kakeya in all dimensions","deck":null,"takeaway":"A 59-page OpenAI manuscript dated 25 Oct claims Sogge's local smoothing conjecture for the wave equation in every dimension n ≥ 2. The stated corollaries include the Bochner–Riesz conjecture, Fourier restriction throughout the interior of the conjectured range, the Kakeya and Nikodym maximal conjectures, and full Hausdorff dimension for every Euclidean Kakeya set. That last one is the Kakeya set conjecture, proved in ℝ³ only in 2025 (Wang–Zahl). OpenAI lists Lean comparator statements for the main estimate and the corollaries. No expert has assessed it yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":5,"filed":"2026-10-26","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-26-openai-yang-mills-qcd-existence-claim","url":"https://postcutoff.com/e/2026-10-26-openai-yang-mills-qcd-existence-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims 4D Yang–Mills and QCD existence","deck":null,"takeaway":"The physics third of OpenAI's 26 Oct announcement claims that certain interacting 4D quantum field theories exist mathematically, among them pure Yang–Mills and quantum chromodynamics. That is the existence half of the Yang–Mills Millennium problem, not the mass gap. A group led by David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, is reviewing it. At the group's request the AI-produced material is not public while they write their own account. Nothing can be checked yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":18,"official":3,"filed":"2026-10-27","updated":"2026-11-13","orgs":["OpenAI"]},{"id":"2026-10-26-three-coloring-n-0-1537-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-26-three-coloring-n-0-1537-gpt-6-astra/","date":"2026-10-26","date_precision":"day","short_title":"3-colorable graphs: O(n^0.1537) colors with GPT-6 Astra","deck":null,"takeaway":"Ken-ichi Kawarabayashi, Kaisei Deguchi, Hirotaka Yoneda and Masataka Yoneda (NII / University of Tokyo) give a polynomial-time algorithm that colors any 3-colorable graph with O(n^0.1537) colors. The previous best was O(n^0.1954) (Bansal–Huang–Lee, SODA '27). By the paper's account, GPT-6 Astra and GPT-6 Pro generated each intermediate algorithm (n^0.1736, n^0.1712, n^(1/6+ε), n^0.16, then n^0.1537), with the authors suggesting ideas and simplifying the analyses. The humans wrote the text.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-27","updated":"2026-10-27","orgs":["National Institute of Informatics","University of Tokyo","OpenAI"]},{"id":"2026-10-26-three-dimensional-su2-yang-mills-continuum-limit-aronow-lopatto-ai","url":"https://postcutoff.com/e/2026-10-26-three-dimensional-su2-yang-mills-continuum-limit-aronow-lopatto-ai/","date":"2026-10-26","date_precision":"day","short_title":"3D SU(2) Yang–Mills continuum limit claimed in five AI-written papers","deck":null,"takeaway":"Building Yang–Mills theory as a rigorous continuum measure is open even in three dimensions, where the theory is superrenormalizable. On 26 Oct P. M. Aronow (Yale) and Patrick Lopatto (UNC Chapel Hill) posted five Hexagon papers. They claim that flowed Wilson loops of 3D SU(2) lattice Yang–Mills on a torus converge as the lattice spacing goes to zero, to a unique measure on the Cao–Chatterjee state space. The proof is a Bałaban-style multiscale renormalization-group analysis. The provenance statement says proof discovery, verification and writing were done by Opus 5.5, GPT 6 Astra and GPT 6.1 Sol, with only 'limited background formalizations' in Lean. Hexagon records human understanding as 'no parts'. It is not the four-dimensional Clay problem. We found no expert check as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","OpenAI","Yale University","University of North Carolina at Chapel Hill"]},{"id":"2026-10-26-ai-assisted-results-october-26-listing","url":"https://postcutoff.com/e/2026-10-26-ai-assisted-results-october-26-listing/","date":"2026-10-26","date_precision":"day","short_title":"AI-assisted results in the 26 Oct 2026 sweep","deck":"Claude-written Lean proof of a frame-indifferent rank-one convex function that is not quasiconvex, a sculptor and Claude Code prove Takemura's conjecture, Claude's contribution to deterministic root-finding, and Zenodo records","takeaway":"Monday's arXiv announcement (papers submitted 22–23 Oct) brought four results with their own entries: OpenAI's propeller conjecture reproved by Heilman, Tokareva's conjecture disproved by MiniMax agents, the triangle lower tail with GPT-6 Astra and Claude, and graph-connectivity communication complexity with ChatGPT. This page lists the next tier, from Lean proofs written entirely by Claude to papers where the AI only checked the text.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":22,"official":22,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Anthropic","OpenAI","various researchers"]},{"id":"2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt","url":"https://postcutoff.com/e/2026-10-25-golomb-welch-conjecture-perfect-lee-codes-chatgpt/","date":"2026-10-25","date_precision":"day","short_title":"Golomb–Welch conjecture (perfect Lee codes) claimed, with ChatGPT","deck":null,"takeaway":"Bocong Chen, Jing Huang and Gaojun Luo posted two arXiv papers on 25 Oct 2026. They claim that Lee balls of radius r ≥ 2 never tile ℤⁿ for n ≥ 3, first for lattice tilings and then for arbitrary centre sets. That is the full Golomb–Welch conjecture from around 1970, the central open problem on perfect Lee codes. Both papers say ChatGPT helped explore proof strategies and check calculations. They are unrefereed, and no expert has commented yet.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-27","updated":"2026-10-27","orgs":["South China University of Technology","Guangzhou University","Nanjing University of Aeronautics and Astronautics","OpenAI"]},{"id":"2026-10-25-formula-minimization-sigma2p-complete-gpt-sol","url":"https://postcutoff.com/e/2026-10-25-formula-minimization-sigma2p-complete-gpt-sol/","date":"2026-10-25","date_precision":"day","short_title":"Formula minimization Σ₂ᵖ-complete (proof found by GPT-Sol-6.1)","deck":null,"takeaway":"Reijo Jaakkola posted a short arXiv note on 25 Oct 2026 proving that the Minimum Equivalent Expression problem is Σ₂ᵖ-complete under polynomial-time many-one reductions. The problem asks whether a Boolean formula has an equivalent formula with at most k variable occurrences, and it has been on Garey and Johnson's open-problem list since 1979. Buchfuhrer and Umans had proved completeness only under Turing reductions, in 2008. The note says the proof 'was discovered by GPT-Sol-6.1', then rewritten by Opus 5.5. It is unrefereed and has no formal proof.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-28","updated":"2026-10-28","orgs":["OpenAI","Anthropic"]},{"id":"2026-10-25-ai-assisted-results-october-25-listing","url":"https://postcutoff.com/e/2026-10-25-ai-assisted-results-october-25-listing/","date":"2026-10-25","date_precision":"day","short_title":"AI-assisted results in the 25 Oct 2026 sweep","deck":"GPT-5.5 Thinking steers a neutral-atom tweezer array, GBClaw grain-boundary agent, ChatGPT Astra-built bone-marrow model","takeaway":"No new arXiv announcement on Sunday. Three leftover 17–20 Oct preprints give an AI model a working role in a physics experiment, a materials simulation campaign and a biomedical model. None is a famous-problem result.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-25","updated":"2026-10-25","orgs":["OpenAI","various researchers"]},{"id":"2026-10-24-ai-assisted-results-october-24-listing","url":"https://postcutoff.com/e/2026-10-24-ai-assisted-results-october-24-listing/","date":"2026-10-24","date_precision":"day","short_title":"AI-assisted results in the 24 Oct 2026 sweep","deck":"ChatGPT 6 Astra removes a Hodge-conjecture assumption, Bake AI's Épi agent, Google's lab-in-the-loop aptamer agents, agentic turbulence closures","takeaway":"No new arXiv announcement on Saturday; this page clears the backlog of AI-disclosing papers from 16–22 Oct that earlier listings did not cite. Two got their own entries: Nakayama's local height note, outlined by ChatGPT 6 Astra, and Bake AI's Épi agent.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":8,"filed":"2026-10-24","updated":"2026-10-24","orgs":["OpenAI","Google Research","Google DeepMind","Anthropic","Bake AI","various researchers"]},{"id":"2026-10-23-ai-assisted-results-october-23-listing","url":"https://postcutoff.com/e/2026-10-23-ai-assisted-results-october-23-listing/","date":"2026-10-23","date_precision":"day","short_title":"AI-assisted results in the 23 Oct 2026 arXiv listing","deck":"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","takeaway":"Friday's arXiv announcement brought a disproof of Barry Simon's L² conjecture and the Critical 2D KPZ process (both with ChatGPT), an expert diagnosis of OpenAI's withdrawn Hodge papers and a hand-written version of OpenAI's Barnette proof. This page lists the next tier.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-23","updated":"2026-10-23","orgs":["Google DeepMind","OpenAI","Anthropic","Harmonic","various mathematicians"]},{"id":"2026-10-23-graph-connectivity-nondeterministic-communication-complexity-chatgpt","url":"https://postcutoff.com/e/2026-10-23-graph-connectivity-nondeterministic-communication-complexity-chatgpt/","date":"2026-10-23","date_precision":"day","short_title":"Graph connectivity communication bound settled, with ChatGPT","deck":null,"takeaway":"Zhijun Zhang (INSAIT) proves an optimal Ω(N log N) nondeterministic lower bound for graph connectivity, up from Raz–Spieker's Ω(N log log N) of 1995. He also gives near-optimal randomized and optimal r-round bounds. The paper's AI section says ChatGPT discovered the edge-colouring argument for Lemma 4.6 and produced a weaker first version of the randomized bound.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-26","updated":"2026-10-26","orgs":["INSAIT","OpenAI"]},{"id":"2026-10-23-propeller-conjecture-openai-proof-heilman-simplification","url":"https://postcutoff.com/e/2026-10-23-propeller-conjecture-openai-proof-heilman-simplification/","date":"2026-10-23","date_precision":"day","short_title":"Heilman simplifies OpenAI's propeller conjecture proof","deck":null,"takeaway":"OpenAI's 6 Oct release contained a Lean-formalized proof of the Gaussian propeller conjecture (Khot–Naor 2008) in every dimension, a claim our dataset had missed. On 23 Oct Steven Heilman (USC), co-author of the 2012 three-dimensional case, posted a shorter version that also settles the equality cases and removes the computer search from the 3D step. It is the first expert follow-up to treat the conjecture as proved.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-10-26","updated":"2026-10-26","orgs":["OpenAI","University of Southern California"]},{"id":"2026-10-23-tokareva-bent-functions-conjecture-disproved-minimax-agents","url":"https://postcutoff.com/e/2026-10-23-tokareva-bent-functions-conjecture-disproved-minimax-agents/","date":"2026-10-23","date_precision":"day","short_title":"Tokareva's bent-function conjecture disproved by LLM agents","deck":null,"takeaway":"Every Boolean function of degree ≤ n/2 is a sum of two bent functions: that conjecture of Natalia Tokareva, listed among Carlet's open problems, fails in every even dimension n ≥ 8. The counterexample x₁x₂(x₃x₄+x₅x₆+x₇x₈) was found by 19 rounds of up to 100 MiniMax M3.1-Flash-Preview agents (1,889 reports) with separate audit rounds. Two key equations are checked in Lean.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-26","updated":"2026-10-26","orgs":["MiniMax","NSUCRYPTO"]},{"id":"2026-10-23-triangle-lower-tail-zhao-conjecture-gpt-6-astra-claude","url":"https://postcutoff.com/e/2026-10-23-triangle-lower-tail-zhao-conjecture-gpt-6-astra-claude/","date":"2026-10-23","date_precision":"day","short_title":"Triangle lower tail: Zhao's conjecture settled with GPT-6 Astra and Claude","deck":null,"takeaway":"Dong Huang and Shuangping Li solve the graphon variational problem behind the lower tail of triangle counts in G(n,p) for n^(-1/2) ≪ p ≪ 1. Minimizers are constant or balanced two-block graphons, with a first-order phase transition at η* = 0.009217…. The authors say GPT-6 Astra helped find key steps and Claude Opus 5.5 helped build the computer-assisted proof. The arguments are formalized in Lean 4.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Tsinghua University","Yale University","OpenAI","Anthropic"]},{"id":"2026-10-22-critical-2d-kpz-process-chatgpt","url":"https://postcutoff.com/e/2026-10-22-critical-2d-kpz-process-chatgpt/","date":"2026-10-22","date_precision":"day","short_title":"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","deck":null,"takeaway":"A long-sought object of stochastic analysis, a non-trivial solution of the two-dimensional KPZ equation at criticality, was built by a team of leading probabilists who say ChatGPT contributed the tightness argument, the convergence proof and the proof of non-Gaussianity under their direction.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-23","updated":"2026-10-23","orgs":["OpenAI"]},{"id":"2026-10-22-simon-l2-conjecture-disproved-chatgpt","url":"https://postcutoff.com/e/2026-10-22-simon-l2-conjecture-disproved-chatgpt/","date":"2026-10-22","date_precision":"day","short_title":"Barry Simon's L² conjecture (Problem 8 of his 2000 list) disproved","deck":"Becker and Yang build a counterexample largely found with ChatGPT","takeaway":"One of Barry Simon's fifteen problems for Schrödinger operators, open since 2000, is false: a small, smooth potential satisfying his L² condition can leave no continuous spectrum at all. The authors say ChatGPT produced the key mechanism and, 'very much to our surprise', extended it to the full spectrum.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-23","updated":"2026-10-23","orgs":["OpenAI"]},{"id":"2026-10-22-ai-assisted-results-october-22-listing","url":"https://postcutoff.com/e/2026-10-22-ai-assisted-results-october-22-listing/","date":"2026-10-22","date_precision":"day","short_title":"AI-assisted results in the 22 Oct 2026 arXiv listing","deck":"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","takeaway":"Thursday's arXiv announcement brought experts' write-ups of two claims from OpenAI's 6 Oct release (Nagata's conjecture, and a Hochster counterexample surveyed by Anghel), a human exposition of the Kozma–Nitzan proof, and a 3D counterexample to Gale's conjecture found with GPT-6 Astra. This page lists the next tier.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":13,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-10-22-microsoft-mai-dxo-science-sequential-diagnosis","url":"https://postcutoff.com/e/2026-10-22-microsoft-mai-dxo-science-sequential-diagnosis/","date":"2026-10-22","date_precision":"day","short_title":"Microsoft MAI-DxO diagnosis paper in Science","deck":null,"takeaway":"The result is now peer-reviewed, and it is model-agnostic: the orchestrator lifts every frontier model tested.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":1,"filed":"2026-10-23","updated":"2026-10-26","orgs":["Microsoft"]},{"id":"2026-10-22-nakayama-local-height-without-hodge-conjecture-chatgpt-astra","url":"https://postcutoff.com/e/2026-10-22-nakayama-local-height-without-hodge-conjecture-chatgpt-astra/","date":"2026-10-22","date_precision":"day","short_title":"ChatGPT 6 Astra outlines the argument that removes the Hodge-conjecture assumption from Nakayama's local height asymptotics","deck":null,"takeaway":"It is another case of a model doing the key step of an arithmetic-geometry argument, with the human doing details and writing.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-24","updated":"2026-10-24","orgs":["OpenAI"]},{"id":"2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition","url":"https://postcutoff.com/e/2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition/","date":"2026-10-21","date_precision":"day","short_title":"Nagata's 1959 conjecture","deck":"Algebraic geometer Jenia Tevelev writes out OpenAI's 'remarkably simple' proof and calls the release 'a stunning achievement'","takeaway":"Nagata's conjecture on plane curves through very general points, open since 1959, was one of the claims in OpenAI's 6 Oct maths release. A UMass Amherst specialist has now reorganized, simplified and published the full proof himself, while criticizing how the manuscripts were dumped on the community.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","UMass Amherst"]},{"id":"2026-10-21-barnette-conjecture-openai-proof-zhentao-li-exposition","url":"https://postcutoff.com/e/2026-10-21-barnette-conjecture-openai-proof-zhentao-li-exposition/","date":"2026-10-21","date_precision":"day","short_title":"Barnette's 1969 Hamiltonicity conjecture","deck":"Graph theorist Zhentao Li publishes a digested, partly re-proved version of OpenAI's proof","takeaway":"OpenAI's 6 Oct release claimed Barnette's conjecture (every cubic 3-connected bipartite planar graph is Hamiltonian), with a Lean proof. A combinatorialist has now written the argument out by hand, replacing its exponential-sum step with explicit sign-reversing involutions.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-10-23","updated":"2026-10-23","orgs":["OpenAI"]},{"id":"2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey","url":"https://postcutoff.com/e/2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey/","date":"2026-10-21","date_precision":"day","short_title":"Hochster's 1973 small Cohen–Macaulay conjecture","deck":"An OpenAI manuscript's counterexample is surveyed, re-proved and extended to every dimension by Cristian Anghel","takeaway":"OpenAI's 6 Oct release included a model-written counterexample to the domain form of Hochster's small Cohen–Macaulay conjecture, a central question of commutative algebra since 1973. A survey by Cristian Anghel, whose September preprints started this line of work, gives a complete proof of it and generalizes it to all dimensions.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","Simion Stoilow Institute of Mathematics of the Romanian Academy"]},{"id":"2026-10-21-ai-assisted-results-october-21-listing","url":"https://postcutoff.com/e/2026-10-21-ai-assisted-results-october-21-listing/","date":"2026-10-21","date_precision":"day","short_title":"AI-assisted results in the 21 Oct 2026 arXiv listing","deck":"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","takeaway":"Wednesday's arXiv announcement brought four results with their own entries: the MDS conjecture in even characteristic, the double-cap conjecture, the quasitrace/monotone completeness counterexamples and the Neumann–Zagier conjecture. An expert verification of Claude's Hadamard matrices was added to that entry. This page lists the next tier, about thirty papers.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":18,"official":18,"filed":"2026-10-21","updated":"2026-10-21","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-10-21-gpt-rosalind-agents-perry-syndrome-lead","url":"https://postcutoff.com/e/2026-10-21-gpt-rosalind-agents-perry-syndrome-lead/","date":"2026-10-21","date_precision":"day","short_title":"Thousands of GPT-Rosalind agents analyse millions of mouse brain cells in 32 minutes and propose a cholesterol-transport lead for Perry syndrome","deck":null,"takeaway":"It is a large public example of massively parallel agent science in biology.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","PerturbAI","Scripps Research"]},{"id":"2026-10-20-double-cap-conjecture-witsenhausen-lean-ai","url":"https://postcutoff.com/e/2026-10-20-double-cap-conjecture-witsenhausen-lean-ai/","date":"2026-10-20","date_precision":"day","short_title":"Double-cap conjecture claimed in all dimensions, with a separate AI-built Lean proof","deck":null,"takeaway":"If confirmed, a 52-year-old geometry problem falls in every dimension, and a machine-checked proof of the statement already exists.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-21","updated":"2026-10-29","orgs":["Universidade Federal do Ceará","Universidade de São Paulo","OpenAI"]},{"id":"2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean","url":"https://postcutoff.com/e/2026-10-20-mds-conjecture-even-characteristic-chatgpt-claude-lean/","date":"2026-10-20","date_precision":"day","short_title":"MDS conjecture settled in even characteristic","deck":"Two independent proofs, one checked in Lean with Claude's help, the other written with ChatGPT","takeaway":"If it holds, this is the biggest advance on the conjecture since Ball's prime-field theorem.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-21","updated":"2026-10-21","orgs":["South China University of Technology","Guangzhou University","OpenAI","Anthropic"]},{"id":"2026-10-20-ai-assisted-results-october-20-listing","url":"https://postcutoff.com/e/2026-10-20-ai-assisted-results-october-20-listing/","date":"2026-10-20","date_precision":"day","short_title":"AI-assisted results in the 20 Oct 2026 arXiv listing","deck":"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","takeaway":"Tuesday's arXiv announcement was the strongest maths batch yet. Seven results have their own entries, among them the Erdős–Szemerédi sunflower conjecture, Erdős–Hajnal for all paths and a non-sofic hyperbolic group. This page catalogues the next tier: about forty papers whose AI disclosures run from 'the proofs were found by ChatGPT-6 Astra' to proofreading.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-20","updated":"2026-10-20","orgs":["OpenAI","Anthropic","DeepSeek","AWS","various mathematicians"]},{"id":"2026-10-20-borodin-kostochka-threshold-4583-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-20-borodin-kostochka-threshold-4583-gpt-6-astra/","date":"2026-10-20","date_precision":"day","short_title":"Borodin–Kostochka conjecture claimed for maximum degree ≥ 4583 (down from 5.3 million), then ≥ 1000 (26 Oct), with GPT-6 Astra assisting","deck":null,"takeaway":"It is another case of a frontier model credited with the technical estimate-tuning in a probabilistic combinatorics proof on a named 1977 conjecture.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-20","updated":"2026-10-26","orgs":["Hong Kong University of Science and Technology","OpenAI"]},{"id":"2026-10-20-claude-scorpion-toxin-binders","url":"https://postcutoff.com/e/2026-10-20-claude-scorpion-toxin-binders/","date":"2026-10-20","date_precision":"day","short_title":"Claude designs binders against scorpion neurotoxins","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-21","updated":"2026-10-21","orgs":["Anthropic","Citrini Research"]},{"id":"2026-10-20-gale-conjecture-r3-counterexample-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-20-gale-conjecture-r3-counterexample-gpt-6-astra/","date":"2026-10-20","date_precision":"day","short_title":"Gale's 1953 conjecture fails in 3D","deck":"A 101-point set found with LLM code evolution and GPT-6 Astra beats the sphere's Borsuk partition ratio","takeaway":"Vsevolod Voronov built a 101-point set in space whose every split into four parts leaves a part of diameter above 0.88928, more than the sphere's 0.88807. That refutes Gale's 1953 conjecture in dimension 3. The search algorithms were developed with GPT-6 Astra, some proposed by the AI, and the set is checkable in exact arithmetic.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":2,"filed":"2026-10-22","updated":"2026-10-29","orgs":["OpenAI","Anthropic"]},{"id":"2026-10-19-erdos-szemeredi-sunflower-conjecture-gpt-6-astra-pro","url":"https://postcutoff.com/e/2026-10-19-erdos-szemeredi-sunflower-conjecture-gpt-6-astra-pro/","date":"2026-10-19","date_precision":"day","short_title":"Erdős–Szemerédi sunflower conjecture proved for every k ≥ 3","deck":"The key vector bound (Theorem 2) was found by ChatGPT 6 Astra Pro (Bucić, He, Lee)","takeaway":"A 1970s Erdős–Szemerédi conjecture is claimed proved for every k: families of subsets of [n] with no k-sunflower have size at most c_k^n with c_k < 2. Before this only k = 3 was known, through the 2016 cap-set breakthrough. The authors say ChatGPT 6 Astra Pro found the proof of the central theorem. The paper also says a Lean check exists, but it has not been released yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["University of Vienna","KAIST","IBS","OpenAI"]},{"id":"2026-10-19-integer-programming-linear-exponent-gpt-astra-ultra","url":"https://postcutoff.com/e/2026-10-19-integer-programming-linear-exponent-gpt-astra-ultra/","date":"2026-10-19","date_precision":"day","short_title":"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)","deck":null,"takeaway":"Ten days after their Subset Sum claim, the same Zhejiang trio claim to settle a basic question in integer programming. With d constraints and entries bounded by N, the running time exponent drops from O(d²) to linear in d. As before, the authors give the idea and say GPT Astra Ultra wrote the complete proofs.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["Zhejiang University","OpenAI"]},{"id":"2026-10-19-perfect-matchings-fpras-general-graphs-openai-liu-meng","url":"https://postcutoff.com/e/2026-10-19-perfect-matchings-fpras-general-graphs-openai-liu-meng/","date":"2026-10-19","date_precision":"day","short_title":"Counting perfect matchings in general graphs: OpenAI FPRAS, simplified proof","deck":null,"takeaway":"Approximately counting perfect matchings is easy for bipartite graphs (Jerrum–Sinclair–Vigoda, 2004) but had no polynomial-time scheme for general graphs. OpenAI's 23 Sept release (family 113) claimed a fully polynomial randomized approximation scheme for all graphs, built on a restricted Poincaré inequality for pairs of matchings, and a Lean development. On 19 Oct Chenghua Liu and Boning Meng posted 'a simplified, self-contained proof of OpenAI's FPRAS'. It keeps the same parameter bounds and avoids the original auxiliary vertices and tree constructions. The same release also claimed the Anari–Oveis Gharan–Vinzant perfect-matching entropy conjecture.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":4,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI"]},{"id":"2026-10-19-quasitrace-monotone-completeness-counterexamples-openai","url":"https://postcutoff.com/e/2026-10-19-quasitrace-monotone-completeness-counterexamples-openai/","date":"2026-10-19","date_precision":"day","short_title":"Kaplansky's quasitrace conjecture reported disproved by an unreleased OpenAI model","deck":"Experts reproduce it and refute the monotone completeness conjecture","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-21","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-19-ai-assisted-results-october-19-listing","url":"https://postcutoff.com/e/2026-10-19-ai-assisted-results-october-19-listing/","date":"2026-10-19","date_precision":"day","short_title":"AI-assisted results in the 19 Oct 2026 arXiv listing","deck":"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","takeaway":"Monday's arXiv announcement brought the first wave of human follow-up papers on OpenAI's October maths release, alongside a dozen routine AI disclosures. The strongest items have their own entries: Brennan, Smith–Toda, lonely runner and tournament Hamiltonian cycles.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":16,"official":16,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","Harmonic","various mathematicians"]},{"id":"2026-10-19-ai-assisted-results-zenodo-october-17-19","url":"https://postcutoff.com/e/2026-10-19-ai-assisted-results-zenodo-october-17-19/","date":"2026-10-19","date_precision":"day","short_title":"AI-assisted maths on Zenodo, 17–19 Oct 2026","deck":"A claimed proof of the 1985 Arad–Herzog conjecture, Messerschmidt's conjecture confirmed with Codex, and a Claude-run study on AI-written science","takeaway":"Zenodo, not arXiv, carried the weekend's AI-assisted results: an unknown author claims the 40-year-old Arad–Herzog conjecture on products of conjugacy classes with agentic OpenAI and Anthropic tools and a literature-relative Lean check, and a Bologna mathematician confirms Messerschmidt's 2018 conjecture with Codex. None is refereed.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":13,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Anthropic","various mathematicians"]},{"id":"2026-10-19-arveson-hyperrigidity-commutative-counterexamples-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-19-arveson-hyperrigidity-commutative-counterexamples-gpt-6-astra/","date":"2026-10-19","date_precision":"day","short_title":"Commutative counterexamples to Arveson's hyperrigidity conjecture","deck":"Proof developed with GPT-6 Astra, finite-dimensional example proposed by it (Bilich, Scherer)","takeaway":"Arveson's hyperrigidity conjecture had already failed for noncommutative C*-algebras. The commutative case, where the known mechanism cannot work, stayed open. Bilich and Scherer now give commutative counterexamples. GPT-6 Astra developed the 30-page initial proof and proposed the finite-dimensional example, and a 40-hour agent search found only an unverified smaller one.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-20","updated":"2026-10-20","orgs":["University of Haifa","Technion","OpenAI","Anthropic"]},{"id":"2026-10-19-fahr-face-cancer-survival-face-photos","url":"https://postcutoff.com/e/2026-10-19-fahr-face-cancer-survival-face-photos/","date":"2026-10-19","date_precision":"day","short_title":"FAHR-Face: Mass General Brigham foundation model estimates biological age and cancer survival from a face photo (Lancet Digital Health)","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":0,"filed":"2026-10-20","updated":"2026-10-20","orgs":["Mass General Brigham"]},{"id":"2026-10-19-lemniscate-irrationality-exponent-chudnovsky-chatgpt","url":"https://postcutoff.com/e/2026-10-19-lemniscate-irrationality-exponent-chudnovsky-chatgpt/","date":"2026-10-19","date_precision":"day","short_title":"Gregory Chudnovsky, with ChatGPT, claims the lemniscate period has irrationality exponent 2, with a Lean 4 proof built on OpenAI's π-exponent library","deck":null,"takeaway":"After OpenAI's claim that π has irrationality exponent 2, a well-known transcendence theorist carries the method to the lemniscate period Ω = 2∫₀¹(1−t⁴)^(−1/2) dt. He says ChatGPT helped with the exposition and the formalization, and he ships a complete Lean 4 proof built on OpenAI's library. Like the π claim, it has not been reviewed independently.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-20","updated":"2026-10-23","orgs":["OpenAI"]},{"id":"2026-10-19-neumann-zagier-conjecture-tree-lattices-ai-proof","url":"https://postcutoff.com/e/2026-10-19-neumann-zagier-conjecture-tree-lattices-ai-proof/","date":"2026-10-19","date_precision":"day","short_title":"Neumann–Zagier 1985 conjecture on tree lattices settled after an AI proof with ChatGPT 5.5 Pro and Opus 4.8","deck":null,"takeaway":"A problem posted as a benchmark task was solved with frontier models and then confirmed and improved by the experts who posed it.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-21","updated":"2026-10-21","orgs":["OpenAI","Anthropic"]},{"id":"2026-10-18-bake-ai-epi-autonomous-research-agent-120-records","url":"https://postcutoff.com/e/2026-10-18-bake-ai-epi-autonomous-research-agent-120-records/","date":"2026-10-18","date_precision":"day","short_title":"Bake AI's Épi agent reports 120 record results with GPT-6 Astra","deck":"Including a narrower n-queens constant and Cohn–Elkies non-sharpness in dimensions 10 and 11","takeaway":"It is one more of the AlphaEvolve-style record-hunting systems, built on OpenAI models by a third party.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-24","updated":"2026-10-24","orgs":["Bake AI","OpenAI"]},{"id":"2026-10-17-2d-euler-unforced-nonuniqueness-lp-vorticity-ai","url":"https://postcutoff.com/e/2026-10-17-2d-euler-unforced-nonuniqueness-lp-vorticity-ai/","date":"2026-10-17","date_precision":"day","short_title":"2D Euler non-uniqueness without force claimed three times, all with AI","deck":null,"takeaway":"Uniqueness of 2D Euler solutions is known for bounded vorticity (Yudovich, 1963). With a force, Vishik showed in 2018 that it fails for vorticity in L^p, p > 2. Without a force this stayed open. Within ten days, three groups posted proofs that it fails there as well. On 17 Oct Umberto Pappalettera and Massimo Sorella posted on Hexagon a manuscript 'developed autonomously by ChatGPT 6.1 Sol'. They claim no authorship and run a reading group to check it. On 22 Oct Matei Coiculescu (NYU) posted Part I of a two-part proof whose details, code and text were 'largely produced with Claude Opus 5.5'. His repository has a Lean 4 formalization of the whole proof, with no sorry. On 27 Oct Hyungjun Choi (Princeton) posted an 86-page construction, with ChatGPT completing technical steps. As of 29 Oct we found no expert assessment of any of the three.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":10,"official":7,"filed":"2026-10-29","updated":"2026-11-10","orgs":["OpenAI","Anthropic","Hexagon Mathematics Foundation"]},{"id":"2026-10-17-erdos-hajnal-all-paths-nguyen-astra-deepmind","url":"https://postcutoff.com/e/2026-10-17-erdos-hajnal-all-paths-nguyen-astra-deepmind/","date":"2026-10-17","date_precision":"day","short_title":"Erdős–Hajnal conjecture proved for every path P_t","deck":"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","takeaway":"Until now the Erdős–Hajnal property was known for paths only up to five vertices (P5, 2023). Oxford's Tung H. Nguyen claims it for every path, with GPT-6 Astra proving parts of the key lemmas. His paper says a Google DeepMind internal model got the P6 case and then all paths on its own, Lean-checked, and that two Claude Opus 5.5 manuscripts on GitHub claim P6 and all paths.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-20","updated":"2026-11-03","orgs":["University of Oxford","OpenAI","Google DeepMind","Google Research","Anthropic"]},{"id":"2026-10-17-non-sofic-hyperbolic-group-sandhu-thom","url":"https://postcutoff.com/e/2026-10-17-non-sofic-hyperbolic-group-sandhu-thom/","date":"2026-10-17","date_precision":"day","short_title":"A non-sofic word-hyperbolic group","deck":"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","takeaway":"Whether every hyperbolic group is sofic was a well-known open question in group theory. Two groups now say no. Both build on the torsion-free, non-residually-finite hyperbolic group in OpenAI's 6 October release. One is co-written by Andreas Thom, an expert in sofic groups. The other, by Douglas and Mghirbi, comes with a Lean formalization.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-20","updated":"2026-10-20","orgs":["TU Dresden","OpenAI"]},{"id":"2026-10-17-ai-assisted-results-october-16-listing","url":"https://postcutoff.com/e/2026-10-17-ai-assisted-results-october-16-listing/","date":"2026-10-17","date_precision":"day","short_title":"arXiv results with disclosed AI help, 16 Oct 2026 announcement","deck":"Suffridge's conjecture, two Wahl conjectures, the 6-D Tarski problem and more","takeaway":"A second read of the 16 Oct arXiv announcement, through the HTML listings because the arXiv API was rate-limited, found three results for their own entries and about fifty more preprints with explicit AI statements, several settling named conjectures.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":16,"official":15,"filed":"2026-10-17","updated":"2026-10-18","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-10-18-ai-assisted-results-october-17-listing","url":"https://postcutoff.com/e/2026-10-18-ai-assisted-results-october-17-listing/","date":"2026-10-17","date_precision":"day","short_title":"AI-assisted results, 16–17 Oct 2026 (Zenodo and arXiv leftovers)","deck":"Ramanujan's prime-counting inequality settled unconditionally with Claude, a Claude Code paper mill on the Hodge conjecture, and LLM agents in materials science","takeaway":"With no new arXiv announcement over the weekend, the notable AI-assisted items came from Zenodo, led by Padhi's Claude-assisted proof that Ramanujan's inequality holds unconditionally for all x ≥ 38,358,837,683.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":14,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Anthropic","various mathematicians"]},{"id":"2026-10-17-cma-fengqing-ai-ensemble-forecast-operational","url":"https://postcutoff.com/e/2026-10-17-cma-fengqing-ai-ensemble-forecast-operational/","date":"2026-10-17","date_precision":"day","short_title":"China's Fengqing AI ensemble weather forecast goes operational","deck":null,"takeaway":"Ensembles give forecasters probabilities, not just one answer.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":0,"filed":"2026-10-18","updated":"2026-10-18","orgs":["China Meteorological Administration"]},{"id":"2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov","url":"https://postcutoff.com/e/2026-10-16-brennan-conjecture-openai-proof-binder-ivrii-prause-smirnov/","date":"2026-10-16","date_precision":"day","short_title":"Brennan's conjecture: experts confirm and shorten OpenAI's proof","deck":null,"takeaway":"The first famous result from OpenAI's October maths release to be checked by leading experts in its field: four complex analysts, including Fields medallist Stanislav Smirnov, found no gap. They say the AI combined known techniques in a way 'missed by the human experts'.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":4,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","University of Toronto","Tel Aviv University","Åbo Akademi University","University of Geneva"]},{"id":"2026-10-16-gauss-circle-problem-209-670-bound","url":"https://postcutoff.com/e/2026-10-16-gauss-circle-problem-209-670-bound/","date":"2026-10-16","date_precision":"day","short_title":"Gauss circle problem: exponent 209/670, below 5/16","deck":null,"takeaway":"The error term in counting lattice points in a circle gets its first exponent below 5/16, a threshold Huxley had suggested might be the limit of the Bombieri–Iwaniec method; the authors disclose ChatGPT and Codex help with strategy and checking.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-17","updated":"2026-10-17","orgs":["OpenAI"]},{"id":"2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt","url":"https://postcutoff.com/e/2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt/","date":"2026-10-16","date_precision":"day","short_title":"Lonely runner: (1+c)/2n bound with ChatGPT 6","deck":null,"takeaway":"A human-led result on the 1967 lonely runner problem in which ChatGPT 6 Pro proposed the GCD-sum route, pointed to Green–Walker and fixed gaps, but 'the models were unable to improve the lonely runner bound on their own'.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","University of Chinese Academy of Sciences","Ohio State University"]},{"id":"2026-10-16-max-hamiltonian-cycles-tournaments-gpt-5-6-wormald","url":"https://postcutoff.com/e/2026-10-16-max-hamiltonian-cycles-tournaments-gpt-5-6-wormald/","date":"2026-10-16","date_precision":"day","short_title":"Tournament Hamiltonian cycles: GPT-5.6 settles Alon's question","deck":null,"takeaway":"Princeton's Divoux and Wang asked GPT-5.6 for a lower bound; a second instance turned the method into the matching upper bound, closing a line of work that runs from Szele's 1943 paper through Alon (1990) and Friedgut–Kahn.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Princeton University"]},{"id":"2026-10-16-openai-2d-area-law-lean-formalized-axiom","url":"https://postcutoff.com/e/2026-10-16-openai-2d-area-law-lean-formalized-axiom/","date":"2026-10-16","date_precision":"day","short_title":"OpenAI's 2D area law proof Lean-verified by Axiom","deck":null,"takeaway":"One of the physics claims in OpenAI's October maths release, an entropy area law for gapped 2D lattice ground states (open since Hastings' 1D proof in 2007), now has a complete Lean 4 proof with a frozen statement layer and comparator check, produced by AI agents under a human auditor.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Axiom Math","OpenAI"]},{"id":"2026-10-16-smith-toda-v4-angeltveit-improves-openai","url":"https://postcutoff.com/e/2026-10-16-smith-toda-v4-angeltveit-improves-openai/","date":"2026-10-16","date_precision":"day","short_title":"Smith–Toda V(4) for p ≥ 11, after OpenAI's p = 1009","deck":null,"takeaway":"A homotopy theorist checked two Smith–Toda papers from OpenAI's maths release, found no errors but 'strange choices' (a suspiciously high prime, needless ultrafilters), and widened the result from one prime to all primes p ≥ 11.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-19","updated":"2026-10-19","orgs":["OpenAI","Australian National University"]},{"id":"2026-10-16-yang-mills-claude-balaban-program-alpoge-sussman","url":"https://postcutoff.com/e/2026-10-16-yang-mills-claude-balaban-program-alpoge-sussman/","date":"2026-10-16","date_precision":"day","short_title":"Alpöge and Sussman say Claude has an (unreadable) existence argument for 4D Yang–Mills on ℝ⁴, by finishing Bałaban's 1980s program","deck":null,"takeaway":"A public repo of raw Claude output claims a construction of four-dimensional SU(2) Yang–Mills theory, the existence half of the Millennium problem but not the mass gap; its authors call it 'slop in progress' that nobody can yet read or check.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Anthropic","Northwestern University"]},{"id":"2026-10-16-ai-assisted-results-october-15-listing","url":"https://postcutoff.com/e/2026-10-16-ai-assisted-results-october-15-listing/","date":"2026-10-16","date_precision":"day","short_title":"arXiv results with disclosed AI help, 15 Oct 2026 listing","deck":"Monodromy conjecture, Fried counterexamples, Godefroy–Lancien–Zizler question and more","takeaway":"One day's maths listing brought five results big enough for their own entries (Smale's mean value constant, Hopf with T³, class numbers ≤ 2000, the Delsarte rate, LLM-found quantum circuits) and about two dozen more with explicit AI statements.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":12,"filed":"2026-10-16","updated":"2026-10-16","orgs":["OpenAI","Anthropic","various mathematicians"]},{"id":"2026-10-16-das-sarma-exciton-superfluidity-claude-opus-codex-referee","url":"https://postcutoff.com/e/2026-10-16-das-sarma-exciton-superfluidity-claude-opus-codex-referee/","date":"2026-10-16","date_precision":"day","short_title":"Das Sarma paper written with Claude Opus 5.5, refereed by Codex","deck":null,"takeaway":"A leading condensed-matter theorist discloses that the analytic derivations, numerics, figures and text of most of a long theory paper were produced by Claude under his direction, with a second AI as internal referee.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-17","updated":"2026-10-17","orgs":["University of Maryland","Anthropic","OpenAI"]},{"id":"2026-10-16-lee-loh-sudakov-directed-cuts-altifina-lean","url":"https://postcutoff.com/e/2026-10-16-lee-loh-sudakov-directed-cuts-altifina-lean/","date":"2026-10-16","date_precision":"day","short_title":"Lee–Loh–Sudakov directed-cut conjecture proved with AI, Lean-verified","deck":null,"takeaway":"A 2013 conjecture on judicious partitions of digraphs, answering Scott's 2005 question for large graphs, was first proved by an AI research framework; the authors say the proofs are Lean-verified.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-17","updated":"2026-10-17","orgs":["Altifina","Charles University","OpenAI","Anthropic"]},{"id":"2026-10-16-sub-riemannian-sard-equiregular-counterexample-claude","url":"https://postcutoff.com/e/2026-10-16-sub-riemannian-sard-equiregular-counterexample-claude/","date":"2026-10-16","date_precision":"day","short_title":"Equiregular counterexample to the sub-Riemannian Sard conjecture","deck":"Claude Opus 5/5.5 wrote the construction and first proofs, Lean-formalized with Codex (Wang, Zelenko)","takeaway":"Lerario, Rizzi and Tiberio had disproved the sub-Riemannian Sard conjecture with a degenerate distribution, which left the natural equiregular case open. Wang and Zelenko close it with a rank-two distribution in dimension nine. Zelenko proposed the strategy; Claude generated the explicit structure and first proofs; the main theorems are fully formalized in Lean.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-20","updated":"2026-10-20","orgs":["Texas A&M University","Anthropic","OpenAI"]},{"id":"2026-10-15-delsarte-lp-bound-exact-rate-salmon-opus","url":"https://postcutoff.com/e/2026-10-15-delsarte-lp-bound-exact-rate-salmon-opus/","date":"2026-10-15","date_precision":"day","short_title":"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","deck":null,"takeaway":"A coding-theory question open since the 1970s, what Delsarte's LP bound converges to, now has a closed answer in elliptic integrals; the author credits Opus 5.5 with the step that made the lower bound work.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-16","updated":"2026-10-16","orgs":["Anthropic","OpenAI"]},{"id":"2026-10-15-hopf-conjecture-t3-symmetry-chatgpt-astra","url":"https://postcutoff.com/e/2026-10-15-hopf-conjecture-t3-symmetry-chatgpt-astra/","date":"2026-10-15","date_precision":"day","short_title":"Hopf's Euler characteristic conjecture proved under T³ symmetry","deck":"The first proof came from ChatGPT-6-Astra 'with minimal human input'","takeaway":"The symmetry needed for Hopf's 1930s conjecture drops from a 4-torus to a 3-torus; the paper says the starting proof of its main theorem was obtained with GPT-6 Astra.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-16","updated":"2026-10-16","orgs":["OpenAI","University of Münster"]},{"id":"2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean","url":"https://postcutoff.com/e/2026-10-15-imaginary-quadratic-fields-class-number-2000-claude-lean/","date":"2026-10-15","date_precision":"day","short_title":"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","deck":null,"takeaway":"The first downstream theorem built on OpenAI's claimed exclusion of Landau–Siegel zeros: the unconditional class-number table jumps from h ≤ 100 (Watkins, 2004) to h ≤ 2000, with the key bounds checked in Lean.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-16","updated":"2026-10-16","orgs":["OpenAI","Anthropic"]},{"id":"2026-10-15-cern-mistral-large-physics-model","url":"https://postcutoff.com/e/2026-10-15-cern-mistral-large-physics-model/","date":"2026-10-15","date_precision":"day","short_title":"CERN and Mistral plan a Large Physics Model","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":2,"filed":"2026-10-16","updated":"2026-10-19","orgs":["CERN","Mistral AI"]},{"id":"2026-10-15-ghys-conjecture-codimension-one-intern-s2-lean","url":"https://postcutoff.com/e/2026-10-15-ghys-conjecture-codimension-one-intern-s2-lean/","date":"2026-10-15","date_precision":"day","short_title":"Ghys's 1995 conjecture proved in the codimension-one case with Shanghai AI Lab's Intern-S2","deck":"Peking University formalizes it in Lean (~550k lines)","takeaway":"Jiesong Zhang's preprint proves that every codimension-one holomorphic Anosov diffeomorphism of a compact complex manifold is conjugate to a hyperbolic torus automorphism, settling Ghys's conjecture in that case and in complex dimension three. It declares that an internal Intern-S2 helped develop the ideas; a Peking University team formalized the main theorems in Lean with its SHEAF agent.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":7,"official":4,"filed":"2026-10-15","updated":"2026-10-15","orgs":["Shanghai AI Laboratory","Peking University"]},{"id":"2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding","url":"https://postcutoff.com/e/2026-10-14-external-memory-integer-sorting-beats-comparison-bound-network-coding/","date":"2026-10-14","date_precision":"day","short_title":"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","deck":null,"takeaway":"A week after the Li–Li network coding conjecture fell, two of its authors turned the refutation into an algorithm: a deterministic external-memory sort of Θ(log N)-bit integers that uses asymptotically fewer block transfers than any comparison sort in some parameter ranges. ChatGPT and Codex 'materially affected' the constructions and analyses.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-15","updated":"2026-10-15","orgs":["University of Toronto","Tsinghua University","OpenAI"]},{"id":"2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu","url":"https://postcutoff.com/e/2026-10-14-hardy-littlewood-chowla-mobius-shifted-primes-lu-wang-wu/","date":"2026-10-14","date_precision":"day","short_title":"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","deck":null,"takeaway":"Three analytic number theorists at Shandong University claim the Möbius function cancels on shifted primes p + 2, a folklore case of the Hardy–Littlewood–Chowla conjecture and a textbook example of the sieve parity barrier. Their method follows an OpenAI Math Release preprint, which ChatGPT helped them understand.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-15","updated":"2026-10-15","orgs":["Shandong University","OpenAI"]},{"id":"2026-10-14-penrose-conjecture-bray-khuri-claude-openai-provenance","url":"https://postcutoff.com/e/2026-10-14-penrose-conjecture-bray-khuri-claude-openai-provenance/","date":"2026-10-14","date_precision":"day","short_title":"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","deck":null,"takeaway":"Leading experts on the Penrose conjecture posted a special-case proof built with Claude's help. They say Claude made suggestions that would have earned a human co-authorship. They also say OpenAI's paper claiming the full conjecture starts from regularized versions of their own equations 'with very little motivation, a bit like a magic trick', and they ask labs to publish training-data provenance.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-15","updated":"2026-10-15","orgs":["Duke University","Stony Brook University","Anthropic","OpenAI"]},{"id":"2026-10-14-ai-assisted-results-mid-october-2026","url":"https://postcutoff.com/e/2026-10-14-ai-assisted-results-mid-october-2026/","date":"2026-10-14","date_precision":"day","short_title":"Mid-October 2026: arXiv results with disclosed AI help","deck":null,"takeaway":"The 13 Oct arXiv listing alone brought a human write-up of GPT-6 Astra's rational exponents proof, Ulam's Lie group question answered with an Astra construction, a better-than-2 k-median algorithm, and about a dozen smaller results whose AI statements range from 'suggested by ChatGPT' to 'GPT 6 Pro was used … to obtain all other new results'.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":20,"official":19,"filed":"2026-10-14","updated":"2026-10-15","orgs":["OpenAI","Anthropic","various mathematicians"]},{"id":"2026-10-14-baker-lab-de-novo-designed-proteases-nature","url":"https://postcutoff.com/e/2026-10-14-baker-lab-de-novo-designed-proteases-nature/","date":"2026-10-14","date_precision":"day","short_title":"AI-designed proteases (Baker lab, Nature)","deck":null,"takeaway":"Earlier de novo enzyme design worked mainly on easy reactions.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":3,"filed":"2026-10-14","updated":"2026-10-14","orgs":["University of Washington","Institute for Protein Design"]},{"id":"2026-10-14-microsoft-probabilistic-bias-correction-subseasonal-forecasts","url":"https://postcutoff.com/e/2026-10-14-microsoft-probabilistic-bias-correction-subseasonal-forecasts/","date":"2026-10-14","date_precision":"day","short_title":"Microsoft PBC improves sub-seasonal weather forecasts (Nature)","deck":null,"takeaway":"Two-to-six-week forecasts matter for farming, water and disaster planning.","category":"science","category_label":"Science & math","importance":2,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-10-15","updated":"2026-10-15","orgs":["Microsoft Research","ECMWF"]},{"id":"2026-10-13-rational-exponents-conjecture-proof-paper-conlon","url":"https://postcutoff.com/e/2026-10-13-rational-exponents-conjecture-proof-paper-conlon/","date":"2026-10-13","date_precision":"day","short_title":"Eight extremal combinatorialists write up GPT-6 Astra's proof of the Erdős–Simonovits rational exponents conjecture (Conlon, Janzer, Liu et al.)","deck":null,"takeaway":"The rational exponents proof that GPT-6 Astra found during Epoch AI's FrontierMath Erdős runs now has a human-written paper by leading experts in the area, who explain and situate it and call it one of 'the recent AI-driven breakthroughs'.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-10-14","updated":"2026-10-14","orgs":["Caltech","EPFL","KAIST","IBS","Emory University","Epoch AI","OpenAI"]},{"id":"2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition","url":"https://postcutoff.com/e/2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition/","date":"2026-10-13","date_precision":"day","short_title":"Thompson's group F is nonamenable","deck":"Specialists Sang-hyun Kim and Yash Lodha publish a human-written version of OpenAI's proof","takeaway":"The amenability of Thompson's group F, open since the 1970s, was one of the headline claims of OpenAI's 6 Oct math release. Two experts on groups of piecewise linear maps have now written the proof out themselves and generalized it, without AI.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-10-14","updated":"2026-10-19","orgs":["OpenAI","KIAS","Purdue University"]},{"id":"2026-10-13-ulam-problem-lie-groups-countably-representable-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-13-ulam-problem-lie-groups-countably-representable-gpt-6-astra/","date":"2026-10-13","date_precision":"day","short_title":"Ulam's question answered","deck":"Every Lie group is a group of permutations of a countable set (Monod & Thom; key construction from GPT-6 Astra)","takeaway":"Nicolas Monod and Andreas Thom prove that every Lie group embeds in the permutation group of a countable set, answering a classical question of Ulam. They say the key deformation was first built by GPT-6 Astra over three weeks of discussions; Thom earlier criticised how OpenAI credited human work.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":1,"filed":"2026-10-14","updated":"2026-10-14","orgs":["EPFL","TU Dresden","OpenAI"]},{"id":"2026-10-13-ai-assisted-results-october-2026","url":"https://postcutoff.com/e/2026-10-13-ai-assisted-results-october-2026/","date":"2026-10-13","date_precision":"day","short_title":"October 2026: the arXiv stream of conjectures settled with disclosed AI help continues","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":17,"official":17,"filed":"2026-10-13","updated":"2026-10-29","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-10-13-k-median-better-than-2-approximation-epfl","url":"https://postcutoff.com/e/2026-10-13-k-median-better-than-2-approximation-epfl/","date":"2026-10-13","date_precision":"day","short_title":"k-median breaks the factor-2 barrier","deck":"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","takeaway":"A human-proved 1.95-LMP algorithm, with a companion paper, gives a better-than-2 approximation for metric k-median. GPT-6 Astra improved the ratio of the same method to 1.844 and proved the method cannot go below 1.8, with Lean certificates. The paper says OpenAI announced other better-than-2 algorithms on 6 Oct.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-14","updated":"2026-10-15","orgs":["EPFL","Google Research","Carnegie Mellon University","New York University","OpenAI"]},{"id":"2026-10-12-bloch-constant-conjecture-disproved","url":"https://postcutoff.com/e/2026-10-12-bloch-constant-conjecture-disproved/","date":"2026-10-12","date_precision":"day","short_title":"Bloch constant conjecture (Ahlfors–Grunsky, 1937) disproved","deck":"Teng Zhang shows B < 0.47186… − 1.3×10⁻⁹, with a Lean proof generated by Codex","takeaway":"The value Ahlfors and Grunsky conjectured for Bloch's constant in 1937 is not the true value. The human proof uses a genus-two translation surface; ChatGPT helped explore approaches and Codex generated the Lean formalization.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-14","updated":"2026-10-14","orgs":["Xi'an Jiaotong University","OpenAI"]},{"id":"2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover","url":"https://postcutoff.com/e/2026-10-12-cureton-thorner-dedekind-zeta-zero-free-axiomprover/","date":"2026-10-12","date_precision":"day","short_title":"Cureton and Thorner, with AxiomProver, extend the quasi-Riemann zero-free half-plane to every Dedekind zeta function (Re s > 31/32, Lean-formalized)","deck":null,"takeaway":"It is one of the first results to build directly on OpenAI's claimed quasi-Riemann breakthrough.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Axiom Math"]},{"id":"2026-10-12-erdos-matching-conjecture-large-matchings-ai","url":"https://postcutoff.com/e/2026-10-12-erdos-matching-conjecture-large-matchings-ai/","date":"2026-10-12","date_precision":"day","short_title":"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","deck":null,"takeaway":"Three Shanghai Jiao Tong mathematicians prove Erdős's 1965 matching conjecture for all n once the matching size is large, by first settling a probability conjecture of Łuczak, Mieczkowska and Šileikis. Their disclosure credits AI with the unified argument for k ≥ 6 and with closing a gap.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-14","updated":"2026-10-29","orgs":["Shanghai Jiao Tong University"]},{"id":"2026-10-12-chow-polynomials-matroids-not-log-concave","url":"https://postcutoff.com/e/2026-10-12-chow-polynomials-matroids-not-log-concave/","date":"2026-10-12","date_precision":"day","short_title":"Chow polynomials of matroids are not log-concave","deck":"Counterexamples found in an LLM search (Claude Opus 5.5, ChatGPT Astra) refute three real-rootedness conjectures","takeaway":"Matroid theorists ran a targeted LLM search after a workshop; within hours Claude Opus 5.5 found explicit families of matroids that break log-concavity of Chow and chain polynomials, disproving conjectures of Ferroni–Schröter, Huh–Stevens and Athanasiadis–Kalampogia-Evangelinou.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":1,"official":1,"filed":"2026-10-14","updated":"2026-10-19","orgs":["Anthropic","OpenAI"]},{"id":"2026-10-12-drouot-claude-opus-proof-bulk-conductances","url":"https://postcutoff.com/e/2026-10-12-drouot-claude-opus-proof-bulk-conductances/","date":"2026-10-12","date_precision":"day","short_title":"\"The proof presented in this paper is due to Claude Opus 5.5\"","deck":"Alexis Drouot publishes Claude's 40-minute elementary proof of his conjecture on geometric bulk conductances","takeaway":"A mathematical physicist posted a paper whose proof he credits almost entirely to Claude Opus 5.5, which produced a 5-page solution in 40 minutes, and wrote that 'LLMs can solve Ph.D.-suited projects in just a few prompts'.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":1,"official":1,"filed":"2026-10-14","updated":"2026-10-14","orgs":["University of Washington","Anthropic"]},{"id":"2026-10-11-landau-siegel-zeros-banks-explicit-account","url":"https://postcutoff.com/e/2026-10-11-landau-siegel-zeros-banks-explicit-account/","date":"2026-10-11","date_precision":"day","short_title":"William Banks writes out OpenAI's Landau–Siegel proof with explicit constants","deck":"(1 − β) log q ≥ 0.003 for every real character, and says he verified it in full","takeaway":"The first detailed human account of OpenAI's claim that Landau–Siegel zeros do not exist gives an explicit, effective constant. Banks used Claude to draft it but says he worked through every line.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":3,"filed":"2026-10-14","updated":"2026-10-23","orgs":["OpenAI","Anthropic","University of Missouri"]},{"id":"2026-10-11-skryabin-conjecture-noetherian-hopf-algebra-antipode","url":"https://postcutoff.com/e/2026-10-11-skryabin-conjecture-noetherian-hopf-algebra-antipode/","date":"2026-10-11","date_precision":"day","short_title":"Skryabin's 2006 conjecture proved","deck":"Every noetherian Hopf algebra has a bijective antipode (proof strategy suggested by GPT-6)","takeaway":"A single-author 8-page paper settles a well-known question about Hopf algebras; the author says the proof strategy came from an iterative dialogue with GPT-6.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-14","updated":"2026-10-14","orgs":["Université libre de Bruxelles","OpenAI"]},{"id":"2026-10-10-vaught-conjecture-infinitary-counterexample-lean","url":"https://postcutoff.com/e/2026-10-10-vaught-conjecture-infinitary-counterexample-lean/","date":"2026-10-10","date_precision":"day","short_title":"Vaught's conjecture fails for infinitary logic","deck":"Ackerman, Freer and Knight post a Lean-verified counterexample, built over 8 months with Claude Code and Codex","takeaway":"A sentence of L_ω1,ω with exactly ℵ1 countable models and no perfect set of them refutes the infinitary Vaught conjecture (1961). It follows Robin Knight's 2002 approach, which was never accepted, and the whole argument is now certified in Lean.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-14","updated":"2026-10-29","orgs":["Harvard University","MIT","University of Oxford","Anthropic","OpenAI"]},{"id":"2026-10-10-erdos-problem-1117-maximum-modulus-gpt-6-codex","url":"https://postcutoff.com/e/2026-10-10-erdos-problem-1117-maximum-modulus-gpt-6-codex/","date":"2026-10-10","date_precision":"day","short_title":"Erdős #1117 on maximum-modulus points answered negatively with GPT-6 help","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":1,"filed":"2026-10-17","updated":"2026-10-17","orgs":["OpenAI","The Chinese University of Hong Kong, Shenzhen"]},{"id":"2026-10-09-geometrization-conjecture-lean-formalization","url":"https://postcutoff.com/e/2026-10-09-geometrization-conjecture-lean-formalization/","date":"2026-10-09","date_precision":"day","short_title":"Thurston's geometrization conjecture formalised in Lean","deck":"The Hamilton–Perelman proof machine-checked with GPT-6, Claude and DeepSeek agents (Chow's group)","takeaway":"Twelve days after the Poincaré conjecture, the same group machine-checked the full geometrization theorem, the result Perelman actually proved. The paper names the models: GPT-6 Astra reviews, and GPT-6 Luna, Sol, Astra, Claude Opus 5.5 and DeepSeek V4.1 Flash build.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-12","updated":"2026-10-12","orgs":["UC San Diego","Cornell University","Princeton University","Tsinghua University","Amazon AGI","NVIDIA Research","University of Oregon"]},{"id":"2026-10-09-kinetic-krylov-safonov-pucci-conjecture-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-09-kinetic-krylov-safonov-pucci-conjecture-gpt-6-astra/","date":"2026-10-09","date_precision":"day","short_title":"\"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)","deck":null,"takeaway":"Six analysts, among them Clément Mouhot and Luis Silvestre, say the missing estimate behind a long-open regularity theorem for kinetic equations was found by GPT-6 Astra Ultra on 6 Sept 2026 with no human contribution. They wrote a human exposition and published the prompts, chats and AI manuscripts, and state that they claim no credit.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-12","updated":"2026-10-29","orgs":["OpenAI","University of Cambridge","University of Chicago","UC Irvine"]},{"id":"2026-10-09-smale-sixth-problem-five-bodies-lean-tomar","url":"https://postcutoff.com/e/2026-10-09-smale-sixth-problem-five-bodies-lean-tomar/","date":"2026-10-09","date_precision":"day","short_title":"Smale's sixth problem claimed for five bodies in the plane","deck":"Finitely many central configurations for all positive masses, AI-found and proved in Lean (Tomar)","takeaway":"An independent researcher claims the planar five-body case of one of Smale's 18 problems for the 21st century, removing the exceptional masses left by Albouy and Kaloshin in 2012. He states that generative AI found and proved the result and wrote the Lean formalization; the paper does not name the models.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-12","updated":"2026-10-19","orgs":["independent"]},{"id":"2026-10-09-tao-math-2-0-caltech-lecture","url":"https://postcutoff.com/e/2026-10-09-tao-math-2-0-caltech-lecture/","date":"2026-10-09","date_precision":"day","short_title":"Terence Tao's Caltech lecture 'Math 2.0'","deck":"AI has ended 'proof scarcity', and blind problem-solving is now 'actively harmful' to mathematics","takeaway":"Tao is the mathematician most often cited on AI and maths, and this is his fullest public statement after OpenAI's release.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":7,"official":4,"filed":"2026-10-10","updated":"2026-10-10","orgs":["UCLA","Caltech"]},{"id":"2026-10-09-arxiv-ai-disclosed-results-early-october-2026","url":"https://postcutoff.com/e/2026-10-09-arxiv-ai-disclosed-results-early-october-2026/","date":"2026-10-09","date_precision":"day","short_title":"arXiv, 5–9 October 2026","deck":"More conjectures settled with disclosed AI help (Quillen's local parameter conjecture in dimension 4, Laughlin gap, poset reconstruction), next to human-only results","takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":14,"filed":"2026-10-12","updated":"2026-10-19","orgs":["OpenAI","Anthropic","various mathematicians"]},{"id":"2026-10-09-deepmind-cryptids-smaller-turing-machines-goldbach-riemann-zf","url":"https://postcutoff.com/e/2026-10-09-deepmind-cryptids-smaller-turing-machines-goldbach-riemann-zf/","date":"2026-10-09","date_precision":"day","short_title":"Gemini Argon shrinks Goldbach, Riemann and ZF Turing machines","deck":null,"takeaway":"Smaller machines tighten the known bounds on where the Busy Beaver function stops being computable from standard mathematics.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-10-16","updated":"2026-10-16","orgs":["Google DeepMind"]},{"id":"2026-10-09-erdos-32-additive-complements-primes-radomskii","url":"https://postcutoff.com/e/2026-10-09-erdos-32-additive-complements-primes-radomskii/","date":"2026-10-09","date_precision":"day","short_title":"Erdős #32: no O(log N) complement to the primes (Radomskii)","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-15","updated":"2026-10-19","orgs":["HSE University","OpenAI"]},{"id":"2026-10-09-nirenberg-conjecture-closed-asymptotic-curves-counterexample","url":"https://postcutoff.com/e/2026-10-09-nirenberg-conjecture-closed-asymptotic-curves-counterexample/","date":"2026-10-09","date_precision":"day","short_title":"Nirenberg's 1962 conjecture on closed asymptotic curves disproved","deck":"Claude and ChatGPT found the first, polygonal counterexample (Banks, Ghomi, Qiu, Raffaelli)","takeaway":"A 64-year-old conjecture from Louis Nirenberg's program on rigidity of surfaces is false. The authors say AI models found the first discrete counterexample, which they then smoothed; the final construction in the paper is classical and checked in Mathematica.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-12","updated":"2026-10-13","orgs":["Georgia Institute of Technology","Anthropic","OpenAI"]},{"id":"2026-10-09-random-permutations-universal-alon-gpt-6-pro","url":"https://postcutoff.com/e/2026-10-09-random-permutations-universal-alon-gpt-6-pro/","date":"2026-10-09","date_precision":"day","short_title":"Random permutations of quadratic length contain every pattern","deck":"Alon's 1000n² problem solved, 'the proof was found by GPT-6 Pro' (Chen, Wang)","takeaway":"The authors of this combinatorics paper state in the abstract that the model found the proof and that they 'do not claim credit for the proof ideas'. It settles the order of magnitude in a problem posed by Noga Alon.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Beijing Institute of Technology","OpenAI"]},{"id":"2026-10-09-subset-sum-near-linear-time-gpt-astra-ultra","url":"https://postcutoff.com/e/2026-10-09-subset-sum-near-linear-time-gpt-astra-ultra/","date":"2026-10-09","date_precision":"day","short_title":"Subset Sum in O(n + N^(1+ε)) time claimed","deck":"Zhejiang team designed the route, GPT Astra Ultra wrote the complete proofs (Chen, Mao, Zhang)","takeaway":"A long-open question in algorithms, whether Subset Sum can be solved in time near-linear in the largest input number, gets a claimed positive answer. The authors designed the approach; GPT Astra Ultra turned it into a 105-page paper and generalized it without guidance.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Zhejiang University","OpenAI"]},{"id":"2026-10-09-ai-weather-models-gray-swan-dubai-2024","url":"https://postcutoff.com/e/2026-10-09-ai-weather-models-gray-swan-dubai-2024/","date":"2026-10-09","date_precision":"day","short_title":"AI weather models forecast Dubai's unprecedented 2024 storm by 'translocation', not extrapolation (University of Chicago, Science Advances)","deck":null,"takeaway":"GraphCast and AIFS forecast it accurately up to 8 days ahead; FuXi underestimated the rain.","category":"science","category_label":"Science & math","importance":2,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":1,"official":1,"filed":"2026-10-12","updated":"2026-10-12","orgs":["University of Chicago"]},{"id":"2026-10-09-ml-assisted-directed-evolution-plant-rubisco","url":"https://postcutoff.com/e/2026-10-09-ml-assisted-directed-evolution-plant-rubisco/","date":"2026-10-09","date_precision":"day","short_title":"Machine-learning-guided evolution makes a plant Rubisco 29% faster (MIT, Stanford, ANU","deck":"Science Advances)","takeaway":"\"A better Rubisco\" is on the list of grand challenges for biology (see the related entry).","category":"science","category_label":"Science & math","importance":2,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":1,"official":1,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Massachusetts Institute of Technology","Stanford University","Australian National University"]},{"id":"2026-10-08-alphaproof-nexus-science-paper","url":"https://postcutoff.com/e/2026-10-08-alphaproof-nexus-science-paper/","date":"2026-10-08","date_precision":"day","short_title":"Google DeepMind's AlphaProof Nexus paper published in Science","deck":"LLM + Lean agent resolved 9 of 353 open Erdős problems and 44 of 492 OEIS conjectures","takeaway":"This is a peer-reviewed, fully formal account of LLM agents solving open research problems at scale and at low cost.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":16,"official":4,"filed":"2026-10-10","updated":"2026-10-10","orgs":["Google DeepMind"]},{"id":"2026-10-08-dualization-eth-lower-bound-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-08-dualization-eth-lower-bound-gpt-6-astra/","date":"2026-10-08","date_precision":"day","short_title":"GPT-6 Astra finds a 3SAT reduction showing monotone Dualization (minimal hypergraph transversals) has no polynomial algorithm under ETH","deck":null,"takeaway":"Whether monotone Boolean dualization, equivalently enumerating all minimal transversals of a hypergraph, can be done in output-polynomial time has been open since the 1990s. A short preprint says no, assuming the Exponential Time Hypothesis, and its authors say the proof 'was originally discovered by GPT-6 Astra'.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-09","updated":"2026-10-13","orgs":["Hokkaido University","Okayama University","Hosei University","OpenAI"]},{"id":"2026-10-08-hyperkahler-syz-conjecture-engel-mauri","url":"https://postcutoff.com/e/2026-10-08-hyperkahler-syz-conjecture-engel-mauri/","date":"2026-10-08","date_precision":"day","short_title":"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","deck":null,"takeaway":"The hyperkähler SYZ conjecture (every nef line bundle on a compact hyperkähler manifold is semiample) has a human-written proof by Philip Engel and Mirko Mauri, which also gives finitely many deformation classes in each dimension when b2 ≥ 5. OpenAI's catalogue claimed the same theorem on 6 Oct; the authors say the proof structures 'appear similar'.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-09","updated":"2026-10-09","orgs":["University of Illinois Chicago","Université Paris Cité","OpenAI"]},{"id":"2026-10-08-erdos-522-littlewood-polynomial-zeros-lean-kawada","url":"https://postcutoff.com/e/2026-10-08-erdos-522-littlewood-polynomial-zeros-lean-kawada/","date":"2026-10-08","date_precision":"day","short_title":"Erdős Problem #522 (1961) resolved with a Lean-verified proof","deck":"Half the zeros of random ±1 polynomials lie in the unit disk, almost surely (Kawada, MIT CSAIL)","takeaway":"A 1961 question of Erdős about the zeros of polynomials with random ±1 coefficients gets a formally verified answer, accepted against the statement in Google DeepMind's Formal Conjectures. The author says he originated the mathematics; unnamed in-house models from MIT's Kellis Lab helped with the Lean formalization and with checking derivations.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":4,"filed":"2026-10-12","updated":"2026-10-19","orgs":["MIT CSAIL","Google DeepMind"]},{"id":"2026-10-08-hilbert-number-lower-bound-gpt-5-6-sol-lean","url":"https://postcutoff.com/e/2026-10-08-hilbert-number-lower-bound-gpt-5-6-sol-lean/","date":"2026-10-08","date_precision":"day","short_title":"Hilbert's 16th problem","deck":"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","takeaway":"The lower bound for the number of limit cycles of degree-d planar polynomial vector fields improves from d² ln d to d² ln² d. The authors say GPT-5.6 Sol developed the proof strategy and arguments, and the construction is formalized in Lean 4.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-09","updated":"2026-10-13","orgs":["Chinese Academy of Sciences","OpenAI"]},{"id":"2026-10-08-holomorphic-cancellation-problem-counterexample","url":"https://postcutoff.com/e/2026-10-08-holomorphic-cancellation-problem-counterexample/","date":"2026-10-08","date_precision":"day","short_title":"Holomorphic Cancellation Problem (1998) answered in the negative, conditional on a newly announced exotic affine space; proofs developed with generative AI (Gaofeng Huang, Harvard)","deck":null,"takeaway":"A space can stop being distinguishable from ordinary complex n-space after one extra dimension is added. The short proof was written with AI help and rests on a result by Auroux, Bai and Pomerleano that has been announced in a lecture but not yet published.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Harvard University"]},{"id":"2026-10-08-karagila-openai-partition-principle","url":"https://postcutoff.com/e/2026-10-08-karagila-openai-partition-principle/","date":"2026-10-08","date_precision":"day","short_title":"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'","deck":null,"takeaway":"OpenAI's catalogue claims the Partition Principle does not imply the Axiom of Choice, one of set theory's oldest open problems (Russell, 1906), with a partial Lean formalization. The best-known expert on the problem, who had promised a bottle of whisky for a solution, called the paper unreadable and declined to check it.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":5,"official":3,"filed":"2026-10-09","updated":"2026-10-09","orgs":["OpenAI"]},{"id":"2026-10-08-lyapunov-spectrum-random-neural-networks-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-08-lyapunov-spectrum-random-neural-networks-gpt-6-astra/","date":"2026-10-08","date_precision":"day","short_title":"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)","deck":null,"takeaway":"A theoretical-neuroscience problem posed in the 1988 paper that founded the field's chaos theory now has an analytic answer. The first derivation came from GPT-6 Astra working alone for 100 minutes; Claude Opus 5.5 wrote all code and figures.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-11","updated":"2026-10-14","orgs":["Flatiron Institute","OpenAI","Anthropic"]},{"id":"2026-10-08-ppt-entanglement-distillation-rains-bound-chatgpt-astra","url":"https://postcutoff.com/e/2026-10-08-ppt-entanglement-distillation-rains-bound-chatgpt-astra/","date":"2026-10-08","date_precision":"day","short_title":"PPT-distillable entanglement can fall strictly below the regularised Rains bound","deck":"Results \"derived almost entirely by ChatGPT Astra\" (Lami)","takeaway":"A quantum information theorist resolves a 2019 open problem about how much entanglement can be distilled, and opens the paper by saying the model derived almost all of the results and techniques. He adds that validating them took 'an order of magnitude fewer' hours than it would have a year earlier.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-12","updated":"2026-10-13","orgs":["Scuola Normale Superiore","OpenAI"]},{"id":"2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar","url":"https://postcutoff.com/e/2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar/","date":"2026-10-08","date_precision":"day","short_title":"Laderman's 23 multiplications are optimal for 3×3 matrices over F_2 and over the integers","deck":"AI-found proof checked in Lean (Tomar)","takeaway":"Whether 3×3 matrices can be multiplied with fewer than Laderman's 23 multiplications has been open since 1976. A preprint claims the answer is no for coefficients in F_2 or the integers, ending a 2026 race in which the lower bound rose from 19 to 20, 21, 22 and now 23. The author says generative AI found and proved the result.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-12","updated":"2026-10-13","orgs":["independent"]},{"id":"2026-10-08-antiquity-ai-archive-search-meteorite-rhinos","url":"https://postcutoff.com/e/2026-10-08-antiquity-ai-archive-search-meteorite-rhinos/","date":"2026-10-08","date_precision":"day","short_title":"AI archive search finds forgotten meteorite and eruptions","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-10","updated":"2026-10-10","orgs":["Anthropic","TypeSafe AI"]},{"id":"2026-10-08-claude-science-complete-uv-sky-map","url":"https://postcutoff.com/e/2026-10-08-claude-science-complete-uv-sky-map/","date":"2026-10-08","date_precision":"day","short_title":"Claude Science builds a complete ultraviolet map of the sky","deck":null,"takeaway":"It is a small but concrete example of agentic AI doing the data-heavy \"infrastructure\" work that scientists often postpone for years.","category":"science","category_label":"Science & math","importance":2,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-09","updated":"2026-10-09","orgs":["Anthropic","Johns Hopkins University"]},{"id":"2026-10-07-colkitt-codex-integer-multiplication-exponent","url":"https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/","date":"2026-10-07","date_precision":"day","short_title":"Outside researchers using AI crowdsource a tighter exponent in OpenAI's integer-multiplication result #109, from 2^-182 to about 2^-8.4 (conditional)","deck":null,"takeaway":"Everything is conditional on OpenAI's unverified manuscript; Lean checks cover only finite certificates and arithmetic, and the gains are asymptotic, not practical speedups.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":39,"official":19,"filed":"2026-10-07","updated":"2026-10-29","orgs":["Independent researchers","OpenAI"]},{"id":"2026-10-07-exact-overlaps-conjecture-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-07-exact-overlaps-conjecture-gpt-6-astra/","date":"2026-10-07","date_precision":"day","short_title":"Exact overlaps conjecture for self-similar measures on the line proved","deck":"GPT-6 Astra found the first proof (Kittle & Kogler), 47,000-line Lean formalization; OpenAI's catalogue claims it independently","takeaway":"A central conjecture of fractal geometry, open in general since Hochman's 2014 breakthrough, now has a proof whose key part came from a GPT-6 Astra session. The authors rewrote and formalized it in Lean, and OpenAI's own release claims the same theorem.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-08","updated":"2026-10-13","orgs":["University College London","Institute for Advanced Study","OpenAI"]},{"id":"2026-10-07-lebrun-salamon-conjecture-proof","url":"https://postcutoff.com/e/2026-10-07-lebrun-salamon-conjecture-proof/","date":"2026-10-07","date_precision":"day","short_title":"Two claimed proofs of the LeBrun–Salamon conjecture","deck":"Hu, Liu & Wan (arXiv, ChatGPT as 'auxiliary tool') and OpenAI's catalogue (family 062)","takeaway":"The 1994 conjecture that every positive quaternion-Kähler manifold is a symmetric Wolf space now has two independent claimed proofs, one by human geometers who used ChatGPT and one in OpenAI's AI-generated catalogue. Neither has been refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-08","updated":"2026-10-13","orgs":["Chongqing University of Technology","OpenAI"]},{"id":"2026-10-07-multiple-unicast-conjecture-false-gpt-6","url":"https://postcutoff.com/e/2026-10-07-multiple-unicast-conjecture-false-gpt-6/","date":"2026-10-07","date_precision":"day","short_title":"Braverman & He refute the 2004 undirected multiple-unicast (network coding) conjecture with a GPT-6-found counterexample on PG(2, 9)","deck":null,"takeaway":"A 22-year-old network coding conjecture is false: on undirected networks, coding can beat routing. The counterexample came from GPT-6, and the Princeton authors checked it with a standalone symbolic verifier.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-08","updated":"2026-10-15","orgs":["Princeton University","OpenAI"]},{"id":"2026-10-07-toms-winter-conjecture-completed-chatgpt-astra-lemma","url":"https://postcutoff.com/e/2026-10-07-toms-winter-conjecture-completed-chatgpt-astra-lemma/","date":"2026-10-07","date_precision":"day","short_title":"Toms–Winter conjecture on C*-algebras completed by two preprints in one week","deck":"Both credit AI with a key lemma (Thiel; Evington, Hua, Schafhauser, Seth, Tikuisis, White)","takeaway":"The Toms–Winter conjecture (2000s) says three very different regularity properties of simple nuclear C*-algebras are equivalent; it was the central structural question of the classification programme. Hannes Thiel proved one missing implication on 5 Oct 2026 and a six-author team the last one on 7 Oct. The team says an initial version of an essential lemma came from ChatGPT 6 Astra in 27 minutes; Thiel says AI collaboration led to his key idea.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-13","updated":"2026-10-13","orgs":["University of Oxford","University of Ottawa","Chalmers University of Technology","OpenAI"]},{"id":"2026-10-07-bolzano-automated-open-problem-solving-charles-university","url":"https://postcutoff.com/e/2026-10-07-bolzano-automated-open-problem-solving-charles-university/","date":"2026-10-07","date_precision":"day","short_title":"Bolzano (Charles University)","deck":"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","takeaway":"A Prague group pointed an open-source prover–verifier loop built on GPT-5.5 and GPT-5.6 Sol at every open question it could extract from recent papers, with no per-problem human guidance. It reports about 214 solutions out of 3,800 attempts, most of them estimates, and four results that the authors of STOC 2026 papers confirmed.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-13","updated":"2026-10-13","orgs":["Charles University","Czech Academy of Sciences"]},{"id":"2026-10-07-chenglong-ma-hovey-strickland-openai-chatgpt-upload-provenance","url":"https://postcutoff.com/e/2026-10-07-chenglong-ma-hovey-strickland-openai-chatgpt-upload-provenance/","date":"2026-10-07","date_precision":"day","short_title":"Hovey–Strickland: author says OpenAI's paper matches his ChatGPT-uploaded draft","deck":null,"takeaway":"OpenAI's 6 Oct maths release includes a paper dated 24 Sept that proves Chai's invariant-ideal conjecture and, through Barthel–Heard–Naumann, the Hovey–Strickland classification of thick tensor ideals of dualizable K(n)-local spectra (family 311). The next day Copenhagen mathematician Chenglong Ma posted his own proof on arXiv. In it he writes that OpenAI's 'proof strategy and specific choices of notation are identical to a preliminary version of the present paper that was uploaded to ChatGPT on September 8, 2026.' OpenAI's paper does not mention him. He also reports an independent proof by Ningyi Li. We found no response from OpenAI as of 29 Oct. The allegation has not been examined publicly beyond a GitHub label and a one-comment Hacker News thread.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","University of Copenhagen"]},{"id":"2026-10-07-cyclicity-conjecture-positive-operators-dobrick","url":"https://postcutoff.com/e/2026-10-07-cyclicity-conjecture-positive-operators-dobrick/","date":"2026-10-07","date_precision":"day","short_title":"Claimed proof of the cyclicity conjecture of Perron–Frobenius theory for positive operators (Dobrick, GPT-5.6 used to simplify arguments)","deck":null,"takeaway":"A single-author arXiv preprint claims to settle the cyclicity conjecture, open since the 1960s: the peripheral spectrum of every positive operator on a complex Banach lattice is cyclic. The author says he used GPT-5.6 models in an OpenCode harness for editing, simplifying some arguments and a literature review. All new contributions are his own. The proof has not been refereed.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-10","updated":"2026-10-13","orgs":["Alexander Dobrick"]},{"id":"2026-10-06-artin-davenport-cubic-forms-ten-variables","url":"https://postcutoff.com/e/2026-10-06-artin-davenport-cubic-forms-ten-variables/","date":"2026-10-06","date_precision":"day","short_title":"Artin–Davenport conjecture proved","deck":"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)","takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-07","updated":"2026-10-13","orgs":["TU Graz","ISTA","Imperial College London","Leibniz Universität Hannover","Academia Sinica","OpenAI"]},{"id":"2026-10-06-free-group-factors-isomorphism-openai","url":"https://postcutoff.com/e/2026-10-06-free-group-factors-isomorphism-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the free group factor problem","deck":"L(F₂) ≅ L(F₃), so all free group factors are isomorphic (Lean-formalized)","takeaway":"The problem has been one of the best-known open questions in operator algebras for decades.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":4,"filed":"2026-10-07","updated":"2026-10-19","orgs":["OpenAI"]},{"id":"2026-10-06-harary-hill-zarankiewicz-crossing-numbers-openai","url":"https://postcutoff.com/e/2026-10-06-harary-hill-zarankiewicz-crossing-numbers-openai/","date":"2026-10-06","date_precision":"day","short_title":"Harary–Hill and Zarankiewicz crossing-number conjectures (OpenAI, Lean)","deck":null,"takeaway":"Two short manuscripts credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claim the exact crossing numbers of complete graphs and complete bipartite graphs. Those are the Harary–Hill conjecture (late 1950s) and Zarankiewicz's 1954 formula for Turán's 1944 brick-factory problem. Lean proves both formulas, and a community comparator run accepted both Lean challenges. On 25 Oct the graph-drawing specialist Martin Balko posted a simplified exposition of the Harary–Hill proof, which he says is based on the polynomial method. Gelasio Salazar is preparing another.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":6,"filed":"2026-10-27","updated":"2026-10-28","orgs":["OpenAI","Charles University"]},{"id":"2026-10-06-hodge-conjecture-cm-abelian-varieties-openai","url":"https://postcutoff.com/e/2026-10-06-hodge-conjecture-cm-abelian-varieties-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI release proves the rational Hodge conjecture for all CM abelian varieties, giving the Tate conjecture for abelian varieties over finite fields; Lean proof added 26 Oct, human-written proof by van Hoften and Sempliner, and Milne calls it proved","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":13,"official":8,"filed":"2026-10-07","updated":"2026-11-10","orgs":["OpenAI"]},{"id":"2026-10-06-openai-anderson-localization-2d-delocalization-3d","url":"https://postcutoff.com/e/2026-10-06-openai-anderson-localization-2d-delocalization-3d/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Anderson localization in 2D and extended states in 3D","deck":null,"takeaway":"Two manuscripts in OpenAI's 6 Oct 2026 math release (family 261, dated 23 Sept) claim the two central conjectures about electrons in a random lattice. In two dimensions, the Anderson model has pure-point spectrum, so all states are localized, at every disorder strength. In three or more dimensions, weak disorder leaves purely absolutely continuous spectrum, meaning extended states, on an energy interval. These are Barry Simon's Problems 1 and 2 of 2000, for uniform disorder. Specialists Jakšić and Last shortened the last step of the 3D proof in an appendix, and a third party has posted a Lean formalization of the 3D result. OpenAI's own Lean covers only the 2D spectrum's location.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-baum-connes-kadison-kaplansky-counterexamples","url":"https://postcutoff.com/e/2026-10-06-openai-baum-connes-kadison-kaplansky-counterexamples/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Baum–Connes and Kadison–Kaplansky counterexamples","deck":null,"takeaway":"Three manuscripts in OpenAI's 6 Oct 2026 math release (family 285, dated 23 Sept) claim the first counterexamples to the Baum–Connes conjecture without coefficients, one of them for a torsion-free group, and a torsion-free group whose reduced C*-algebra has a nontrivial projection, which would refute the Kadison–Kaplansky conjecture. Until now, counterexamples were known only for the version with coefficients. On 26 Oct Guoliang Yu, a leading specialist, posted his own construction of projections of every trace in (0,1), saying it develops arguments from OpenAI's irrational-trace paper. None of the three papers has a Lean proof.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-borel-conjecture-dimension-four-free-group-not-good","url":"https://postcutoff.com/e/2026-10-06-openai-borel-conjecture-dimension-four-free-group-not-good/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims 4D Borel counterexample and that F₂ is not good","deck":null,"takeaway":"Four manuscripts in OpenAI's 6 Oct 2026 math release (families 305 and 320) make three claims in 4-manifold topology. Freedman's disk embedding theorem fails without a fundamental-group hypothesis, so the free group F₂ is not 'good', the central open question of topological 4-manifold surgery. A four-dimensional Poincaré duality group has no closed aspherical manifold model, against Wall's question. And two closed aspherical 4-manifolds with a hyperbolic fundamental group are homotopy equivalent but not homeomorphic, against the topological Borel conjecture. None has a Lean proof or a published expert check as of 29 Oct.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-borsuk-conjecture-dimension-nine-counterexample","url":"https://postcutoff.com/e/2026-10-06-openai-borsuk-conjecture-dimension-nine-counterexample/","date":"2026-10-06","date_precision":"day","short_title":"Borsuk's conjecture fails in dimension 9 (OpenAI claim, Lean-checked)","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 maths release (family 156, dated 23 Sept) claims that the compact set of rank-one projectors onto lines in ℝ⁴, a copy of real projective 3-space in a 9-dimensional space, cannot be cut into ten pieces of smaller diameter. That would make 9 the smallest known dimension in which Borsuk's 1933 conjecture fails. The record had been 63, set in May 2026 with a finite 321-point set. The main theorem has a Lean proof, which a community Comparator run accepted. Gil Kalai called it 'a major progress' and said the topological proof 'looks very beautiful', but added that it still needs careful checking. The example is infinite, so for finite sets the record is still 63. Gábor Damásdi showed that every finite subset of it can be split into eight smaller pieces.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":18,"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-cannon-conjecture-proof-claim","url":"https://postcutoff.com/e/2026-10-06-openai-cannon-conjecture-proof-claim/","date":"2026-10-06","date_precision":"day","short_title":"Cannon's conjecture claimed (OpenAI, Lean); Bonk's seminar talk","deck":null,"takeaway":"A 31-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims Cannon's conjecture: every word-hyperbolic group whose boundary is a 2-sphere acts properly and cocompactly by isometries on hyperbolic 3-space. For torsion-free groups this makes them fundamental groups of closed hyperbolic 3-manifolds, a group-theoretic counterpart of geometrization. The proof bounds combinatorial modulus and then applies a Bourdon–Kleiner criterion. The Lean statement was accepted in a community comparator run. On 22 Oct Mario Bonk, who with Bruce Kleiner proved a key special case, gave a seminar talk titled 'The AI Proof of Cannon's conjecture'.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-catalan-constant-irrational","url":"https://postcutoff.com/e/2026-10-06-openai-catalan-constant-irrational/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI: Catalan's constant is irrational (Lean)","deck":null,"takeaway":"Family 005 of OpenAI's 6 Oct 2026 release is a 44-page manuscript, dated 24 Sep, proving that Catalan's constant G = 1 − 1/9 + 1/25 − 1/49 + … is irrational, one of the classical open questions about special values. Its Lean statement is one line on top of Mathlib and passed the community comparator runs. Frank Calegari sees 'a bunch of new ideas, or at least new optimizations', and outside authors have already turned the method into explicit irrationality measures.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-deligne-drinfeld-conjecture-proof","url":"https://postcutoff.com/e/2026-10-06-openai-deligne-drinfeld-conjecture-proof/","date":"2026-10-06","date_precision":"day","short_title":"Deligne–Drinfeld conjecture proved; experts simplify and reprove OpenAI's proof","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 maths release (family 008, dated 23 Sept) proves the Deligne–Drinfeld conjecture. It says that the Grothendieck–Teichmüller Lie algebra grt₁ is free on one generator in each odd weight 3, 5, 7, …. The question goes back to Deligne (1989) and Drinfeld (1990). Thomas Willwacher (ETH Zurich), whose July 2026 paper had extended Brown's half of the conjecture, wrote that 'the remaining part … was solved by OpenAI'. He posted a much shorter version of the proof, about eight pages instead of about forty. Minoru Hirose then gave an alternative proof through multiple zeta values. Specialists studied the proof at a Trinity College Dublin workshop in mid-October. OpenAI's Lean proof of the main statement was accepted in a community Comparator run. This is one of the best-confirmed famous results of the release.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":13,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","ETH Zurich","Kagoshima University"]},{"id":"2026-10-06-openai-erdos-arithmetic-progressions-conjecture-claim","url":"https://postcutoff.com/e/2026-10-06-openai-erdos-arithmetic-progressions-conjecture-claim/","date":"2026-10-06","date_precision":"day","short_title":"Erdős's arithmetic-progression conjecture claimed (OpenAI, Lean)","deck":null,"takeaway":"A 198-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arbitrarily long arithmetic progressions, the problem for which Erdős offered $5000. It does so by claiming quasipolynomial bounds in Szemerédi's theorem for every progression length. A Lean proof of the qualitative conjecture was accepted in a community comparator run. Thomas Bloom of erdosproblems.com reads the Lean code as proving a weaker bound than the paper claims, and describes the proof as a combination of existing techniques. No expert has published a check of the 198 pages.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":10,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-finitely-presented-burnside-periodic-group-claim","url":"https://postcutoff.com/e/2026-10-06-openai-finitely-presented-burnside-periodic-group-claim/","date":"2026-10-06","date_precision":"day","short_title":"Finitely presented Burnside problem answered negatively? (OpenAI, Lean)","deck":null,"takeaway":"Two manuscripts credited to 'OpenAI' claim an infinite group with a finite presentation in which every element has finite order. Whether such a group exists, the finitely presented form of Burnside's 1902 question, had stayed open even after the infinite finitely generated examples of Golod–Shafarevich and Novikov–Adian. The claimed group is the Steinberg group St₁₂(R) over a characteristic-two algebra and also has Kazhdan's property (T). A companion dated 5 Oct makes it residually finite, with all element orders powers of 2. The existence statement is formalized in Lean and was accepted in a community comparator run; no group theorist has published an assessment.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":9,"filed":"2026-10-29","updated":"2026-10-31","orgs":["OpenAI"]},{"id":"2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc","url":"https://postcutoff.com/e/2026-10-06-openai-gromov-lawson-aspherical-conjecture-psc/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims the aspherical positive-scalar-curvature conjecture","deck":null,"takeaway":"Two manuscripts in OpenAI's 6 Oct 2026 math release (family 335) claim that a closed manifold with positive scalar curvature is rationally inessential, in every dimension and without spin assumptions. That would prove the Gromov–Lawson (Schoen–Yau) conjecture that closed aspherical manifolds, such as tori and hyperbolic manifolds, admit no metric of positive scalar curvature. It was known in dimensions up to five (Chodosh–Li, Gromov). A second paper proves Gromov's proposed integral bound of simplicial volume by negative scalar curvature. Roman Sauer, whose technique appears in the proofs, wrote 'I am stunned.' No Lean proof; no expert verification yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":2,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-hadwiger-conjecture-counterexample-claim","url":"https://postcutoff.com/e/2026-10-06-openai-hadwiger-conjecture-counterexample-claim/","date":"2026-10-06","date_precision":"day","short_title":"Hadwiger's conjecture disproved? (OpenAI, Lean)","deck":null,"takeaway":"A 104-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims arbitrarily large graphs whose chromatic number exceeds their largest clique minor. That would refute Hadwiger's 1943 conjecture, a far-reaching generalization of the four colour theorem known to hold only up to K6. The examples have no three pairwise non-adjacent vertices and also beat the fractional weakening. A Lean formalization of the ordinary counterexample, added in OpenAI's 7 Oct update, was accepted in a community comparator run, and an independent Lean project derived the fractional version. The graphs come from a random construction; no explicit example has been published, and no expert has published an assessment.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-heisenberg-ferromagnet-bose-einstein-condensation","url":"https://postcutoff.com/e/2026-10-06-openai-heisenberg-ferromagnet-bose-einstein-condensation/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims quantum ferromagnet order and interacting Bose condensation","deck":null,"takeaway":"Two families in OpenAI's 6 Oct 2026 math release claim long-open theorems of quantum statistical mechanics. Family 271: the quantum Heisenberg ferromagnet in dimension 3 or more has a spontaneously magnetized equilibrium state at low temperature, plus Bloch's T^3/2 law with its exact coefficient. Family 267: a dilute gas of hard-sphere bosons Bose-condenses in its ground state and at a fixed positive temperature. The magnetization and ground-state condensation statements have Lean proofs that passed an independent Comparator run. OpenAI published a reasoning summary for 271, and Scientific American listed both among the release's most important claims. No expert verification yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-hilbert-sixteenth-uniform-limit-cycle-bound-claim","url":"https://postcutoff.com/e/2026-10-06-openai-hilbert-sixteenth-uniform-limit-cycle-bound-claim/","date":"2026-10-06","date_precision":"day","short_title":"Hilbert's 16th problem: uniform limit-cycle bound claimed (OpenAI)","deck":null,"takeaway":"A 160-page manuscript credited to 'OpenAI' (dated 24 Sept, public on 6 Oct) claims that for every degree d there is a finite B(d) bounding the number of limit cycles of every real planar polynomial vector field of degree at most d. That is the uniform-boundedness assertion in the second part of Hilbert's sixteenth problem (1900), open for every degree above one. The bound is not effective. None of the release's Lean statements covers it: only a companion result, at most two limit cycles for quintic Liénard systems, is formalized. No dynamical-systems expert has published an assessment.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-hilbert-smith-conjecture-claim","url":"https://postcutoff.com/e/2026-10-06-openai-hilbert-smith-conjecture-claim/","date":"2026-10-06","date_precision":"day","short_title":"Hilbert–Smith conjecture claimed by OpenAI, Lean-checked by a separate AI proof","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release (family 304, dated 23 Sept) claims the Hilbert–Smith conjecture, the manifold-action form of Hilbert's fifth problem, in every dimension. Only dimensions up to three were known (Pardon, 2013). OpenAI's paper has no Lean proof. On 8 Oct a GitHub author posted a separate 17-page proof written by 'A. Dabrowski's AI agents', with a Lean 4 formalization that the Palomar registry lists since 16 Oct. We found no public expert assessment of either proof as of 29 Oct.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-hilbert-tenth-problem-rationals-claim","url":"https://postcutoff.com/e/2026-10-06-openai-hilbert-tenth-problem-rationals-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Hilbert's tenth problem over ℚ is undecidable","deck":null,"takeaway":"Family 004 of OpenAI's 6 Oct 2026 release claims that no algorithm decides whether an integer polynomial has a rational zero, the rational-number version of Hilbert's tenth problem, open since Matiyasevich settled the integer case in 1970. The 62-page proof avoids defining the integers inside the rationals; it reduces integer solvability to infinitely many finite rational tests. It is not formalized and leans on three companion manuscripts. Frank Calegari said it 'threw me for a loop', and Hector Pasten called the strategy 'original'.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-kadison-similarity-problem-lean","url":"https://postcutoff.com/e/2026-10-06-openai-kadison-similarity-problem-lean/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Kadison's similarity problem (Lean)","deck":null,"takeaway":"Family 288 of OpenAI's 6 Oct 2026 release (paper dated 23 Sept) claims a positive answer to Kadison's similarity problem of 1955. The claim: every bounded unital homomorphism from a unital C*-algebra into the operators on a Hilbert space becomes a *-homomorphism after a bounded change of coordinates. The paper also claims one hyperreflexivity constant valid for every von Neumann algebra. Before, only special cases were known: nuclear algebras, cyclic representations, algebras without tracial states, and II₁ factors with property Γ. Lean statements of the similarity theorem, the hyperreflexivity bound and the underlying commutator estimate were accepted in two independent comparator runs. As of 29 Oct we found no assessment by an operator algebraist.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-kaplansky-zero-divisor-direct-finiteness-counterexamples-claim","url":"https://postcutoff.com/e/2026-10-06-openai-kaplansky-zero-divisor-direct-finiteness-counterexamples-claim/","date":"2026-10-06","date_precision":"day","short_title":"Kaplansky's zero-divisor and direct-finiteness conjectures refuted? (OpenAI, Lean)","deck":null,"takeaway":"Two families of OpenAI's 6 Oct release claim to refute group-ring conjectures from Kaplansky's 1950s list. One claims a finitely presented torsion-free group G whose group algebra F₂[G] has zero divisors, where the conjecture predicted none. The other claims group algebras in characteristic 2, in odd characteristic and over a torsion-free group in which ab = 1 but ba ≠ 1. Such examples need non-sofic groups, so they would also refute Gottschalk's 1973 surjunctivity conjecture. A further paper refutes the unrestricted Determinant Conjecture. Lean covers the zero-divisor, the torsion examples and the determinant claim, not the torsion-free direct-finiteness example. Specialists have commented on the presentation, not on correctness.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-mahler-conjecture-lean","url":"https://postcutoff.com/e/2026-10-06-openai-mahler-conjecture-lean/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI proves the Mahler conjectures (Lean)","deck":null,"takeaway":"Family 087 of OpenAI's 6 Oct 2026 release claims Mahler's conjecture for symmetric convex bodies (volume product at least 4ⁿ/n!, equality exactly for Hanner polytopes) and for general convex bodies (equality exactly for simplices) in every dimension, plus a symplectic statement that gives a second proof of the symmetric case. The symmetric case was known only up to dimension 3. Both main theorems have Lean proofs whose statements read as the standard conjectures, and the community comparator run accepted them. No expert has yet published a full check.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":8,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-matrix-multiplication-omega-9-4","url":"https://postcutoff.com/e/2026-10-06-openai-matrix-multiplication-omega-9-4/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI proves ω ≤ 9/4 for matrix multiplication","deck":null,"takeaway":"Family 107 of OpenAI's 6 Oct 2026 release proves that two n × n matrices can be multiplied with O(n^(9/4+ε)) arithmetic operations, the largest drop in the exponent in decades; the best bound before was ω < 2.371177, set in August 2026 with AlphaEvolve. The 13-page proof is non-constructive and comes with a Lean formalization. By 21 Oct, Amanov, van den Berg and Zuiddam had re-proved it over every field and Rafael Pass had given an explicit construction. The resulting algorithms are galactic and give no practical speedup.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":13,"official":9,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-plane-not-five-colourable-hadwiger-nelson-claim","url":"https://postcutoff.com/e/2026-10-06-openai-plane-not-five-colourable-hadwiger-nelson-claim/","date":"2026-10-06","date_precision":"day","short_title":"Chromatic number of the plane is 6 or 7, OpenAI claims (Lean)","deck":null,"takeaway":"A 62-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims that every 5-colouring of the plane has two points at distance one with the same colour, with no regularity assumed on the colour classes. That raises de Grey's 2018 lower bound of five to six, so the chromatic number of the plane is 6 or 7. The proof first transfers arbitrary colourings to a weak measurability condition, then rules out five colours under it. It is formalized in Lean and passed a community comparator run and a stricter independent check. Mathematicians called the method unlike anything in the prior literature; by compactness a finite non-5-colourable unit-distance graph must exist, but none has been displayed.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":10,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-relativistic-vlasov-maxwell-global-solutions-lean","url":"https://postcutoff.com/e/2026-10-06-openai-relativistic-vlasov-maxwell-global-solutions-lean/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims global smoothness for 3D relativistic Vlasov–Maxwell (Lean)","deck":null,"takeaway":"Family 362 of OpenAI's 6 Oct 2026 release (paper dated 23 Sept) concerns the three-dimensional one-species relativistic Vlasov–Maxwell system, the basic model of a collisionless plasma. The claim: every smooth, compactly supported particle density with finite-energy fields generates a unique global smooth solution, with no smallness or symmetry assumption. Glassey and Strauss showed in 1986 that solutions stay smooth as long as particle momenta stay bounded. Since then the large-data problem had been settled only under size, symmetry or dimension restrictions. A Lean statement of the full theorem was accepted in two independent comparator runs. OpenAI's published reasoning summary says the final assertion rests on new estimates that the cited continuation theorem does not supply. We found no assessment by a kinetic theorist as of 29 Oct.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-strong-cosmic-censorship-near-kerr","url":"https://postcutoff.com/e/2026-10-06-openai-strong-cosmic-censorship-near-kerr/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims strong cosmic censorship near Kerr","deck":null,"takeaway":"Three manuscripts in OpenAI's 6 Oct 2026 math release (family 264, dated 23 Sept) claim local strong cosmic censorship for rotating Kerr black holes, with no symmetry assumption. For generic small perturbations of two-ended Kerr data, the spacetime cannot be extended past its maximal development with a continuous metric and square-integrable connection. That is the weak-solution formulation Christodoulou proposed in 2009. Berkeley's Sung-Jin Oh, who has worked toward this problem for thirteen years, wrote that the proof takes a route different from the field's and organised a reading group. No Lean proof; no expert verdict yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-yau-nodal-conjecture-surfaces-counterexamples","url":"https://postcutoff.com/e/2026-10-06-openai-yau-nodal-conjecture-surfaces-counterexamples/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI: Yau's nodal bound holds on surfaces, fails in higher dimensions","deck":null,"takeaway":"Three manuscripts in OpenAI's 6 Oct 2026 math release (family 350, dated 23 Sept) claim to settle Yau's 1982 conjecture on the size of nodal sets of Laplace eigenfunctions for smooth metrics. On every smooth closed surface the nodal length is at most C√λ, which completes the conjecture there. In dimensions 3 and 4, explicit smooth metrics, including some arbitrarily close to the round 3-sphere, violate the upper bound, and on S⁴×S¹ it fails by a power of λ. Lean proofs of all three statements passed an independent Comparator run. Whether the formal statements match the papers has not been checked by specialists.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-quasi-riemann-hypothesis-openai","url":"https://postcutoff.com/e/2026-10-06-quasi-riemann-hypothesis-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the quasi-Riemann hypothesis","deck":"No zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized)","takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":19,"official":14,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-unique-games-conjecture-openai","url":"https://postcutoff.com/e/2026-10-06-unique-games-conjecture-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims a proof of Khot's Unique Games Conjecture, with Lean formalization","deck":null,"takeaway":"If OpenAI's claimed proof holds, it settles the Unique Games Conjecture, a central open problem in hardness of approximation since 2002. It comes with a Lean formalization, but experts have not yet checked it.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":16,"official":5,"filed":"2026-10-07","updated":"2026-11-12","orgs":["OpenAI"]},{"id":"2026-10-06-finite-index-over-treeable-jkl-problem","url":"https://postcutoff.com/e/2026-10-06-finite-index-over-treeable-jkl-problem/","date":"2026-10-06","date_precision":"day","short_title":"Chen, Tserunyan & Tucker-Drob solve the Jackson–Kechris–Louveau 'finite index over treeable' problem (2002); ChatGPT 6 Astra Ultra supplied a 'crucial observation'","deck":null,"takeaway":"A countable Borel equivalence relation that contains a treeable subrelation of finite index is itself treeable. This answers Open Problem 6.4(B) of Jackson–Kechris–Louveau (2002), open even in the index-2 measure-preserving case. The authors had the strategy; GPT supplied the observation that made the stages fit together.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-09","updated":"2026-10-13","orgs":["University of Warsaw","McGill University","University of Florida","OpenAI"]},{"id":"2026-10-06-gch-above-strongly-compact-woodin-question","url":"https://postcutoff.com/e/2026-10-06-gch-above-strongly-compact-woodin-question/","date":"2026-10-06","date_precision":"day","short_title":"Woodin's question answered","deck":"GCH below a strongly compact cardinal implies GCH everywhere (Zhixing You, 4-page proof with GPT-6 Astra)","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-07","updated":"2026-10-13","orgs":["Hebrew University of Jerusalem","OpenAI"]},{"id":"2026-10-06-generalized-lax-conjecture-openai-counterexample-kummer-netzer-generic-case","url":"https://postcutoff.com/e/2026-10-06-generalized-lax-conjecture-openai-counterexample-kummer-netzer-generic-case/","date":"2026-10-06","date_precision":"day","short_title":"Generalized Lax conjecture: OpenAI counterexample claim; AI-found generic case (Kummer–Netzer)","deck":null,"takeaway":"The generalized Lax conjecture says every hyperbolicity cone is spectrahedral, a slice of a positive semidefinite cone. On 21 Sept 2026 Mario Kummer and Tim Netzer posted 'a proof found by an artificial intelligence system' for the generic case, strictly hyperbolic polynomials (arXiv 2609.24542). Family 095 of OpenAI's 6 Oct maths release (manuscripts of 24 Sept and 5 Oct) claims the general conjecture is false. It gives hyperbolicity cones that are not even spectrahedral shadows, which also disproves the projected Lax conjecture, and the main statement is in Lean. Kevin Shu's exposition on the Quasicoherent blog accepts the counterexample. Neither result is refereed. We found both only on 31 Oct, through a citation.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-31","updated":"2026-10-31","orgs":["OpenAI"]},{"id":"2026-10-06-hironaka-resolution-of-singularities-lean-fable-agents","url":"https://postcutoff.com/e/2026-10-06-hironaka-resolution-of-singularities-lean-fable-agents/","date":"2026-10-06","date_precision":"day","short_title":"Hironaka's resolution of singularities formalized in Lean by agents","deck":null,"takeaway":"Resolution (resolution.org) reports that an orchestrated team of agents, mostly Claude Fable 5.1 with occasional checks by GPT-6 Astra, autoformalized Hironaka's 1964 theorem in characteristic zero between 4 and 10 Sept 2026, together with refinements by Kollár, Włodarczyk and Bierstone–Milman. That took about 100B input tokens, 300M output tokens and roughly $135k. A human mathematician intervened about 30 times. The repository says neither library contains a sorry.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":4,"filed":"2026-10-26","updated":"2026-10-29","orgs":["Resolution","Anthropic"]},{"id":"2026-10-06-irrationality-exponent-pi-openai","url":"https://postcutoff.com/e/2026-10-06-irrationality-exponent-pi-openai/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims the irrationality exponent of π is exactly 2 (Lean-formalized), which implies the Flint–Hills series converges","deck":null,"takeaway":"Since Mahler's 1953 bound of 42, mathematicians had lowered the irrationality exponent of π only to about 7.1. OpenAI's release claims the conjectured value 2 in a 23-page paper with a Lean proof. Six days on, no expert has publicly assessed it; an outsider has already adapted the Lean proof to log 2.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":8,"filed":"2026-10-12","updated":"2026-10-13","orgs":["OpenAI"]},{"id":"2026-10-06-openai-abelian-zilber-pink-claim","url":"https://postcutoff.com/e/2026-10-06-openai-abelian-zilber-pink-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims abelian Zilber–Pink","deck":null,"takeaway":"Family 016 of OpenAI's 6 Oct 2026 release claims the abelian case of the Zilber–Pink conjecture on unlikely intersections: every subvariety of an abelian variety over a number field has only finitely many maximal atypical subvarieties. By a 2022 theorem of Barroero and Dill this would give Zilber–Pink for abelian varieties over any field of characteristic zero. Three more papers claim the curve case in the Siegel threefold A2. Nothing is formalized; the first specialist papers cite it as an 'announced' AI-generated proof, and one group says the release pushed them to post their own related work a day later.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":6,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-artin-primitive-root-conjecture-infinitude","url":"https://postcutoff.com/e/2026-10-06-openai-artin-primitive-root-conjecture-infinitude/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Artin's primitive root conjecture (infinitude)","deck":null,"takeaway":"Family 029 of OpenAI's 6 Oct 2026 release claims that every integer other than −1 and the squares is a primitive root modulo infinitely many primes, at least c·x/(log x)² of them in each interval (x, 2x). Before, this was known only under the generalized Riemann hypothesis (Hooley, 1967) or for all but at most two prime bases (Heath-Brown, so one of 2, 3, 5 works), not for any single named base such as 2. The argument rests on the release's own zero-free half-planes. Frank Calegari calls the argument interesting but 'not at all surprising' once quasi-RH is known. Manjul Bhargava has posted a positive-density strengthening.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-bloch-conjecture-surfaces-claim","url":"https://postcutoff.com/e/2026-10-06-openai-bloch-conjecture-surfaces-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Bloch's conjecture for surfaces","deck":null,"takeaway":"Family 040 of OpenAI's 6 Oct 2026 release claims Bloch's conjecture: on every smooth complex projective surface without holomorphic 2-forms (p_g = 0), zero-cycles modulo rational equivalence are completely described by their degree and Albanese image. Bloch, Kas and Lieberman proved this in 1976 for surfaces not of general type; for general type only special families were known. The 45-page proof uses virtual localization on Quot schemes. It is not formalized, and the only public expert comment so far, from Daniel Litt, flags a citation problem he thinks is harmless.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":2,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim","url":"https://postcutoff.com/e/2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims BSD in Selmer corank ≤ 1 and Goldfeld's conjecture","deck":null,"takeaway":"Families 002 and 006 of OpenAI's 6 Oct 2026 maths release claim the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose p-power Selmer group has corank 0 or 1 at some prime, and Goldfeld's conjecture that half of all quadratic twists have analytic rank 0 and half rank 1. Together they would give full BSD for 100% of the quadratic twists of every curve. None of it is formalized, and the experts who have commented read it as a strong technical result in the rank ≤ 1 range that leaves the general conjecture untouched.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-closed-geodesics-spheres-infinitely-many","url":"https://postcutoff.com/e/2026-10-06-openai-closed-geodesics-spheres-infinitely-many/","date":"2026-10-06","date_precision":"day","short_title":"Infinitely many closed geodesics on every sphere, claimed and re-proved","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release (family 345, dated 24 Sept) claims that every smooth Riemannian metric on the n-sphere has infinitely many geometrically distinct closed geodesics, for every n ≥ 2. Only the 2-sphere was known (Bangert, Franks, Hingston, early 1990s). It also covers every closed 3-manifold. On 15 Oct Zhiqi Chen, Shaoqiang Deng and Hui Zhang posted a proof that does not use OpenAI's lemmas, written with OpenAI Codex and checked by them, and extended it to the complex, quaternionic and Cayley projective spaces. They credit OpenAI's paper as the source of the strategy.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Nankai University","Guangdong University of Technology","Southeast University"]},{"id":"2026-10-06-openai-de-giorgi-conjecture-dimension-eight","url":"https://postcutoff.com/e/2026-10-06-openai-de-giorgi-conjecture-dimension-eight/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims De Giorgi's conjecture in dimension eight","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release (family 375, dated 26 Sept) claims to complete De Giorgi's conjecture. Every bounded entire solution of the Allen–Cahn equation Δu = u³ − u in ℝ⁸ that is strictly increasing in one direction is a one-dimensional tanh profile. Dimensions 2 and 3 were known, as was Savin's result up to dimension 8 under an extra limit assumption, and counterexamples exist from dimension 9. The paper also classifies stable solutions in ℝ⁷ without energy bounds. A Chinese group has already built on it. No expert assessment and no Lean proof are known as of 28 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-28","updated":"2026-10-28","orgs":["OpenAI"]},{"id":"2026-10-06-openai-falconer-distance-conjecture-lean","url":"https://postcutoff.com/e/2026-10-06-openai-falconer-distance-conjecture-lean/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI proves the Falconer distance conjecture (Lean)","deck":null,"takeaway":"Family 073 of OpenAI's 6 Oct 2026 release proves that every compact set in ℝᵈ (d ≥ 2) of Hausdorff dimension greater than d/2 has a distance set of positive length. This is Falconer's 1985 conjecture, the continuous analogue of Erdős's distinct-distances problem; the best planar threshold before was 5/4. The 64-page proof comes with a Lean formalization whose statement uses Mathlib's own Hausdorff dimension and Lebesgue measure, and the community comparator run accepted it. A first follow-up already builds on its method.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":6,"filed":"2026-10-29","updated":"2026-11-12","orgs":["OpenAI"]},{"id":"2026-10-06-openai-hot-spots-conjecture-simply-connected-lean","url":"https://postcutoff.com/e/2026-10-06-openai-hot-spots-conjecture-simply-connected-lean/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims the planar hot spots conjecture (Lean)","deck":null,"takeaway":"Family 369 of OpenAI's 6 Oct 2026 release (paper dated 24 Sept) concerns smooth bounded simply connected planar domains. It claims that on every such domain each first nonconstant Neumann eigenfunction has no interior critical point, so its maximum and minimum occur only on the boundary. This is the strict form of the simply connected case of Rauch's hot spots problem (1974), stated as a conjecture by Krzysztof Burdzy in 2005. It includes smooth convex domains, long a main open case. The Lean statement passed the community comparator runs. Burdzy (8 Oct) and Rodrigo Bañuelos (23 Oct), whose 1999 paper set out the strict and non-strict forms of the problem, each used ChatGPT to extend the result to nonsmooth domains. Both took OpenAI's theorem as an input they had not checked. No specialist has published a check of the proof as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e","url":"https://postcutoff.com/e/2026-10-06-openai-k-median-optimal-approximation-1-plus-2-over-e/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI model claims tight 1 + 2/e approximation for k-median","deck":null,"takeaway":"Family 125 of OpenAI's 722-manuscript release claims a deterministic polynomial-time (1 + 2/e + ε)-approximation for metric k-median, which would settle the problem's approximability under P ≠ NP. OpenAI lists Lean comparator statements for it; no independent expert check has been published yet.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-15","updated":"2026-10-15","orgs":["OpenAI"]},{"id":"2026-10-06-openai-l-equals-rl-equals-bpl-lean-claim","url":"https://postcutoff.com/e/2026-10-06-openai-l-equals-rl-equals-bpl-lean-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims L = RL = BPL with a Lean proof","deck":null,"takeaway":"OpenAI's 6 Oct 2026 math release (family 103, manuscript dated 23 Sept) claims L = RL = BPL. That would mean randomness does not help logarithmic-space computation, a central open problem of complexity theory. The best unconditional bound was Saks and Zhou's O(log^{3/2} n) space, slightly improved by Hoza. A Lean formalization added on 7 Oct proves the equality in an explicit Turing-machine model. The community comparator run of 26 Oct accepted it. A re-check found the proof's definitions identical to the statement file's and the proof built on standard axioms only. It notes that it does not show 'that the machine model is the standard one'. No complexity theorist has assessed it publicly as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-log-abundance-nonvanishing-claim","url":"https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-claim/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims the abundance conjecture; Jihao Liu raises priority","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release, 'Log abundance in characteristic zero' (dated 24 Sept), claims the log abundance conjecture in every dimension in characteristic zero, and with it canonical nonvanishing for smooth projective varieties. Both are central open problems of the minimal model program, proved only up to dimension three. Companion papers claim Iitaka's subadditivity conjecture and related results. On 7 Oct Jihao Liu (Peking University) submitted a human-led proof that nonvanishing implies abundance. arXiv announced it only on 28 Oct. His comment line says he was 'forced to submit here due to an announcement of the proof of the abundance conjecture by OpenAI'. In a remark he suggests that an earlier draft, sent to 'codex chat' before July 2026, may have fed OpenAI's work. No birational geometer has assessed the claims publicly, there is no Lean proof, and OpenAI has not responded.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Peking University"]},{"id":"2026-10-06-openai-modularity-elliptic-curves-imaginary-quadratic","url":"https://postcutoff.com/e/2026-10-06-openai-modularity-elliptic-curves-imaginary-quadratic/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI: elliptic curves over imaginary quadratic fields are modular","deck":null,"takeaway":"Family 030 of OpenAI's 6 Oct 2026 release is a 30-page manuscript proving that every elliptic curve over every imaginary quadratic field is modular. Caraiani and Newton had done this for a positive proportion of fields; the new step handles curves whose mod-3 and mod-5 representations are both reducible. Frank Calegari calls it 'a really nice trick rather than a fundamental breakthrough'. Within two days Bao Le Hung posted a note saying the same idea also proves modularity over all totally real fields, itself a long-open problem. Nothing is formalized or refereed.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-nearby-lagrangian-conjecture-counterexample","url":"https://postcutoff.com/e/2026-10-06-openai-nearby-lagrangian-conjecture-counterexample/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims a counterexample to the nearby Lagrangian conjecture","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release (family 340, dated 23 Sept) claims to disprove the nearby Lagrangian conjecture, a central conjecture of symplectic topology posed by Arnold, Lalonde and Sikorav. For some large even N, it builds a closed exact Lagrangian in the cotangent bundle of S⁹×S^{N−1} that is diffeomorphic to the base but cannot be moved onto the zero section by a Hamiltonian isotopy. Earlier work had shown such Lagrangians are always simple homotopy equivalent to the base (Abouzaid–Kragh). On 15 Oct Wenyuan Li gave a seminar talk on how the construction works, 'assuming the counterexample is correct'. No Lean proof; no expert verdict.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":1,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-periodic-tiling-conjecture-dimension-three","url":"https://postcutoff.com/e/2026-10-06-openai-periodic-tiling-conjecture-dimension-three/","date":"2026-10-06","date_precision":"day","short_title":"Periodic tiling conjecture fails in dimension 3 (OpenAI claim, Lean-checked)","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 maths release (family 155, dated 23 Sept) claims a finite set of cubes in ℤ³ whose translates tile space, while no tiling by them is periodic. The same holds for its solid version in ℝ³. Greenfeld and Tao had disproved the periodic tiling conjecture only in some very high dimension. In one and two dimensions the conjecture is true, so three would be the lowest dimension where it fails. The Lean proof, which also covers the lower dimensions, was accepted in a community Comparator run. Erik Demaine and Stefan Langerman reused the paper's gadgets to prove that deciding whether one connected polycube tiles 3D space is undecidable. Chaim Goodman-Strauss said the paper has novel ideas but is 'unnecessarily difficult to read'. No published expert check of the whole proof was found as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":9,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-polyakov-o3-sigma-model-mass-gap-claim","url":"https://postcutoff.com/e/2026-10-06-openai-polyakov-o3-sigma-model-mass-gap-claim/","date":"2026-10-06","date_precision":"day","short_title":"2D O(n) mass gap (Polyakov) and O(3) continuum limit claimed (OpenAI)","deck":null,"takeaway":"Five manuscripts credited to 'OpenAI' (23 Sept to 5 Oct, public on 6 Oct) claim the lattice and continuum versions of a problem that physicists call the warm-up for Yang–Mills. The 2D classical O(n) model with n ≥ 3 is claimed to have exponentially decaying correlations at every positive temperature, as Polyakov predicted in 1975. A massive continuum O(3) field theory satisfying the Osterwalder–Schrader axioms, with a unique vacuum and a positive gap, is also claimed, along with the exact O(4) mass asymptotics. Only the 20-page decay theorem is in Lean. Roland Bauerschmidt called the first paper groundbreaking and comprehensible and the second 'horrible'. Zohar Komargodski noted the decay was known from numerics for decades.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-release-3d-restriction-bochner-riesz-claims","url":"https://postcutoff.com/e/2026-10-06-openai-release-3d-restriction-bochner-riesz-claims/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims 3D restriction and Bochner–Riesz conjectures","deck":null,"takeaway":"Beside quasi-RH and Unique Games, OpenAI's 6 Oct catalogue claims a whole cluster of central harmonic-analysis conjectures in three dimensions: Stein's restriction conjecture for the sphere, the diagonal extension conjecture, Bochner–Riesz, Sogge's local smoothing and the planar endpoint of Carleson's Schrödinger problem. None has a Lean formalization and no expert review has been published; the first arXiv paper to cite one appeared on 15 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":7,"official":7,"filed":"2026-10-19","updated":"2026-10-26","orgs":["OpenAI"]},{"id":"2026-10-06-openai-sidorenko-conjecture-counterexample-claim","url":"https://postcutoff.com/e/2026-10-06-openai-sidorenko-conjecture-counterexample-claim/","date":"2026-10-06","date_precision":"day","short_title":"Sidorenko's conjecture disproved? (OpenAI, Lean)","deck":null,"takeaway":"A 51-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims that Sidorenko's 1986 conjecture is false. Sidorenko predicted that every bipartite graph H appears in any host graph at least as often as in a random graph of the same edge density. The counterexample H is explicit: the incidence graph of 22 triples on 13 points, with 35 vertices and 66 edges. The host graph is shown to exist by sampling from a finite weighted kernel but is not written down. Lean proves the existence statement and it passed a community comparator run. The same H also refutes the Skokan–Thoma forcing conjecture.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-spin-one-haldane-gap-proof","url":"https://postcutoff.com/e/2026-10-06-openai-spin-one-haldane-gap-proof/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI proves the spin-one Haldane gap (periodic chains)","deck":null,"takeaway":"Two manuscripts in OpenAI's 6 Oct 2026 math release (dated 24 Sept) prove that the spin-one antiferromagnetic Heisenberg chain has a spectral gap that stays positive as even periodic rings grow. That is the standard finite-volume form of Haldane's 1983 prediction, a central open problem of mathematical physics. Hal Tasaki, a leading expert on the problem, studied the analytic part. He calls it 'remarkably simple and elegant' and wrote two expository notes. He has not checked the computer-assisted estimates. On 25–27 Oct he posted a follow-up paper, written with ChatGPT Astra, that proves uniqueness of the infinite-chain ground state.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":4,"filed":"2026-10-28","updated":"2026-10-28","orgs":["OpenAI"]},{"id":"2026-10-06-openai-unit-distance-upper-bound-power-saving-claim","url":"https://postcutoff.com/e/2026-10-06-openai-unit-distance-upper-bound-power-saving-claim/","date":"2026-10-06","date_precision":"day","short_title":"Unit-distance bound below n^(4/3) claimed (OpenAI, Lean)","deck":null,"takeaway":"A 53-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims that n points in the plane determine at most O(n^(4/3−δ)) pairs at distance one, for some unspecified δ > 0. The bound n^(4/3) of Spencer, Szemerédi and Trotter had not been improved since 1984. A companion proves the weak pinned Erdős distance conjecture. Both are formalized in Lean and passed a community comparator run, and Terence Tao's optimization-constants repository now lists the new exponent. On 25 Oct a commenter posting as WillSawin explained the argument on erdosproblems.com and said he sees how to make its ineffective step effective. Since OpenAI's May counterexample gave a lower bound of n^(1+c), the exponent now lies strictly between 1 and 4/3.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-doubleai-high-girth-regular-graphs-8-5","url":"https://postcutoff.com/e/2026-10-06-doubleai-high-girth-regular-graphs-8-5/","date":"2026-10-06","date_precision":"day","short_title":"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)","deck":null,"takeaway":"Amnon Shashua's start-up doubleAI posted a 40-page paper credited only to its AI 'Double Agent': for every prime power q, explicit (q³+1)-regular graphs with girth ≥ (8/5)·log_{q³} n − O(1), improving the 4/3 constant that had stood since 1988. A Lean proof is posted; the company says it ran autonomously for under $2,000 of compute. Not peer-reviewed.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":3,"filed":"2026-10-15","updated":"2026-10-15","orgs":["doubleAI"]},{"id":"2026-10-06-eleven-square-packing-optimality-lean","url":"https://postcutoff.com/e/2026-10-06-eleven-square-packing-optimality-lean/","date":"2026-10-06","date_precision":"day","short_title":"Walter Trump's 1979 packing of 11 unit squares proved optimal, with a full Lean verification (AI involvement reported, unconfirmed)","deck":null,"takeaway":"It settles a well-known open case in packing theory, widely known from xkcd.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":5,"filed":"2026-10-07","updated":"2026-10-13","orgs":["Open-source community"]},{"id":"2026-10-06-explicit-tensors-border-rank-3n-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-06-explicit-tensors-border-rank-3n-gpt-6-astra/","date":"2026-10-06","date_precision":"day","short_title":"Explicit tensors of border rank 3n − o(n), past the 'linear flattening barrier'","deck":"Proofs obtained with GPT-6 Astra (Benjamin Lovitz)","takeaway":"The record lower bound for the border rank of an explicit tensor moves from about 2n to about 3n, and the standard method is shown to be unable to pass 2n − 1. The author says GPT-6 Astra was used to obtain the proofs of the main results.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Concordia University","OpenAI"]},{"id":"2026-10-06-multiplier-associativity-hard-for-resolution-gpt-5-6-sol","url":"https://postcutoff.com/e/2026-10-06-multiplier-associativity-hard-for-resolution-gpt-5-6-sol/","date":"2026-10-06","date_precision":"day","short_title":"Why SAT solvers fail on multiplication","deck":"First exponential resolution lower bound for multiplier associativity; 'main ideas and the core proof … are due to GPT-5.6-Sol' (Liew)","takeaway":"SAT solvers are known to choke on multiplication, but nobody had proved that short proofs do not exist. Vincent Liew's preprint proves an exponential lower bound for resolution proofs that n-bit multiplication is associative, resolving an open problem he had posed with Paul Beame. He credits the main ideas and the core proof to GPT-5.6 Sol; a Lean 4 formalization was generated by Claude Fable 5.1.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-13","updated":"2026-10-13","orgs":["OpenAI","Anthropic"]},{"id":"2026-10-06-polynomial-littlewood-offord-gpt-6-pro","url":"https://postcutoff.com/e/2026-10-06-polynomial-littlewood-offord-gpt-6-pro/","date":"2026-10-06","date_precision":"day","short_title":"Optimal bound for the polynomial Littlewood–Offord problem, 'discovered autonomously by GPT-6 Pro' (exposition by Grebennikov)","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":1,"filed":"2026-10-07","updated":"2026-10-13","orgs":["OpenAI"]},{"id":"2026-10-06-random-3-sat-threshold-upper-bound-4-268-llm","url":"https://postcutoff.com/e/2026-10-06-random-3-sat-threshold-upper-bound-4-268-llm/","date":"2026-10-06","date_precision":"day","short_title":"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'","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-13","updated":"2026-10-13","orgs":["Independent"]},{"id":"2026-10-05-claude-refutes-3sum-apsp-hypotheses","url":"https://postcutoff.com/e/2026-10-05-claude-refutes-3sum-apsp-hypotheses/","date":"2026-10-05","date_precision":"day","short_title":"Claude discovers an algorithm that refutes the 3SUM and APSP hypotheses","deck":"Truly subquadratic 3SUM and truly subcubic APSP (Alman & Vassilevska Williams, Lean-verified)","takeaway":null,"category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-06","updated":"2026-10-13","orgs":["Anthropic","Columbia University","MIT"]},{"id":"2026-10-05-random-k-sat-satisfiability-conjecture-carenini-openai","url":"https://postcutoff.com/e/2026-10-05-random-k-sat-satisfiability-conjecture-carenini-openai/","date":"2026-10-05","date_precision":"day","short_title":"Random k-SAT threshold exists for every k ≥ 3: a human proof and an OpenAI Lean-checked proof","deck":null,"takeaway":"The satisfiability conjecture says that random k-SAT switches from satisfiable to unsatisfiable at one limiting clause density α_k. Friedgut proved a sharp but possibly n-dependent threshold in 1999. Ding, Sly and Sun proved the conjecture for large k (Annals, 2022). On 5 Oct 2026 Cambridge PhD student Gaia Carenini posted on ECCC a proof for every fixed k ≥ 3. Her report states that none of its arguments was generated by a language model. The next day OpenAI's maths release included a separate proof, dated 25 Sept, which credits Carenini with priority. Its Lean statement was accepted in the community Comparator run, and it adds Θ(n) hitting-time variance and computability of the 3-SAT threshold. Neither proof has had a published expert assessment as of 29 Oct. Neither determines the value of α_3.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-29","updated":"2026-10-29","orgs":["University of Cambridge","OpenAI"]},{"id":"2026-10-05-deepmind-alphaprotein-novo-enzyme-design","url":"https://postcutoff.com/e/2026-10-05-deepmind-alphaprotein-novo-enzyme-design/","date":"2026-10-05","date_precision":"day","short_title":"Google DeepMind's AlphaProtein Novo designs new-to-nature enzymes from scratch (piperidine synthesis, DEHP degradation), with Frances Arnold's lab","deck":null,"takeaway":"Most industrial enzymes start from natural proteins and are improved by mutation.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":5,"official":4,"filed":"2026-10-06","updated":"2026-10-07","orgs":["Google DeepMind","Caltech","University of Pittsburgh"]},{"id":"2026-10-05-rhombicosidodecahedron-not-rupert-claude","url":"https://postcutoff.com/e/2026-10-05-rhombicosidodecahedron-not-rupert-claude/","date":"2026-10-05","date_precision":"day","short_title":"Teen's Claude-coded proof claims rhombicosidodecahedron is not Rupert","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":3,"filed":"2026-10-14","updated":"2026-10-19","orgs":["Anthropic"]},{"id":"2026-10-05-decreasing-diagrams-complete-for-confluence-rta-problem-56","url":"https://postcutoff.com/e/2026-10-05-decreasing-diagrams-complete-for-confluence-rta-problem-56/","date":"2026-10-05","date_precision":"day","short_title":"Decreasing diagrams are complete for confluence","deck":"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)","takeaway":"Is one proof technique enough to certify every confluent rewriting system? Van Oostrom asked in 1993 and conjectured no. Three researchers prove yes, with only three labels, for systems of any cardinality. The mathematics is theirs; the two formalizations were developed with Anthropic's Claude.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-13","updated":"2026-10-13","orgs":["Vrije Universiteit Amsterdam","Taras Shevchenko National University of Kyiv","Anthropic"]},{"id":"2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-04-linear-hadwiger-conjecture-proved-gpt-6-astra/","date":"2026-10-04","date_precision":"day","short_title":"Linear Hadwiger conjecture proved","deck":"K_t-minor-free graphs are O(t)-colourable; proof found by GPT-6 Astra (Norin & Steiner), Lean-verified by Codex","takeaway":"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.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-10-06","updated":"2026-10-19","orgs":["OpenAI","McGill University","ETH Zurich"]},{"id":"2026-10-04-vals-claude-agents-luttinger-compensated-magnetic-semiconductors","url":"https://postcutoff.com/e/2026-10-04-vals-claude-agents-luttinger-compensated-magnetic-semiconductors/","date":"2026-10-04","date_precision":"day","short_title":"Claude agents propose room-temperature magnetic semiconductors","deck":null,"takeaway":"Vals AI reports two candidate magnets for spintronic memory found by a team of Claude Opus 5.5 agents running DFT (PBE+U and HSE06). The first is a new design, YBaMnFeO₅, predicted to order up to about 420–490 K. The second is a Prussian-blue cyanide made in 1999 that nobody had identified as this kind of magnet. Nothing has been measured yet, and the new compound may be hard to make in its ordered form.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":1,"filed":"2026-10-26","updated":"2026-10-26","orgs":["Vals AI","Anthropic"]},{"id":"2026-10-02-kls-conjecture-proved-ai-assisted","url":"https://postcutoff.com/e/2026-10-02-kls-conjecture-proved-ai-assisted/","date":"2026-10-02","date_precision":"day","short_title":"Kannan–Lovász–Simonovits conjecture proved in two AI-assisted preprints","deck":"Song–Zhang's O(1) bound, then Bizeul–Klartag–Lehec ('most proofs … found by ChatGPT')","takeaway":"KLS was among the best-known open problems in high-dimensional geometry, with consequences for sampling algorithms on convex bodies.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-06","updated":"2026-10-13","orgs":["Microsoft Research","Weizmann Institute of Science","OpenAI","Anthropic"]},{"id":"2026-10-02-epsilon-dvoretzky-conjecture-klartag-moshe","url":"https://postcutoff.com/e/2026-10-02-epsilon-dvoretzky-conjecture-klartag-moshe/","date":"2026-10-02","date_precision":"day","short_title":"Klartag and Moshe prove the ε-Dvoretzky conjecture","deck":"The key probabilistic idea came from a ChatGPT discussion","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-05","updated":"2026-10-05","orgs":["Tel Aviv University","Weizmann Institute of Science","OpenAI"]},{"id":"2026-10-02-linnik-constant-3-99-naslund-claude-gpt-astra","url":"https://postcutoff.com/e/2026-10-02-linnik-constant-3-99-naslund-claude-gpt-astra/","date":"2026-10-02","date_precision":"day","short_title":"Linnik's constant ≤ 3.99, written by Claude Opus 5.5 and GPT Astra","deck":null,"takeaway":"Linnik proved in 1944 that the least prime congruent to a mod q is at most q^L for an absolute constant L. Heath-Brown brought L down to 5.5 in 1992 and Xylouris to 5. On 2 Oct Eric Naslund posted on Hexagon a computer-assisted proof that L ≤ 3.99. It rests on a weighted form of a Heath-Brown lemma that Heath-Brown said no neat way of achieving 'seems available'. The paper says it was written, and the research carried out, by Claude Opus 5.5 and GPT Astra 6 under Naslund's prompting and supervision, with some formalization by Grok. Two components are Lean-verified on Palomar; the final bound itself is not. Hexagon lists human understanding as 'some parts', and we found no expert assessment as of 29 Oct.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["Anthropic","OpenAI","xAI"]},{"id":"2026-10-02-meta-muse-spark-six-math-papers","url":"https://postcutoff.com/e/2026-10-02-meta-muse-spark-six-math-papers/","date":"2026-10-02","date_precision":"day","short_title":"Meta publishes six maths papers made by mathematicians working with Muse Spark in ordinary meta.ai chat, saying five answer open questions","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":8,"filed":"2026-10-03","updated":"2026-10-13","orgs":["Meta"]},{"id":"2026-10-02-tomon-triangle-free-box-graphs-astra","url":"https://postcutoff.com/e/2026-10-02-tomon-triangle-free-box-graphs-astra/","date":"2026-10-02","date_precision":"day","short_title":"ChatGPT Astra finds geometric triangle-free graphs with near-optimal chromatic number, the first improvement on Burling's 1965 box bound","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-05","updated":"2026-10-05","orgs":["Umeå University","OpenAI"]},{"id":"2026-10-01-lehmer-permutation-conjecture-opus-5-5","url":"https://postcutoff.com/e/2026-10-01-lehmer-permutation-conjecture-opus-5-5/","date":"2026-10-01","date_precision":"day","short_title":"Lehmer's 1965 permutation conjecture proved","deck":"Claude Opus 5.5 found the short hypercube proof","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":1,"filed":"2026-10-02","updated":"2026-10-09","orgs":["Eindhoven University of Technology","Anthropic","Harmonic"]},{"id":"2026-10-01-bootloops-claude-shaped-science","url":"https://postcutoff.com/e/2026-10-01-bootloops-claude-shaped-science/","date":"2026-10-01","date_precision":"day","short_title":"Harvard physicist Matthew Schwartz releases BootLoops, an open-source harness that let Claude produce 36 manuscripts in 18 fields in three months","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":7,"official":4,"filed":"2026-10-05","updated":"2026-10-29","orgs":["Harvard University","Anthropic"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":5,"filed":"2026-09-30","updated":"2026-10-19","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-09-30-mt0-svg-ai-generated-tammes-15-robbins-frog-model-lean","url":"https://postcutoff.com/e/2026-09-30-mt0-svg-ai-generated-tammes-15-robbins-frog-model-lean/","date":"2026-09-30","date_precision":"day","short_title":"mt0-svg: AI-made Tammes N=15, Robbins V>2, frog model (Lean)","deck":null,"takeaway":"Between Sept 28 and Oct 18, 2026 one pseudonymous GitHub account released six papers whose 'results, proofs and code were found and written by AI', each with a Lean 4 proof and comparator check: the first new case of the Tammes problem since N=14 (2015), a 1996 question on Robbins' problem, and both open cases of a 2017 frog-model conjecture. No model is named and no expert review is known.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":11,"official":10,"filed":"2026-10-18","updated":"2026-10-26","orgs":["mt0-svg (pseudonymous)"]},{"id":"2026-09-30-barvinok-log-concavity-totally-unimodular-astra","url":"https://postcutoff.com/e/2026-09-30-barvinok-log-concavity-totally-unimodular-astra/","date":"2026-09-30","date_precision":"day","short_title":"GPT-6 Astra proves Barvinok's log-concavity question for contingency tables on lines, giving lattice-point bounds for all totally unimodular polytopes","deck":null,"takeaway":"It is a named question in a central area of combinatorial counting, with the core proof credited to the model.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-05","updated":"2026-10-05","orgs":["OpenAI"]},{"id":"2026-09-30-google-era-flusight-flu-forecasting","url":"https://postcutoff.com/e/2026-09-30-google-era-flusight-flu-forecasting/","date":"2026-09-30","date_precision":"day","short_title":"Google's ERA-built model ranks #1 of 39 in the CDC FluSight 2025–26 flu-hospitalization forecasting challenge","deck":null,"takeaway":"It is a real-world, prospectively scored test of AI-generated scientific software, not a retrospective benchmark.","category":"science","category_label":"Science & math","importance":2,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":1,"official":1,"filed":"2026-10-01","updated":"2026-10-01","orgs":["Google Research"]},{"id":"2026-09-29-list-total-colouring-conjecture-false-astra","url":"https://postcutoff.com/e/2026-09-29-list-total-colouring-conjecture-false-astra/","date":"2026-09-29","date_precision":"day","short_title":"List Total Colouring Conjecture (late 1990s) disproved","deck":"Counterexample found by ChatGPT 6 Astra Ultra 'with little input'","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":2,"filed":"2026-10-05","updated":"2026-10-07","orgs":["OpenAI","University of Victoria"]},{"id":"2026-09-29-lincodeevolve-record-binary-codes","url":"https://postcutoff.com/e/2026-09-29-lincodeevolve-record-binary-codes/","date":"2026-09-29","date_precision":"day","short_title":"EPFL's LinCodeEvolve (Viazovska, Abbe) finds seven record-breaking binary linear codes with LLM-guided program search","deck":null,"takeaway":"Viazovska, who solved sphere packing in dimensions 8 and 24, is now co-authoring LLM-search papers.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-09-30","orgs":["EPFL"]},{"id":"2026-09-29-lovasz-conjecture-nearly-linear-bound-gpt-5-6-sol","url":"https://postcutoff.com/e/2026-09-29-lovasz-conjecture-nearly-linear-bound-gpt-5-6-sol/","date":"2026-09-29","date_precision":"day","short_title":"GPT-5.6 Sol supplies key steps in a nearly linear bound for the Lovász path conjecture","deck":null,"takeaway":"This is a large quantitative step on a famous 57-year-old problem in graph theory.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-05","updated":"2026-10-05","orgs":["OpenAI"]},{"id":"2026-09-29-microsoft-quine-biology-world-model","url":"https://postcutoff.com/e/2026-09-29-microsoft-quine-biology-world-model/","date":"2026-09-29","date_precision":"day","short_title":"Microsoft Research unveils Quine, a multimodal 'world model of biology' validated in pancreatic-cancer screens with the Broad Institute","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-10-05","updated":"2026-10-05","orgs":["Microsoft Research","Broad Institute"]},{"id":"2026-09-28-four-body-convex-central-configurations-unique-tomar","url":"https://postcutoff.com/e/2026-09-28-four-body-convex-central-configurations-unique-tomar/","date":"2026-09-28","date_precision":"day","short_title":"Four-body problem: uniqueness of convex central configurations for all masses claimed (Simó–Yoccoz conjecture), AI-found and formalized in Lean (Tomar)","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-13","updated":"2026-10-19","orgs":["independent"]},{"id":"2026-09-28-schiffer-pompeiu-convex-higher-dimensions","url":"https://postcutoff.com/e/2026-09-28-schiffer-pompeiu-convex-higher-dimensions/","date":"2026-09-28","date_precision":"day","short_title":"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","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Jizhou Guo"]},{"id":"2026-09-28-mathathon-joint-statement","url":"https://postcutoff.com/e/2026-09-28-mathathon-joint-statement/","date":"2026-09-28","date_precision":"day","short_title":"Caltech 'Mathathon' is reworked into 'Old Problems, New Proofs' after an open letter from mathematicians","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":2,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Caltech","XTX Markets","SAIR"]},{"id":"2026-09-27-erdos-993-forest-independence-polynomials-unimodal","url":"https://postcutoff.com/e/2026-09-27-erdos-993-forest-independence-polynomials-unimodal/","date":"2026-09-27","date_precision":"day","short_title":"Erdős Problem #993 (1987)","deck":"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)","takeaway":"A 1987 question of Alavi, Malde, Schwenk and Erdős asks whether the numbers of independent sets of each size in a tree or forest always rise and then fall. Tong Zhang and Wei Li claimed a computer-assisted proof on 27 Sept 2026 with help from GPT-6 Astra. A second proof (arXiv, 6 Oct) says AI systems produced all of its new mathematics, its Lean proof and most of its text.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-10-13","updated":"2026-10-16","orgs":["Northwestern Polytechnical University","University of Toledo","OpenAI","Anthropic"]},{"id":"2026-09-27-poincare-conjecture-lean-formalization","url":"https://postcutoff.com/e/2026-09-27-poincare-conjecture-lean-formalization/","date":"2026-09-27","date_precision":"day","short_title":"Perelman's proof of the Poincaré conjecture formalised in Lean, with AI-generated code","deck":null,"takeaway":"After Fermat's Last Theorem (Claude, Sept 2026), this is the second landmark formalization of a famous proof in a month.","category":"science","category_label":"Science & 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(1989), citing OpenAI's '100 open problems' claim","deck":null,"takeaway":"If correct, it would settle one of the best-known open problems in the calculus of variations.","category":"science","category_label":"Science & math","importance":4,"confidence":"low","status":{"key":"pending","labels":["Awaiting review"]},"sources":1,"official":1,"filed":"2026-09-30","updated":"2026-10-19","orgs":["Francesco Deangelis","University of Münster","OpenAI"]},{"id":"2026-09-22-odlyzko-poonen-conjecture-proved","url":"https://postcutoff.com/e/2026-09-22-odlyzko-poonen-conjecture-proved/","date":"2026-09-22","date_precision":"day","short_title":"Odlyzko–Poonen conjecture (1993) proved unconditionally","deck":"'The proofs are due to GPT-6 Astra', which also wrote a 22,000-line Lean formalisation","takeaway":"The unconditional case had stayed open after Breuillard and Varjú's conditional proof in 2019.","category":"science","category_label":"Science & 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All three teams disclose AI help; Google's 250-page proof says 'the overwhelming majority of the novel ideas' came from AI and is checked end-to-end in Lean.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":8,"official":6,"filed":"2026-10-09","updated":"2026-10-13","orgs":["Google","Chinese University of Hong Kong","FPT University","OpenAI"]},{"id":"2026-09-21-openai-100-open-problems-claim","url":"https://postcutoff.com/e/2026-09-21-openai-100-open-problems-claim/","date":"2026-09-21","date_precision":"day","short_title":"OpenAI says an internal model resolved 100+ long-standing open problems in 24 days of training (list released 6 Oct: 722 manuscripts)","deck":null,"takeaway":"If substantiated, it would mean open problems are being resolved at industrial scale.","category":"science","category_label":"Science & 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The winning pair computed their polynomials without AI, while 'Almost all the other participants relied on the technology'.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-10-14","updated":"2026-10-14","orgs":["SAIR Foundation","LMFDB","Paderborn University","RPTU Kaiserslautern"]},{"id":"2026-09-10-pi-irrationality-measure-bai-gpt","url":"https://postcutoff.com/e/2026-09-10-pi-irrationality-measure-bai-gpt/","date":"2026-09-10","date_precision":"day","short_title":"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","deck":null,"takeaway":"The best known bound on how well π can be approximated by fractions moved for the first time since 2019, in a preprint whose author says OpenAI models helped with parameter search, proofs and computation. Claude-assisted papers then moved the ζ(3) and ζ(2) records the same way. OpenAI's 6 Oct release goes much further, claiming the conjectured exact value μ(π) = 2 with a Lean proof, which would make all these upper bounds obsolete.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":8,"official":7,"filed":"2026-10-11","updated":"2026-10-12","orgs":["OpenAI","Anthropic"]},{"id":"2026-09-09-pierce-birkhoff-conjecture-disproved","url":"https://postcutoff.com/e/2026-09-09-pierce-birkhoff-conjecture-disproved/","date":"2026-09-09","date_precision":"day","short_title":"Pierce–Birkhoff conjecture disproved by a multi-agent GPT + Claude harness","deck":"Counterexamples Lean-verified","takeaway":"Pierce–Birkhoff is a classic problem in real algebraic geometry and ordered rings, open for 70 years.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-10-19","orgs":["University of Chicago","Lek-Heng Lim","Zehua Lai","Junyu Ren","OpenAI","Anthropic"]},{"id":"2026-09-09-google-nasa-jpl-mapl-emit-methane","url":"https://postcutoff.com/e/2026-09-09-google-nasa-jpl-mapl-emit-methane/","date":"2026-09-09","date_precision":"day","short_title":"Google and NASA JPL release MAPL-EMIT, an AI model that maps methane plumes worldwide","deck":"It reads data from the EMIT space instrument; the paper is in PNAS.","takeaway":"Methane is a strong short-term warming gas (Google cites about 30x the warming potential of CO2 over 100 years).","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":4,"filed":"2026-10-01","updated":"2026-10-01","orgs":["Google Research","NASA JPL"]},{"id":"2026-09-08-openai-navier-stokes-blowup","url":"https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/","date":"2026-09-08","date_precision":"day","short_title":"OpenAI claims a Millennium Prize problem","deck":"10,000 AI agents prove forced Navier–Stokes blow-up; priority dispute erupts","takeaway":"It is the first credible AI claim on a Clay Millennium Prize problem, even if only a technically permitted variant.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"disputed","labels":["Disputed"]},"sources":42,"official":4,"filed":"2026-09-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-09-08-alphagenome-atlas","url":"https://postcutoff.com/e/2026-09-08-alphagenome-atlas/","date":"2026-09-08","date_precision":"day","short_title":"AlphaGenome Atlas predicts the effect of all ~9 billion possible single-letter human DNA variants","deck":null,"takeaway":"Like the AlphaFold database for proteins, it turns a model into a lookup resource that could speed up rare-disease diagnosis.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-09-29","updated":"2026-09-30","orgs":["Google DeepMind"]},{"id":"2026-09-07-koethe-conjecture-disproved","url":"https://postcutoff.com/e/2026-09-07-koethe-conjecture-disproved/","date":"2026-09-07","date_precision":"day","short_title":"Pre-release GPT-6 Astra disproves the Köthe conjecture with a Lean-verified counterexample","deck":null,"takeaway":"If it survives review, it resolves one of the most famous open problems in ring theory, found autonomously and verified formally.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI","Epoch AI"]},{"id":"2026-09-07-anandkumar-euler-singularity-r3","url":"https://postcutoff.com/e/2026-09-07-anandkumar-euler-singularity-r3/","date":"2026-09-07","date_precision":"day","short_title":"Caltech team (Anandkumar) reports a stable self-similar singularity candidate for the unforced 3D Euler equations on R³, found with PINNs and LLM help","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Caltech"]},{"id":"2026-09-07-erdos-1122-monotone-additive-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-07-erdos-1122-monotone-additive-gpt-6-astra/","date":"2026-09-07","date_precision":"day","short_title":"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","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-10","updated":"2026-10-10","orgs":["OpenAI"]},{"id":"2026-09-06-collatz-positive-density-log-time-mazur-codex","url":"https://postcutoff.com/e/2026-09-06-collatz-positive-density-log-time-mazur-codex/","date":"2026-09-06","date_precision":"day","short_title":"Collatz positive-density convergence, AI-developed, Lean-checked","deck":null,"takeaway":"The first linear lower bound for Collatz-convergent integers (earlier bounds were X^0.84, then X^0.90): an explicit positive proportion of n < X reach 1 within 10.46 ln n steps. Developed mainly by AI agents, Lean-checked twice, not yet peer reviewed. The Collatz conjecture itself stays open.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":3,"filed":"2026-10-18","updated":"2026-10-19","orgs":["ProofAtlas","OpenAI","Axiom Math"]},{"id":"2026-09-06-odin-ai-agent-positive-curvature-s3xs3-hopf-sign-conjecture","url":"https://postcutoff.com/e/2026-09-06-odin-ai-agent-positive-curvature-s3xs3-hopf-sign-conjecture/","date":"2026-09-06","date_precision":"day","short_title":"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)","deck":null,"takeaway":"If correct, a 1930s conjecture of Hopf falls in dimension 6, with the construction and proof credited to an AI agent; no geometer has confirmed it yet.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":6,"filed":"2026-10-16","updated":"2026-10-16","orgs":["Odin Automatic AI Research Agent","Harvard T.H. Chan School of Public Health"]},{"id":"2026-09-05-i3322-bell-inequality-infinite-dimensions","url":"https://postcutoff.com/e/2026-09-05-i3322-bell-inequality-infinite-dimensions/","date":"2026-09-05","date_precision":"day","short_title":"The I3322 Bell inequality needs infinite dimensions","deck":"Pál–Vértesi conjecture (2010) proved from an approximate proof by GPT-5.5 Pro","takeaway":"It settles a basic question about how much entanglement quantum correlations can require, in the smallest possible Bell scenario.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Andrea Coladangelo","University of Washington","OpenAI"]},{"id":"2026-09-04-claude-formalizes-fermats-last-theorem","url":"https://postcutoff.com/e/2026-09-04-claude-formalizes-fermats-last-theorem/","date":"2026-09-04","date_precision":"day","short_title":"Claude produces the first complete machine-checked proof of Fermat's Last Theorem in Lean, in 11 days","deck":null,"takeaway":"Formalising FLT had been a flagship multi-year human project.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":2,"filed":"2026-09-29","updated":"2026-11-09","orgs":["Anthropic"]},{"id":"2026-09-04-reed-solomon-list-decoding-capacity-ai","url":"https://postcutoff.com/e/2026-09-04-reed-solomon-list-decoding-capacity-ai/","date":"2026-09-04","date_precision":"day","short_title":"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)","deck":null,"takeaway":"In early September 2026 two preprints settled long-standing problems about Reed–Solomon codes, the most widely used error-correcting codes.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":8,"filed":"2026-10-07","updated":"2026-10-29","orgs":["Simons Institute","OpenAI","Ethereum Foundation"]},{"id":"2026-09-03-dying-percolation-theta-pc-zero","url":"https://postcutoff.com/e/2026-09-03-dying-percolation-theta-pc-zero/","date":"2026-09-03","date_precision":"day","short_title":"Claude-written Lean proof claims the dying percolation conjecture θ(p_c)=0 in every dimension","deck":null,"takeaway":"If the formal statement matches the intended theorem, a famous problem in mathematical physics is settled by machine-written formal mathematics.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":6,"filed":"2026-09-29","updated":"2026-10-22","orgs":["Anthropic","OpenAI"]},{"id":"2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-erdos-sos-conjecture-proved-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"GPT-6 Astra proves the Erdős–Sós conjecture with a short counting argument","deck":"Mathematicians race to simplify and extend it","takeaway":"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\".","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":10,"official":9,"filed":"2026-09-30","updated":"2026-09-30","orgs":["OpenAI","Epoch AI"]},{"id":"2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1","url":"https://postcutoff.com/e/2026-09-03-frontiermath-erdos-astra-disproves-erdos-problem-1/","date":"2026-09-03","date_precision":"day","short_title":"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","deck":null,"takeaway":"Problem #1 is probably the longest-standing open Erdős problem.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":10,"official":7,"filed":"2026-09-30","updated":"2026-10-16","orgs":["OpenAI","Epoch AI"]},{"id":"2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra","url":"https://postcutoff.com/e/2026-09-03-smale-mean-value-conjecture-disproved-gpt-6-astra/","date":"2026-09-03","date_precision":"day","short_title":"GPT-6 Astra autonomously disproves Smale's mean value conjecture in Lean","deck":"HKU mathematicians build on it to show Smale's constant exceeds 1.1018","takeaway":"A Smale problem from 1981 fell to an unattended model run in an Epoch AI benchmark, with a Lean proof; six weeks later two complex analysts turned the construction into a quantitative bound M > 1.1018.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":9,"official":9,"filed":"2026-10-16","updated":"2026-11-12","orgs":["OpenAI","Epoch AI","University of Hong Kong"]},{"id":"2026-09-03-weathernext-3","url":"https://postcutoff.com/e/2026-09-03-weathernext-3/","date":"2026-09-03","date_precision":"day","short_title":"Google DeepMind's WeatherNext 3 learns from live satellite data","deck":"Hourly 5 km global forecasts and up to 60% better precipitation skill","takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":5,"official":4,"filed":"2026-09-29","updated":"2026-10-07","orgs":["Google DeepMind","Google Research"]},{"id":"2026-09-03-male-fruit-fly-connectome","url":"https://postcutoff.com/e/2026-09-03-male-fruit-fly-connectome/","date":"2026-09-03","date_precision":"day","short_title":"Complete male fruit fly nervous-system connectome (166,000 neurons), reconstructed with Google AI, published in Cell","deck":null,"takeaway":"Connectomics is one of the clearest cases where AI turns an impossible manual task into a feasible one.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result 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Atomics"]},{"id":"2026-09-02-nvidia-nemotron-ioi-2026","url":"https://postcutoff.com/e/2026-09-02-nvidia-nemotron-ioi-2026/","date":"2026-09-02","date_precision":"month","short_title":"NVIDIA's Nemotron-3-Ultra-CC outscores every human at IOI 2026","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":3,"filed":"2026-09-29","updated":"2026-09-29","orgs":["NVIDIA"]},{"id":"2026-09-02-krenn-nature-designing-physics-experiments-ai","url":"https://postcutoff.com/e/2026-09-02-krenn-nature-designing-physics-experiments-ai/","date":"2026-09-02","date_precision":"day","short_title":"Nature: 'Designing physics experiments with artificial intelligence' (Krenn group) on AI-found setups that beat human designs","deck":null,"takeaway":"It shows AI in science moving from analysing data to designing the instruments and experiments themselves.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":3,"filed":"2026-09-30","updated":"2026-09-30","orgs":["University of Tübingen","TU Wien","University of Vienna"]},{"id":"2026-08-30-bounded-prime-gaps-186","url":"https://postcutoff.com/e/2026-08-30-bounded-prime-gaps-186/","date":"2026-08-30","date_precision":"day","short_title":"GPT-6 Astra lowers the bounded prime gaps record from 246 to 186","deck":null,"takeaway":"Bounded prime gaps were one of the celebrated stories of 2013–14.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["OpenAI"]},{"id":"2026-08-28-matrix-spencer-conjecture-proved","url":"https://postcutoff.com/e/2026-08-28-matrix-spencer-conjecture-proved/","date":"2026-08-28","date_precision":"day","short_title":"Matrix Spencer conjecture proved","deck":"Authors credit GPT-5.6 Sol Pro with 'the heavy-lifting' for the key lemma","takeaway":"Matrix Spencer was one of the headline open problems in discrepancy theory.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Emrullah Akbas","Suvrit Sra","OpenAI"]},{"id":"2026-08-27-antigravity-teamwork-open-problems","url":"https://postcutoff.com/e/2026-08-27-antigravity-teamwork-open-problems/","date":"2026-08-27","date_precision":"day","short_title":"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","deck":null,"takeaway":"It is another data point in the summer-2026 wave of AI-assisted results on open problems.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":6,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Google","Google DeepMind"]},{"id":"2026-08-26-erdos-rankin-large-prime-gaps-improved","url":"https://postcutoff.com/e/2026-08-26-erdos-rankin-large-prime-gaps-improved/","date":"2026-08-26","date_precision":"day","short_title":"GPT-5.6 improves the Erdős–Rankin / Ford–Green–Konyagin–Maynard–Tao bound for large prime gaps","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":2,"filed":"2026-09-29","updated":"2026-10-15","orgs":["OpenAI"]},{"id":"2026-08-23-hopf-problem-s6-complex-structure","url":"https://postcutoff.com/e/2026-08-23-hopf-problem-s6-complex-structure/","date":"2026-08-23","date_precision":"day","short_title":"Claude-assisted construction claims a complex structure on the 6-sphere, answering Hopf's 1947 problem (pending verification)","deck":null,"takeaway":"The existence of a complex structure on S⁶ is one of the best-known open problems in geometry.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":3,"filed":"2026-09-29","updated":"2026-10-30","orgs":["Anthropic"]},{"id":"2026-08-23-elliptic-curve-rank-record-31","url":"https://postcutoff.com/e/2026-08-23-elliptic-curve-rank-record-31/","date":"2026-08-23","date_precision":"day","short_title":"Claude-assisted search breaks the elliptic curve rank record","deck":"Rank 30, then 31","takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Anthropic"]},{"id":"2026-08-19-caratheodory-loewner-smooth-counterexamples-alpoge-claude","url":"https://postcutoff.com/e/2026-08-19-caratheodory-loewner-smooth-counterexamples-alpoge-claude/","date":"2026-08-19","date_precision":"day","short_title":"Smooth Carathéodory and Loewner conjectures disproved with Claude","deck":null,"takeaway":"On 19 Aug 2026 Levent Alpöge (Anthropic) posted an explicit formula on X: a smooth strictly convex closed surface in ℝ³ with exactly one umbilic point, against Carathéodory's conjecture that every such surface has at least two, and umbilics of index 1 + k/2, against Loewner's index ≤ 1. He credited John-Paul Smith 'and of course claude'. On 25 Oct the umbilic specialists Naoya Ando and Masaaki Umehara posted a simpler, self-contained construction on arXiv, found with GPT-6.1's help. The analytic case, the classical form of both conjectures, is not affected.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":3,"filed":"2026-10-27","updated":"2026-10-27","orgs":["Anthropic","Kumamoto University","Institute of Science Tokyo","OpenAI"]},{"id":"2026-08-19-yau-tian-donaldson-cscK-disproof-claim","url":"https://postcutoff.com/e/2026-08-19-yau-tian-donaldson-cscK-disproof-claim/","date":"2026-08-19","date_precision":"day","short_title":"Peking University preprint claims an AI-found disproof of the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics","deck":null,"takeaway":"The Yau–Tian–Donaldson programme is central to modern complex geometry, and the paper is explicit that its main result is AI-generated.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-10-07","orgs":["Peking University","Jihao Liu","Anthropic","OpenAI"]},{"id":"2026-08-18-claude-protein-binders-wet-lab","url":"https://postcutoff.com/e/2026-08-18-claude-protein-binders-wet-lab/","date":"2026-08-18","date_precision":"day","short_title":"Claude autonomously designs protein binders that work in the lab against 14 of 15 targets","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":4,"official":1,"filed":"2026-09-30","updated":"2026-10-07","orgs":["Anthropic","Adaptyv Bio","Twist Bioscience"]},{"id":"2026-08-18-palomar-lean-registry","url":"https://postcutoff.com/e/2026-08-18-palomar-lean-registry/","date":"2026-08-18","date_precision":"day","short_title":"Palomar launches: a registry of Lean-verified mathematics to curb misrepresented AI proof claims","deck":null,"takeaway":"Formal verification became the main way to trust AI mathematics in 2026.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":6,"official":6,"filed":"2026-09-29","updated":"2026-10-29","orgs":["Lean FRO","ICARM"]},{"id":"2026-08-17-alphaevolve-matrix-multiplication-exponent","url":"https://postcutoff.com/e/2026-08-17-alphaevolve-matrix-multiplication-exponent/","date":"2026-08-17","date_precision":"day","short_title":"AlphaEvolve helps lower the matrix multiplication exponent ω to below 2.371177","deck":null,"takeaway":"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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Google DeepMind","MIT"]},{"id":"2026-08-13-banach-isometric-conjecture-solved","url":"https://postcutoff.com/e/2026-08-13-banach-isometric-conjecture-solved/","date":"2026-08-13","date_precision":"day","short_title":"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","deck":null,"takeaway":"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.","category":"science","category_label":"Science & 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physics.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":1,"official":1,"filed":"2026-09-30","updated":"2026-09-30","orgs":["Philip Arathoon","Gavin Ball","Matthew D. 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