AI-assisted results, 30 Oct: arXiv announcement of 29–30 Oct

Your AI and this story

Key facts

• Own entries: Murty–Simon conjecture: proof for all n, bound found by GPT-5.6 Sol (2610.39032), Spectral isometry conjecture for C*-algebras: GPT-6 Astra proof, Lean (2610.40470), Online Steiner forest: greedy claimed Θ(log k log log k), with GPT Astra (2610.40569), Laczkovich’s Lipschitz mapping problem: claimed answer, ChatGPT key step (2610.38834), Pure semisimplicity conjecture: claimed counterexample, GPT-6 Astra, Lean (2610.39883), Petty’s conjecture in dimension 5: proof by Tencent’s hyprover agent (2610.39967), OpenConjecture: GPT models on 200 random arXiv conjectures (2610.40382). Added to an existing entry: Viaclovsky’s infinitely many inequivalent complex structures from the Alpöge–Claude construction (2610.39102) → Hopf S⁶ entry
• Alon–Seymour–Thomas conjecture (Xinheng Lin, Fuzhou University; 2610.40270): every n-vertex K_h-minor-free graph has a balanced separator of size ≤ min{n, 21h√n}, the O(h√n) bound conjectured in 1990 (earlier bounds cited in the paper include Kawarabayashi–Reed’s O(h√n + f(h)) and a 2026 preprint by H. Le, arXiv 2605.05494). ‘This manuscript presents a proof of the Alon–Seymour–Thomas conjecture developed under the author’s guidance using ChatGPT 6pro. The author has checked every formula and theorem by hand … GPT 6.1 Sol (Max) was also used to check the proof repeatedly.’ Single author, 17 pages, famous conjecture: treat as a claim until specialists comment
• Borsuk’s problem in ℝ⁴ (Andrii Arman, Andriy Bondarenko, Andriy Prymak, Danylo Radchenko; 2610.39078): every set of diameter 1 in ℝ⁴ is covered by seven balls of diameter < 0.9999, so b₄ ≤ 7, via contact points of the circumball and an exact computation over 23,494 parameter boxes (github.com/andriyprm/4d7balls). ‘An earlier version of this work established the weaker bound g₄ ≤ 8 … We subsequently explored the problem with ChatGPT Astra, initially seeking purely analytic arguments … This led to the approach based on the circumradius and substantially reduced the computational search space’
• Fiedorowicz’s conjecture (Oren Ben-Bassat; 2610.38760): every finite simply connected CW complex is homotopy equivalent to the classifying space of a finite monoid; explicit 1795-element monoid for ℂP². ‘The key idea of realizing CP² by a finite monoid arose in a conversation between the author and Anthropic’s Claude Opus 5.5. Starting from that idea, the author supplied all of the theoretical arguments … All of the content in the GitHub repository was written by Claude’ (github.com/OrenXGitHub/monoid-computations)
• Kroó’s 2004 conjecture (Matthew J. Colbrook; 2610.38877): on every origin-symmetric convex body, continuous functions on the boundary are uniform limits of sums of homogeneous polynomials of degrees n and n+1; also Kroó’s 2019 weighted conjecture. ‘the author interacted with and guided ChatGPT 6, which suggested proof strategies and intermediate arguments … ChatGPT 6, accessed through Codex, also assisted with the formalisation in Lean’
• Algebraic geometry: Kawatani–Okawa conjecture in dimensions 3 and 4 (Dongjian Wu, Nantao Zhang; 2610.39838; ‘The authors manually proved the conjecture in dimension three and developed a strategy for the dimension-four case. ChatGPT 6.0 was used to fill in the gaps’); Witt liftability of quasi-F-split Calabi–Yau varieties (Teppei Takamatsu; 2610.38427; ‘The proofs of Theorems A and B, as well as nearly all of the other original mathematical arguments in this paper, were developed by the author through extensive discussions with OpenAI’s ChatGPT (GPT-6 Astra)’); balancing tautological intersection numbers (2610.39462, GPT-6 Astra organised the proofs of Sections 4–6); Abhyankar rectifiability with ChatGPT Sol 6.1 (2610.38474)
• Number theory: Borisov–Nuer conjecture settled, with Ulrich line bundles on unnodal Enriques surfaces (Haowu Wang, Shaoyun Yi; 2610.38414; ‘Generative AI was used interactively in the subsequent mathematical exploration, including the development, testing, and refinement of parts of the construction and proof’); Cavallar–Lemmermeyer’s list of 58 norm-Euclidean complex cubic fields is complete (Gustav Kjærbye Bagger, Andrew R. Booker, Tim Trudgian; 2610.38073; ‘We used Claude Opus 5 to solve this problem through a series of prompts over two days in late July 2026’); Evan O’Dorney, van der Waerden’s problem for cubic surfaces, G_S ≠ W(E₆) for O(H^(19+1/21+ε)) forms (2610.37945; the title page says ‘Written by ChatGPT 6’); Ken Ono, asymptotic Jensen–Pólya criterion for GRH for every number field, with Theorem 3 ‘formalized in Lean 4 using AxiomProver’ (2610.38870)
• Combinatorics: off-diagonal book Ramsey numbers r(B_⌈αn⌉, B_n) determined asymptotically for all α (Dai, Fan, Lin; 2610.38515; ‘used ChatGPT to explore this direction’); hook monomial immanants are Kazhdan–Lusztig positive, the hook case of Stembridge’s and Haiman’s conjectures (Keith McNulty; 2610.39021; Claude Code wrote and ran the software and drafted text); e-positive caterpillars have maximum degree ≤ 3, the caterpillar case of the Dahlberg–She–van Willigenburg conjecture (2610.39481; GPT-5.5, GPT-5.6, GPT-6 Astra and Codex); diameter-two graphs with χ_μ = χ beyond complete bipartite (Siddhesh Chaubal; 2610.39430; Claude Opus 5.5 ‘designed and ran the computer searches that found G₁₁ … produced the proofs of Theorems 3.4 and 3.8’); every group is the automorphism group of a set-theoretic Yang–Baxter solution (G. Militaru; 2610.39228; Claude Opus 5.5 ‘to explore and develop constructions and proofs’); Liu–Luo–Luo–Rao Liouville property for Delaunay triangulations (2610.39242; ‘The key idea of the proof of Theorem 1.1 is produced with the assistance of ChatGPT’)