AI-assisted results, 31 Oct: arXiv of 30 Oct and Hexagon of 29–31 Oct

Your AI and this story

Key facts

• Own entries from this sweep: Mézard–Parisi formula: Panchenko’s proof ‘found by ChatGPT Astra’ (arXiv 2610.39545, 2610.39542), Number of C₄-free graphs is 2^{(1+o(1))ex(n,C₄)} (claim, GPT-6 Pro) (2610.39595), Right-angled hyperbolic polytopes exist only up to dimension 8 (claim) (2610.38900), Bhargava: elliptic curves have cyclic reduction at a positive proportion of primes (claim) (Hexagon 2610.00699), Generalized Lax conjecture: OpenAI counterexample claim; AI-found generic case (Kummer–Netzer) (2609.24542, found through 2610.37850), AFEM rate-optimal for any bulk parameter θ < 1 (ChatGPT framework) (2610.39719), Onsager hard rods: AI-written first-order transition claim, not human-verified (Hexagon 2610.00667). Added to existing entries: Hexagon 2610.00672 (Tulsyan) and Hexagon 2610.00692 (Stein Naves de Brito) → Anthropic ETH entry; Kourovka 8.52 → finitely presented Burnside entry
• Ramsey numbers: András Pongrácz (Debrecen), ‘An exact computer-assisted certificate for R(5,6)≤84’ (Hexagon 2610.00693, 31 Oct): linear inequalities on any 84-vertex graph with no K₅ and no independent 6-set, summed with explicit rational weights to a contradiction, using McKay’s catalogues and published small Ramsey bounds (R(4,6) ≤ 40, R(5,5) ≤ 46). Provenance: ‘Primarily AI-generated … AI assistance was used in developing the computational argument, verification programs, and exposition … Human review of the complete argument is ongoing.’ Tools: ChatGPT GPT-6 Astra. We did not confirm the previous upper bound for R(5,6) in the material we read
• Infinite graphs: Nathan Bowler and Max Pitz (arXiv 2610.40458) answer negatively Diestel’s 1992 question (end-faithful spanning trees from one-ended subgraphs), Diestel–Kühn’s 2004 question (topological spanning trees in every connected graph) and Carmesin’s 2019 question (thin-end-faithful spanning trees), around Halin’s 1964 conjecture. ‘It was Claude (Anthropic) that observed that our published graph G has no end-faithful and no topological spanning tree … A first version of G♭ … was found by Claude … The remaining two lemmas of this section, which show that S misses an end, were found by Claude and rewritten by us’
• Probability: Mark Holmes and Thomas S. Salisbury (arXiv 2610.40464) prove the limiting velocity of nearest-neighbour random walk in i.i.d. random environment is deterministic when the local environment takes two values, without ellipticity: ‘The proof uses a clever adaption (suggested by ChatGPT) of coupling arguments previously developed by the authors … A version of this argument was obtained by prompting ChatGPT version GPT-6 Astra Pro.’ Patrick Lopatto, smallest bulk eigenvalue gaps of Wigner matrices without smoothness, including Bernoulli matrices, converge after N^{3/2} scaling to the GOE Poisson process (Hexagon 2610.00675; ‘Proof discovery, verification, and writing were done with GPT 6 Astra, GPT 6.1 Sol, and Claude Opus 5.5’, primarily AI-generated). Bencs–Huang–Lee–Liu–Regts, first polynomial-time algorithm for the antiferromagnetic Ising partition function on random d-regular graphs up to Kesten–Stigum (arXiv 2610.40598; ‘GPT 5.5 Pro was used … for only a few minor technical aspects’; an attempt to have GPT 5.6 Codex merge notes ‘was very far from satisfactory’; Chen–Wang–Zhang proved it independently)
• Graph colouring: Matthew Bowen, András Máthé and Oleg Pikhurko (Warwick), local rules giving independent sets of density (1−o(1)) ln D/D in triangle-free graphs and (1+ε)D/ln D colourings in LOCAL, measurable, Baire and Borel settings (Hexagon 2610.00677; ChatGPT Astra 6; ‘This AI-assisted draft’; ‘The contributors have convinced themselves that all main results (Theorems 1.1-1.3) are correct but have not verified all intermediate auxiliary steps to the letter’)
• Extremal combinatorics: Chen–Gong–Lu–Tu–Yuan, maximum number of triangles in k-regular graphs, confirming the Petingi–Rodríguez conjecture and the triangle case of Cambie–de Joannis de Verclos–Kang for odd k, with counterexamples for even k ≥ 18 (arXiv 2610.39461; ‘the authors used OpenAI Codex to assist with the development of some mathematical proofs’); Sun–Yuan, Turán number of the Cartesian product of two stars (arXiv 2610.39486, Codex, same wording); Li–Su answer many questions of Kra–Moreira–Richter–Robertson on infinite sumsets, main theorem in Lean (arXiv 2610.39637; ‘substantial assistance from AI systems (OpenAI’s GPT models and Anthropic’s Claude), including the development of the proofs’); Yuda Chen, crescent configurations in ℝ³ for every n (arXiv 2610.38368; ‘GPT-5.5 Pro assisted in the search for the constructions’); Nenadov’s simplified dense model theorem, where ‘Astra then suggested’ the container notion (arXiv 2610.39176); Coutinho–Higino, lifted entropy splitting from a ‘somewhat convoluted solution’ by Claude Opus 5.5, counterexample by ChatGPT Astra 6 (arXiv 2610.40145); Raychev’s guaranteed-delivery pursuit game, where ‘Astra 6, operating through the Codex harness, made substantial contributions’ (arXiv 2610.39469, Lean)
• Codes and computation: Harada, weight enumerators of binary extremal self-dual codes of length 60, codes constructed with GPT-6 Astra and checked in Magma (arXiv 2610.38323); Kurz, optimal self-orthogonal binary codes, some proofs ‘found with AI tools’ (arXiv 2610.39870); Hugo Nielsen, 2×2 matrix multiplication with 5n^{log₂7} − 4n² + (3/4)n² log₂ n operations under Karstadt–Schwartz basis change, previous last term (3/2)n² log₂ n, ‘GPT-6-Astra was used to conduct the computational search that identified the main result’ (Hexagon 2610.00671); John Tripodo, covering number C(13,7,4) = 30 with Lean, ‘None of this contribution has been checked and understood by any human’ (Hexagon 2610.00669); Alben Weeks, Jacobsthal’s function on primorials h(65)…h(71) exact (Hexagon 2610.00660; ChatGPT and Gemini for the solver); Joseph Shunia, deterministic primality test with one binomial congruence, bit complexity L^{32+o(1)}, ‘OpenAI’s GPT-6 Astra developed and wrote the proofs’ (Hexagon 2610.00697); Firsov, order maintenance in worst-case constant time verified in 78,000 lines of Lean with Claude Opus 5.5 and Astra 6 (arXiv 2610.38616)
• Number theory and algebra: Jacob Crim, a linear-in-rank bound for integral arithmetic progressions on elliptic curves from ‘many conversations with Claude opus 5.5’ (Hexagon 2610.00665); David Niedbala Giraudin proves Ehbauer–Grishkov–Logachev’s Conjecture 3.7 on uniformizable Anderson t-motives, no human check (Hexagon 2610.00686); Chao Liu’s zero-sum papers (Hexagon 2610.00599 = arXiv 2610.38373, Hexagon 2610.00643 = arXiv 2610.38447, Hexagon 2610.00673; ‘AI tools developed by OpenAI were used in this work’); Qiyue Tang disproves log-concavity and unimodality conjectures of Chen, Chen–Sun and Chen–Huang–Sun for cluster monomials of type A₅ (Hexagon 2610.00668, Hexagon 2610.00670); Pedro Caceres verifies Suzuki’s two-sided RH margin to 2.15·10²⁹ (Hexagon 2610.00661, Hexagon 2610.00688; Claude Opus 5.5 and Fable 5); Zhiqiang Yu, pre-modular fusion categories of global dimension 2p (arXiv 2610.39318); Li–Ye, special cycles on Baily–Borel compactifications, LLM ‘suggested the key idea’ (arXiv 2610.38187); Belotto da Silva–Kiro, Denjoy–Carleman noetherianity, ChatGPT Astra 6 Pro ‘contributed to the development of mathematical ideas’ (arXiv 2610.40265); Bunina–Lee–Miasnikov–Muliarchyk, failure of Kaplansky’s direct finiteness in every odd characteristic, ‘OpenAI’s ChatGPT was used extensively for mathematical exploration’ (arXiv 2610.22536, via oai-math-revised notes)