October 2026: the arXiv stream of conjectures settled with disclosed AI help continues

This catalogue continues the July–September catalogue. It lists arXiv preprints from the 2–9 October 2026 listings that settle or advance a named conjecture or open question and state an AI contribution. For each paper the PDF’s AI statement was read; quotes are from the papers, and nothing here was independently verified. The blocks below were written during the update runs of 6–13 October and are kept in the order and wording in which they were added (they were moved here from the summer entry on 13 Oct 2026 without changes, apart from one cross-reference). A smaller batch from 5–9 October is in a separate entry.

Oct 2–5 listing (manual arXiv scan, Oct 6 morning; 132 PDFs, 70+ with AI statements)

Own entries: 3SUM/APSP hypotheses refuted by an algorithm Claude found (Alman & Vassilevska Williams, 2610.06783; Claude discovers an algorithm that refutes the 3SUM and APSP hypotheses), the linear Hadwiger conjecture proved by GPT-6 Astra (Norin & Steiner, 2610.05291; Linear Hadwiger conjecture proved) and the KLS conjecture (Song & Zhang 2610.01447; Bizeul, Klartag & Lehec 2610.05474; Kannan–Lovász–Simonovits conjecture proved in two AI-assisted preprints).

• Golod–Shafarevich towers of arithmetic surfaces over Z, answering questions of Bost–Charles, Ihara, Frey–Kani–Völklein and Venkatesh (2022, linear conductor growth). Keller & Pagano, arXiv 2610.064612: worked on it since 2022 and “were able to bring it to fruit only after a collaboration with GPT5.6 Sol, Fable and Aletheia a Gemini-powered internal agent developed at Google DeepMind”.
• Amicable numbers: A(x) ≤ x·exp{−(1/2+o(1)) log x log₃x / log₂x}. Pollack, arXiv 2610.056613: “We report on a proof, found by ChatGPT Astra”.
• First FPRAS for counting common bases of two matroids (and Hadamard products of curvature-nonnegative measures). Chen & Liu, arXiv 2610.06724: after a prompt to “keep trying”, ChatGPT “found a solution”; the authors then recognised its stochastic-gradient-descent framework and generalised it with ChatGPT’s help.
• Simple nil ring (Kaplansky’s 1956 problem, solved by Smoktunowicz 1999): a substantially simpler construction “found by Aristotle (Harmonic) with the assistance of ChatGPT”. Gubarev, arXiv 2610.05384.
• Benson–Etingof–Ostrik conjecture for finite symmetric tensor categories. Coulembier & Flake, arXiv 2610.04096: main ideas of Sections 3–4 “originally obtained using GPT-6 Astra”, from a formula found by numerical experiments with Claude 4.6–5 Opus.
• LeBrun–Salamon conjecture for quaternionic dimension ≤ 14 (real dimension ≤ 56). Daura Serrano, Gohr, Lawn & Schedler, arXiv 2610.06336: LLMs “used extensively”, mainly GPT-6-Astra and GPT-5.6-Sol, with occasional Claude Opus 5/5.5 and Claude Fable.
• Automorphic commutator relation (Liu–Wang, relative geometric Langlands) without extra hypotheses. Shurui Liu, arXiv 2610.05781: draft produced by the agentic system Rethlas in about 1 hour with OSVerify; Danus on GPT-6-Astra failed in 5 hours.
• Gaussian elimination growth factor under partial pivoting for Gaussian matrices. Urschel, arXiv 2610.06785: “the result of human–AI interaction involving ChatGPT Sol 5.6 & Astra 6”.
• Strong freezing in the binary perceptron (Krauth–Mézard prediction). Gong & Li, arXiv 2610.06275: GPT-6 suggested the restricted partition function that completes the monotonicity argument.
• Tight quantum query lower bounds for XEB sampling (proves an earlier conjecture for Haar-random states). Bhateja, Esmaili & Mantri, arXiv 2610.06721: “ChatGPT Astra 6.1 produced the main results for Fourier sampling” (note: GPT-6.1 Astra was never released publicly).
• Noncommutative AM–GM inequality holds for n ≥ 2⌈m/2⌉². Zhong, arXiv 2610.04874: “The proof of Theorem 1.2 was found by GPT6 Astra gradually.”
• Two-dimensional majority rule is P-complete. arXiv 2610.04311: results “assisted by AI agents” (Claude Code with Sonnet/Opus 5.5; Codex with GPT-6 Astra and GPT-6.1 Sol).
• Bernstein–Sato zero loci need not be linear (counterexample found with ChatGPT Pro; Shi & Zuo, 2610.04408); Suciu’s conjecture on hyperplane arrangements (key idea proposed by GPT-5.6 Sol; 2610.06683); Boolean Hidden Matching generalisations (two key ideas from GPT-6 Astra; 2610.06789); stabilizer extent / Clifford rank bounds (GPT-6 Astra and GPT-5.6 Sol “used extensively, including to generate mathematical arguments”; 2610.06819); Max-k-Cut games counterexample (developed with GPT-6 Astra; 2610.04948); (q−1)-nests (“comparable work by a human would warrant co-authorship” for ChatGPT 5.6; 2610.05514); Poly–Raby theorem in infinite dimensions (example obtained directly with GPT-6 Astra; 2610.06595); non-plain virtually free group counterexample suggested by a commercial LLM (2610.04491); Borwein–Bradley-type identities for 1/π⁴ (Claude Opus 5.5 and GPT-6; 2610.04770); OEIS conjectures of Kimberling and others proved with Claude (2610.04842, 2610.05776); steady-state response curves parameters found by Claude Opus 5.5 (2610.04029); isosceles-free grid colouring as an AI-assisted proof case study (2610.04194).
• Benchmarks/tools: VeriContest’s 1,325 theorems all proved via a Lean-backed Rust verifier, 1,259 in one run on Claude Opus 5.5 at a median $1.17 per proof (2610.03994); AIProver auto-formalization agent lifts Claude Code/Codex semantic correctness to 79.8%/62.4% (2610.05367).
• Smaller AI roles (exploration, checking, editing): Prym–Green up to genus 30, GW/DT4 on local P², DML(1) property (proof of Prop. 1.6 “learned from ChatGPT 6 Astra”), Binder–DeMarco, Godbersen, Newtonian capacities (auxiliary function from GPT-6.0), Chern-form positivity, ×2×3 entropy (Claude line-by-line checking), and about 25 others.

Oct 5–6 listing (sweep Oct 7; PDFs read)

Own entries: the Artin–Davenport conjecture (cubic forms in 10 variables; key lemma via ChatGPT 6.0; Bonolis, Browning, Glas & Wang, 2610.08226; Artin–Davenport conjecture proved), Woodin’s GCH question for strongly compact cardinals (You, 2610.07537, GPT-6 Astra; Woodin’s question answered), the polynomial Littlewood–Offord bound “discovered autonomously by GPT-6 Pro” (Grebennikov, 2610.08708; Optimal bound for the polynomial Littlewood–Offord problem, ‘discovered autonomously by GPT-6 Pro’ (exposition by Grebennikov)) and Reed–Solomon list decoding up to capacity / proximity gaps (Sept 4–5, GPT-5.6 Sol and ChatGPT 5.6 Pro; surfaced by the TIFR exposition 2610.08610; 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)). A second Poincaré formalization (Frenzymath, 2610.08329) was added to Perelman’s proof of the Poincaré conjecture formalised in Lean, with AI-generated code.

• Erdős–Gallai cycle decomposition conjecture (1960s): every n-vertex graph splits into O(n) cycles and edges. Jaehoon Kim (KAIST), arXiv 2610.07840: “The author made extensive use of ChatGPT (OpenAI) and Claude (Anthropic) in developing the arguments and preparing the text and figures.” The same theorem is family 181 of OpenAI’s 6 Oct release (manuscript dated 24 Sep; see OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems).
• Mumford–Tate conjecture in every degree for hyper-Kähler varieties (plus weight–monodromy). Ito & Zou, arXiv 2610.07890. Most results were human and pre-August, but the reduction in Proposition 2.2 “was pointed out by ChatGPT 5.6 Sol in August 2026” and made the full conjecture available; Proposition 5.7 and Example 5.11 “were found with” ChatGPT 6 Astra’s help.
• Keevash–Mubayi simplex-cluster conjecture in its full range (a common strengthening of the Erdős–Chvátal simplex conjecture and Mubayi’s cluster conjecture). Wu & Feng, arXiv 2610.08567: ChatGPT used “to discuss mathematical details, assist in developing and refining proofs”. The Erdős–Chvátal simplex conjecture itself had been claimed in full by Jing Huang (arXiv 2609.14470, 13 Sep; no AI statement).
• Continuous Aldous–Lyons conjecture fails (non-co-sofic invariant random subgroups in rank-one Lie groups, p-adic groups and Aut(T)); answers a question in Gelander’s ICM paper and builds on the new non-sofic groups (including OpenAI’s, OpenAI’s unreleased ‘Astra’ model claims ten advances in maths and theoretical CS, with Lean proofs). Gelander, arXiv 2610.08581: “The author acknowledges the assistance of OpenAI’s GPT-6 Astra in developing the arguments and preparing the manuscript.”
• McLeod’s 1959 conjecture on complete homogeneous symmetric ratios. Sra, arXiv 2610.07302: “developed through extensive use of an AI assistant (OpenAI’s GPT Pro). None of the claims were obtained zero-shot”; the assistant’s literature survey surfaced the queuing-theory interpretation, while the tensorization trick is the author’s.
• Dirac Faber–Krahn conjecture (strict, non-negative mass, infinite-mass boundary). Ammari & Qiu, arXiv 2610.08665: numerics from ChatGPT 6 Astra suggested the key non-vanishing property, after which “it provides the detailed proof” of Theorems 5.4 and 6.1; the authors simplified and rewrote it.
• Random-oracle unitary synthesis is impossible (and scalable ROM-PRUs ruled out; an O(N)-design construction). Huang, Ramkumar & Wright, arXiv 2610.06700: after the humans set up the definition, “GPT 5.6 Sol … was able to solve the problem” within several hours; GPT-6 Astra helped simplify it. Contrast: OpenAI’s family 283 claims the worst-case Aaronson–Kuperberg unitary synthesis problem has a positive answer.
• Strang’s conjecture (dimension of C¹ cubic splines): a D₃-symmetric counterexample with no quasi-singular vertices. Potu, arXiv 2610.08470: “The search for this triangulation relied heavily on … Claude Fable 5.1.”
• Adámek–Rosický problem (2001): cocompletion under sifted colimits need not be finitely accessible. Kawase, arXiv 2610.08548: GPT-6.1 Sol used “to explore and refine candidate constructions following a strategy devised by the author”.
• Carlet’s cyclic-additive conjecture for Kasami monomials (an NSUCRYPTO 2019 problem). Nagy, McNeil & Vajda, arXiv 2610.07905: Claude Fable 5, Claude Opus 5 and ChatGPT 6 Pro in “sustained mathematical discussions”; Lean 4 formalisation with Aristotle.
• Asymptotic Makar-Limanov rank conjecture. Yan, Shen & Yang, arXiv 2610.06800: main results “obtained through an interaction between the authors and an artificial intelligence agent system, MechMath Agent Team (MMAT)”, with GPT-6 Astra.
• Hendrey–Norin–Wood question: mad is partitionable, giving the clustered chromatic number of bounded-mad graphs. Grzesik et al., arXiv 2610.04645: the potential function “was initially suggested by ChatGPT-5.6 Sol”, but its proof relied on a lemma “whose validity was difficult to verify”, so the authors wrote their own proof.
• Other substantive AI roles: Kakeya compression example answering a question of arXiv 2607.27629 (Du, 2610.05404; ChatGPT 5.6 Sol and 6 Astra); Pólya’s inequality for every Neumann eigenvalue of annuli with Codex-built certificates (Dai & Sun, 2610.05146); absolutely continuous radial convex-order counterexample “found using generative AI tools” (Guéant, 2610.07404); Kuperberg’s quadrilateral “chord conjecture” note from “continuous collaboration” with OpenAI and Anthropic models (Vígh, 2610.08168); Hardy–Littlewood maximal operator not weak (1,1) on lamplighter, BS(1,n) and Sol (Fujiwara, 2610.08121; GPT-5.6 Sol for Lemma 3.1); sharply repulsive points for stochastic subgradient descent (Li & Hu, 2610.07777); 1,142 weight-five bivariate bicycle codes from LLM program evolution (IBM, 2610.06623).
• Smaller AI roles (checking, code, editing): Binder–DeMarco conjecture disproved on an open set (2610.05602; ChatGPT numerics), Prym–Green in even genera 20–30 (Payne & Willwacher, 2610.05646; ChatGPT Pro and Codex), GW/DT4 on local P² (2610.06024; ChatGPT-6 Astra), Godbersen’s conjecture (2610.06806; ChatGPT for checks), SABR right-wing conjectures of Henry-Labordère (2610.04209; Codex and Claude, Lean checks), Szykuła’s Figure 3 automata (2610.07044; Codex), Iwahori Gaiotto conjecture (2610.08243; ChatGPT found three errors), Butler’s conjecture and N!/k bounds (2610.07461), Clunie–Sheil-Small coefficient-difference conjecture disproved (2610.07770; ChatGPT for editing and discussion), FEEC dynamo spectra (2610.01501; ChatGPT and Claude Opus 5.5). Explicitly AI-free ideas: quantum gap amplification (FOCS 2026, 2610.04952; ChatGPT 5.4/Claude Opus 4.6 for elementary calculations only) and Cheq non-finite axiomatizability (2609.39577).

Early October additions (arXiv, 7 Oct)

Own entries: exact overlaps conjecture (Exact overlaps conjecture for self-similar measures on the line proved), LeBrun–Salamon (Two claimed proofs of the LeBrun–Salamon conjecture), multiple-unicast conjecture refuted (Braverman & He refute the 2004 undirected multiple-unicast (network coding) conjecture with a GPT-6-found counterexample on PG(2, 9)).

• Explicit counterexample to Tsirelson’s problem (approximation form; Cqa ≠ Cqc). Gao, Yu & Zhi (CAS), arXiv 2610.102484. A binary linear system game with 1,417,152 equations in 1,889,684 variables has a perfect commuting-operator strategy, while every finite-dimensional quantum strategy succeeds with probability strictly below 1. The full system and the separation are formalized in Lean 4 with Mathlib (repo5). Tsirelson’s problem was already answered negatively by MIP* = RE (2020), but non-constructively. The paper builds on Wang & Zhi’s explicit Connes-embedding polynomial (in the summer catalogue) and on Kun–Thom and Thom (2026). Disclosure: “Generative AI tools (GPT 6 Astra and Fable 5.1) were used to assist with mathematical exploration and proof development, the implementation and computational verification of the explicit construction, the writing and debugging of Lean 4 code, and the drafting and revision of this manuscript.”
• Lê’s conjecture (1982; injective germs ℂ² → ℂ³), reduction to orders 2, 3, 4. Portilla Cuadrado, arXiv 2610.104186. The paper itself is human-led (ChatGPT 5.6 for references and writing), but its disclosure says that when the draft was handed to ChatGPT, “it found a flaw in the argument. After several iterations and a final long session, ChatGPT found a fix for the case mn = 4 and proved the conjecture, also, in the two remaining cases, mn = 2, 3. These cases will appear in another paper”, in which the author “has minor intellectual contribution”. If that follow-up holds, Lê’s conjecture would be settled with the decisive cases due to ChatGPT.
• Polishchuk–Van den Bergh conjecture on curves: explicit S₃-action counterexamples. Pan, arXiv 2610.10147. ChatGPT used “as a tool for exploring proof strategies, refining arguments, and drafting and revising the manuscript”.
• Centroaffine Bernstein conjecture disproved (complete centroaffine extremal hypersurfaces via Calabi compositions). Xing, Sun & Xu, arXiv 2610.09332. “The authors used AI models to assist with proving Theorem 1.1” (model not named).
• Hassanzadeh–Nasrollah Nejad–Simis conjecture on Euler conductors proved, and Ma–Zuo’s integral-closure strengthening disproved. Zhang & Zuo, arXiv 2610.09833. Thanks “AI models, particularly ChatGPT 6 Astra, for valuable assistance in exploring some of the proof strategies and calculations”.
• Coarse graph theory. Georgakopoulos (Warwick), arXiv 2610.09992. “A mid-range AI agent made available by the University of Warwick … CheapGPT, was used to prove Lemma 8.2” (minor role).
• Tamagawa products in BSD, statistically (Angdinata, Lau, Ono, Swaminathan, Vemulapalli, arXiv 2610.10243). Not an open-problem resolution: all 21 formal targets were “autoformalized in Lean by AxiomProver” (Axiom Math; repo).
• Minor-role disclosures the same day: Type-ω Vaught’s conjecture for ω-stable theories (2610.09314, ChatGPT-5.6 Sol proofreading), geometric Arveson–Douglas conjecture via Veronese embeddings (2610.09836, Claude Sonnet 4.6 for spelling), Pappas’s conjecture on regular integral models (2610.09475, ChatGPT found a sign error), the Cyclicity Conjecture (2610.10093, GPT-5.6 via OpenCode for simplifications and literature review; now own entry Claimed proof of the cyclicity conjecture of Perron–Frobenius theory for positive operators (Dobrick, GPT-5.6 used to simplify arguments)).

6–8 Oct additions (sweep 2026-10-09)

Own entries: finite index over treeable, the Jackson–Kechris–Louveau problem (Chen, Tserunyan & Tucker-Drob solve the Jackson–Kechris–Louveau ‘finite index over treeable’ problem (2002); ChatGPT 6 Astra Ultra supplied a ‘crucial observation’), Dualization lower bound under ETH found by GPT-6 Astra (GPT-6 Astra finds a 3SAT reduction showing monotone Dualization (minimal hypergraph transversals) has no polynomial algorithm under ETH), Hilbert number Ω(d² ln² d) with GPT-5.6 Sol and Lean (Hilbert’s 16th problem), and the human-written hyperkähler SYZ proof (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). Found late: the Courtade–Kumar conjecture, proved three times on 21–22 Sep with AI (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).

• Erdős–Sós bipartite-link conjecture disproved: 3-graphs on 497k vertices whose links are all bipartite, with edge density 1/4 + 25/420 903 336 > 0.250000356 (the conjecture said 1/4 + o(1); flag-algebra upper bound 0.258295). Tianchi Yang (Georgia Tech), arXiv 2610.116428: “The author used GPT-6.1 to assist with exploring proof strategies, checking proofs, and improving the exposition.”
• Mongardi’s conjecture (finite symplectic group actions on K3^[n]-type hyperkähler manifolds: rk(S) + ℓ(A_S) < 24 for the Leech coinvariant lattice). Fu, Wang & Zheng, arXiv 2610.11109: “Large language models were used to explore proof strategies, assist in the development and refinement of mathematical arguments, and draft and revise the text.”
• Two compactness conjectures of Brudnyi (Conjectures 2.5 and 2.11) refuted by a translation-invariant, order-continuous quasi-Banach function lattice on ℝ. Huang & Xu, arXiv 2610.11066: “The authors acknowledge the assistance of ChatGPT 6 Astra. Suggestions obtained through interactions with this AI system were subsequently examined, reformulated, incorporated into the manuscript, and independently verified.”
• Bermond–Thomassen conjecture, asymptotically: minimum out-degree 2k + O(k^{3/4} log k) forces k disjoint directed cycles. Bang-Jensen, Wang & Wang, arXiv 2610.11517: “AI tools aided the construction of the sequence Z_j(v) in Lemma 5.5.”
• P. A. Smith’s conjecture on local presentations of groups (a group is recovered from every generating subset iff it is trivial or cyclic of order 2 or 3). Yu Pan, arXiv 2610.10586: ChatGPT used “for exploration of proof strategies and assistance with developing and checking arguments”.
• Smaller roles: triangular prism equations for near-group categories, linking unitarity to a Stark-type real-multiplication conjecture (He, Liu, Lu, Palcoux & Ren, 2610.11652; “The AI tool was used as a research assistant”); Volčič’s rank Nullstellensatz conjecture (2610.01804, AI-tools assistance acknowledged) and Lefschetz multisections (2610.01776, AI for consistency checks and GAP code), both posted 1 Oct; vector parking functions (2610.05665, LLM for literature and figures); Hedetniemi’s conjecture for uncountable complementary graphs (2610.08027, ChatGPT for grammar); found late from the leads queue: quantitative explosion of the divisible sandpile (Bou-Rabee & Panagiotis, 2609.02829, 2 Sep; “We acknowledge ChatGPT 5.6 for help with the proof of Proposition 4.7”). Galt–Tyutyanov conjecture on Skiba’s question (2610.12295): “AI is not used in proving the conjecture”, ChatGPT only for language.
• Related human work (no AI statement): Vu Khac Ky’s balanced Hellinger conjecture for antipodal Boolean functions (2610.10618, citing OpenAI family 119); an independent computer-assisted proof of the Chen–Raspaud conjecture for k = 4 (Nazarczuk, 2610.07968; research done in 2025, concurrent with Choi’s computer-free proof).
• Research on the process: “Viva La Vida” (Xu, Zheng & Han, Berkeley, 2610.0482912) instrumented a multi-agent prover on an open problem (186 hours, $5,694). A unanimity verifier silently blocked acceptance after API failures, one verifier approved three claimed solutions that GPT rejected, and 26% of the lemma library came from rejected arguments with their refutational context removed.

8–9 Oct additions (manual arXiv scan, 12 Oct evening; PDFs read)

Five results from this listing have their own entries: GPT-6 Astra’s kinetic Krylov–Safonov theorem and Pucci’s conjecture, Smale’s sixth problem for five bodies, the rank of 3×3 matrix multiplication over F_2, Erdős Problem #522 and PPT entanglement distillation. Also in the listing:

• Forcade–Pollington conjecture (1990): if an injective map from {1, …, k} to a group of order k respects products, the group is abelian (listed as unsolved by Richard Guy). John M. Campbell (Dalhousie), arXiv 2610.120877: the proof is “based on our extensive interactions with GPT-5.6 Pro”.
• Shamir secret sharing, the “one-half barrier”: one-bit local leakage resilience for rates k/n ≥ 1/2 − δ, and improved worst-case Optimal Polynomial Intersection. Yihang Sun and Mary Wootters (Stanford), arXiv 2610.12357: ChatGPT 6-Astra “produced the key truncation idea in Step 5” and, prompted with the authors’ idea, “obtained a version of the balanced ensemble and Theorem 1.12”; ChatGPT 5.6-Sol “was heavily consulted” earlier.
• Hot spots on non-smooth domains: Krzysztof Burdzy, arXiv 2610.1262211, extends the smooth-domain hot-spots theorem from OpenAI’s 6 Oct collection to arbitrary bounded simply connected planar domains with a simple first eigenvalue (non-strict form). “This paper was generated by ChatGPT in response to multiple prompts from the human author. KB has not checked every detail, because the theorem is, to a human, an obvious corollary.” Rodrigo Bañuelos generated “an almost identical theorem and proof” at the same time; neither will submit to a journal. Burdzy adds that “The OpenAI model that generated [3] missed the theorem presented in this note” and that OpenAI’s paper failed to cite de Dios Pont’s counterexample for convex domains in high dimensions.
• Quasi-Riemann hypothesis, constants: Baiying Liu, arXiv 2610.12234, re-tunes two parameters of OpenAI’s 7/8 argument to get zero-free half-planes Re s > 0.874975 and Re s > 0.874957…, formalized in Lean. “OpenAI Codex (GPT 6.1 Sol Medium-Ultra) assisted with drafting, source checking, and the search for improved bounds. Anthropic Claude (Opus 5.5 Medium) assisted with auditing.”
• Eleven-variable Vasc inequality (a cyclic sum is nonnegative for all positive 11-tuples): exact rational certificates and a Lean 4 proof of 2,031 source files that trusts native evaluation, by the “MechMath Agent Team” of the Key Laboratory of Mathematics Mechanization (Guo, Qiu, Cao, Feng, Gao), arXiv 2610.13463. The paper does not describe the agents’ role.
• Smaller roles: fast and stable approximate matrix multiplication, with a better border-rank bound for ⟨2,4,5⟩ (Dumitriu, Holtz, Schneider et al., 2610.13905; “An LLM also provided feedback on an earlier draft … highlighting an error in the proof of Theorem 4.6, which was corrected by a human”).
• Related human work (AI for wording only, or stated not used): the 4-to-1 Games Conjecture with perfect completeness, Khot 2002 (Fei, Minzer & Wang, 2610.12378; “the mathematical ideas and the overall approach presented in this paper are due to the authors”); Problem 101 of Yau’s 1982 list, negatively curved minimal surfaces in the round 4-sphere (Caniato, 2610.12238; “all the proofs were developed without the use of AI tools”); the four-loop pressure of cold QED, open since 1977 (Kärkkäinen et al., 2610.13873; “No AI tools were used to perform or validate the calculations”); a gravitational path-integral proof of the OSV conjecture (Bobev, Boruch, Iliesiu & Maskalaniec, 2610.12550; “The manuscript was written by the authors”).

5–8 Oct additions (lead from the 13 Oct sweep; PDFs read 13 Oct)

Five results from this batch have their own entries: Erdős Problem #993, unimodality of forest independence polynomials, the Toms–Winter conjecture, Bolzano’s unattended run on 3,800 open problems, multiplier associativity is hard for resolution and completeness of decreasing diagrams. Knausgård’s 83.9% distinct-zeros paper is in the zeta-zeros entry.

• Jackson–Ordaz conjectures on Hamilton cycles in digraphs: a near-linear Chvátal–Erdős condition, f_2(a) = O(a (log a)^4 / (log log a)^2), towards the conjectured f_2(a) ≤ a + 1, and a disproof of the Jackson–Ordaz pancyclicity conjecture. Chengli Li and Bo Ning, arXiv 2610.10292. “Early discussions with ChatGPT 5.6 explored several approaches … but yielded only partial results. … The main progress came on 9 September 2026, when extensive author-guided discussions with ChatGPT 6 led to the near-linear bound stated in Theorem 1.1.”
• Colonel Blotto games: computing an inverse-polynomial approximate Nash equilibrium of two-player general-sum Blotto is PPAD-hard even with two battlefields, “resolving an open question of [KPF+25]”; constant-rank cases are tractable. Pollatos and Kontogiannis, arXiv 2610.05956: “Initial versions of the proofs were developed with the assistance of ChatGPT 5.6 Sol and Astra.”
• Finite-time blowup for the unforced 2D incompressible porous media equation (smooth odd initial density on the torus). Mimi Dai, arXiv 2610.06712, 51 pages. The pressure-Hessian control follows “an idea from the Euler construction of OpenAI”; acknowledgement: “Publicly available ChatGPT was used to understand the finite-time blowup construction of Open AI [14] for the Euler equations, and to assist with some of the computations in the manuscript.” A human-led result that transfers a technique from an AI-written manuscript to another equation.
• Papers building on OpenAI’s unrefereed manuscripts (each result is conditional on, or derived from, an OpenAI preprint that has not been reviewed):
  • de Bruijn–Newman constant Λ ≤ 0.13963, “assuming that ζ(s) has no nontrivial zero with Re s > 7/8, as asserted in a recent unrefereed preprint of OpenAI”. Hakimoglu-Brown, arXiv 2610.12803: “ChatGPT assisted with the replacement strategy, analytic derivations, exact scalar and Bernstein-coefficient checks, interval verification of the stepped tables, and drafting.” “The seven-eighths hypothesis remains an assumption.”
  • L^3 reverse square function estimate for the light cone in four dimensions, obtained “from the critical local-smoothing and round-cone propagation statements in the September 2026 OpenAI manuscript”. Temur, arXiv 2610.12681; no AI-use statement found.
  • Ranks of the twin-prime elliptic curves y² = x(x−2)(x−p): equality of analytic and algebraic rank and the exact BSD formula for p ≡ 3, 5, 7 (mod 8), “assuming the low-corank Birch–Swinnerton-Dyer statement announced in the October 2026 OpenAI mathematics release”. Ruihan Chen, arXiv 2610.09370; no AI-use statement found.
  • Anderson model: Jaksic and Last, arXiv 2610.11611, a human note on the Bethe lattice, add an appendix written “one day after OpenAI publicly released the preprint … claiming to prove purely absolutely continuous spectrum at weak disorder for the Anderson model on Z^d, d ≥ 3”, observing that their structural argument shortens the last step of OpenAI’s proof.

6–9 Oct additions (sweep 2026-10-13, first arXiv fetch after the rate-limit outage; PDFs read 13 Oct)

No new own entries from this batch. All items are preprints and none has been reviewed.

• Isoperimetric problem in the three-dimensional cube: the complete relative isoperimetric profile of the unit cube with a classification of all minimizers (“corner eighth-balls, edge quarter-cylinders, and coordinate slabs, with transitions at 4π/81 and 1/π”) and quantitative stability with the optimal exponent two; the authors say they “resolve the isoperimetric conjecture in Milman’s formulation” for all cuboids (0, β) × (0, 1)² and “the three-dimensional case of Brezis’s Open Problem 10.1”. Bin Deng, Jiahuan Li and Yilu Liu, arXiv 2610.08644, 63 pages. “The authors used ChatGPT (OpenAI) as an auxiliary tool for language editing, exploration of some of the proof strategies and calculations.” A well-known open problem if the proof holds; human-led, no expert check found yet.
• László’s conjecture on the Toeplitz Böttcher–Wenzel form is false: for an explicit family of Toeplitz pairs the quartic form is not a sum of squares at all sufficiently large orders; it is a sum of squares for every 2 ≤ N ≤ 50 (Lean 4 through order 20), and the large-order negative theorem “has also been formally verified in Lean 4”. Wenqi Zhu and Ping Nie, arXiv 2610.089809: the authors “used large language model systems (Claude, Anthropic; Codex models, OpenAI) to generate candidate constructions and code, to run computations, to draft mathematical arguments”. Lean-verified in part; repository github.com/erenup/toeplitz-bw-not-sos.
• Sampling the Sherrington–Kirkpatrick model: polynomial-time simulated annealing with inverse-polynomial total-variation error for every β < 1, through a weak Poincaré inequality. Zhe Hou, Jingcheng Liu and Yixiao Yu, arXiv 2610.0988210. The disclosure is unusually detailed: key ideas “were selected among a few alternatives found by GPT 5.6 Pro. GPT 5.6 Pro also claimed a false proof of an FPRAS extension …, which GPT 6 Astra Pro managed to fix. GPT 6 Astra Pro also found a separate deterministic FPTAS for the SK model, which we had to defer to a separate manuscript in view of OpenAI’s recent announcements.” “We verified all the proofs produced by AI.”
• Dynamic matching with abandonment (Arnosti, Niazadeh, Nuti, Simon, arXiv 2610.12991): “One result in this paper was obtained with the help of an AI system.” The polynomial-time computability of a restricted fixed point (Proposition 5.4) “was first found by OpenAI’s Codex Astra, after several rounds of prompting and interaction with the authors, and was subsequently verified and simplified by the authors”.
• Deb–Sokal conjectures on higher-order Stirling triangles: row log-concavity of the higher-order Stirling cycle and subset triangles, plus a combinatorial interpretation of the subset case. William Y.C. Chen and Elena L. Wang, arXiv 2610.09362; the proofs rely on “strengthened log-concavity inequalities discovered with the assistance of ChatGPT 5.6”.
• Hurwitz vector fields with periodic orbits (Problem 19 in Gasull’s list of open problems on low-dimensional dynamics): Castañeda and Rebollo-Perdomo, arXiv 2610.09176. “The construction presented in this paper emerged from an iterative and exploratory dialogue between the authors and OpenAI’s ChatGPT.”
• Building on OpenAI’s unrefereed manuscripts (results inherit that status):
  • Irrationality exponent of log r is 2 for every positive rational r ≠ 1: Liu, Jiang and Zhang, arXiv 2610.10192, adapting “the separated-weight method” of OpenAI’s π manuscript; no AI-use statement. Details in the π entry.
  • Radius of comparison equals half the mean dimension for minimal free actions of amenable groups with the uniform Rokhlin property and Cuntz comparison of open sets: George A. Elliott and Zhuang Niu, arXiv 2610.11143. “The arguments in both cases are based on the recent preprint of OpenAI dealing with Z-actions”; “The work was carried out using ChatGPT.”
  • Erdős similarity problem for null sequences with adjacent ratios bounded below: Kan Jiang and Lifeng Xi, arXiv 2610.11449, which “adapts a finite-tree construction for geometric sequences” and cites two unpublished OpenAI manuscripts (dyadic and geometric cases); no AI-use statement found.
• Smaller roles: a sharp minimum for cube-apex bodies with Lean 4 verification (Shen and Dubovski, 2610.11014; “OpenAI Codex assisted mathematical exploration, Lean proof development, literature checking and manuscript preparation”); r-neutralized Pfister–Sullivan entropy (Chen, Xie, Zhou, 2610.11639; GPT-6 helped “in the construction of Example 3.12”); sharp Hilbert transform norms on Schatten classes (Long, 2610.10158; “OpenAI models as assistive tools”); constant-round approximation of dominating set (Amiri and Siebertz, 2610.09618; “AI agents to fill in technical details … and check the correctness of the proofs”); bi-contact plugs (Hozoori, 2610.11243; an LLM supplied “some computational details” of Lemma 5.8).

9 Oct addition (leads run, 13 Oct evening; PDF read)

• Quantitative Eshelby stability (solid mechanics; a stability version of the Eshelby conjecture that uniform interior strain characterizes ellipsoidal inclusions): a nonlinear two-sided relation between interior strain heterogeneity and distance from the ellipsoid family, plus a finite-shape theory checked by finite elements. Bo Ni, arXiv 2610.1340013, 33 pages. The paper presents itself as a case of “human-led, AI-assisted theoretical mechanics”: “Generative AI supported literature organization, candidate formulation, analytical exploration, symbolic and numerical checking, code development, partial formalization, and manuscript preparation”, while the researchers set objectives and validation; “Selected mathematical steps were reconstructed by an AI reviewer given the definitions and target before seeing the candidate proof”, and the Lean 4 work “is not described as complete formal verification”. It adds: “No claim of autonomous discovery, measured research acceleration, or AI correctness probability is made.” The model is not named. The paper also introduces a curated list, Open Problems in Mechanics.

In the first nine days of October 2026 the stream did not slow down: each arXiv listing brought several named conjectures with a disclosed AI role, and two new patterns appeared. Authors now describe the model’s work in detail, including its false proofs, and other mathematicians have started to build on OpenAI’s 6 October manuscripts before anyone has refereed them, so errors there would propagate. For a model with an older cutoff, the practical lesson is the same as for the summer: check the current status of a conjecture before calling it open, and check whether a claimed resolution has been verified.