AI-assisted results, 2 Nov: arXiv of 2 Nov and Hexagon of 1–2 Nov

Your AI and this story

Key facts

• Own entries from this sweep: Kung–Traub conjecture proved, Lean check with ChatGPT 6 (arXiv 2610.40660), Alon–Krivelevich conjecture disproved with GPT-6 Astra (arXiv 2610.42094), Hopfield model thermodynamic limit, AI-generated proofs (arXiv 2610.41938)
• Operator algebras: Rui Okayasu (arXiv 2610.41746) proves a von Neumann algebra N is injective iff ‖T‖dec = ‖T‖cb for every map ℓ∞₃ → N, answering Haagerup’s 1985 Problem 3.8 (recorded as open by Pisier). AI statement: ‘This work was carried out in collaboration with an AI system (Claude Opus 5.5). The main argument was developed in discussions between the author and the AI system, and the manuscript was drafted mainly by the AI system. The author has verified all proofs’
• Quantum information geometry: Suvrit Sra, ‘A Proof of Petz’s Conjecture on the Scalar Curvature of the Kubo–Mori Metric’ (Hexagon 2610.00717): Petz (1994) conjectured the scalar curvature increases as the state becomes more mixed; Dittmann (2000) reduced it to concavity of a function h, proved here via a Stieltjes-function argument. Provenance: mixed, all parts understood by a human; ‘developed with an AI assistant (OpenAI’s GPT Pro). None of the claims were obtained zero-shot’. Sra posted 21 more Hexagon works on 1–2 Nov (Hexagon 2610.00718–2610.00732: q-Foulkes problem I–III, Macdonald (q,t)-permanent I–II, a Koteljanskii inequality for permanents, a sharp Brégman–Minc bound, Wigner–Yanase–Dyson metrics with monotone curvature, p-Bures–Wasserstein distance, lower bounds for the quantum permanent, Turán-ratio monotonicity, theta-kernel sign regularity, Jack/Macdonald majorization, Schur forms; GPT Pro, GPT-6 Astra, Claude Opus 5.5, Fable 5.1)
• Representation theory: Qiyue Tang (Tsinghua), ‘Equality of the Mirković–Vilonen and theta bases’ (Hexagon 2611.00006) claims Kamnitzer’s ICM 2022 conjecture. Provenance: seven ChatGPT-6-Pro conversations (~23 h), Pharos, Codex; ‘This was the first time Rethlas-OSV identified a nontrivial error in my work that invalidated the proof of the main theorem’; ChatGPT-6-Pro repaired it in six hours, a second Rethlas-OSV review found further errors including ‘one that again blocked the proof as written’, which were corrected; only ‘some parts’ checked by a human. Also Tang: full Fock–Goncharov criterion via positive polytopes (Hexagon 2611.00015; the GHKK interior criterion, ChatGPT-6-Pro session of ~270 min), Huang–Wang conjectures on projected stated SL_n skein algebras (Hexagon 2611.00013), Markov–Euclid asymptotics (Hexagon 2611.00002); see also Hexagon 2610.00587 (Davison–Mandel Conjectures 1.4, 2.17, 3.13 and Davison’s ICM 2026 two-wall conjecture, GPT-6-Astra-Pro, passed Rethlas-OSV)
• Motivic homotopy: Niels Feld (Rennes), ‘Gersten injectivity and Morel’s purity conjecture’ (Hexagon 2611.00017): failure of Gersten injectivity at a henselian regular local ring gives a counterexample to Morel’s purity conjecture. Provenance: primarily AI-generated, all parts understood by a human; drafted with GPT-6 Astra, revised with Claude Opus 5.5, adversarial referee report by GPT-6 Astra
• Graph colouring: Narayanan Narayanan, Suchismita Mishra, Umesh Shankar (IIT Madras), claw-free perfect graphs satisfy ch(G) = χ(G) = ω(G) (Hexagon 2610.00713; ‘The proofs have been written by hand and verified by the authors’; Claude and GPT-6 helped formalize some arguments). Also arXiv: Cranston–Li–Su–Wang–Xu (2026) conjecture ‘done overwhelmingly by AI models’ (2610.40863); treedepth NP-hard on planar max-degree-3 graphs, ‘The idea for the reduction was suggested by Claude Opus 5.5’ (2610.42033)
• Other arXiv items with AI statements: entropic c-theorem proof (hep-th 2610.40874; ‘Claude and ChatGPT for development of arguments’); metric Zariski multiplicity conjecture in dimension 4 (2610.41954; Claude and Gemini ‘to refine arguments’); Graham’s rearrangement problem in finite abelian groups (2610.40712; ChatGPT for the absorption argument); maximal determinant of order 15 over third roots of unity (2610.40679); Haiman’s conjecture for n ≤ 10 by computation (2610.41555); a Tasaki LLM-generated extension cited in a weak-Gibbsianity counterexample (2610.41871); a Nagata–Biran–Szemberg note written while studying OpenAI manuscripts (2610.41931); Novikov-conjecture paper used ChatGPT only for language (2610.40062)
• Hexagon, AI-run deposits with no human check (‘understanding: none’), stated as claims only: David Niedbala Giraudin, Korenblum’s constant c₂ ≥ 0.446 (2611.00014), trefoil distortion bound (2611.00012), Vitorino–Vukusic Problem 5 (2611.00009), torus-knot ropelength (2611.00008), q-Markoff coincidences answering Labbé–Lapointe–Steiner (2611.00005), Chvátal–Sankoff upper bounds (2610.00709), Yamada’s constant 283 lowered to 69 (2610.00707) and three p-adic linear-forms papers (2610.00704, 2610.00706, 2610.00708); Qidong He, hard-sphere density coexistence (2610.00734); Alon Schechter, illumination bounds (2611.00007, 2611.00010, 2611.00011); Daniel Adam Pyle with ‘Lyra’ (2610.00700, 2610.00701)