AI-assisted maths on Zenodo, 17–19 Oct 2026

Your AI and this story

Key facts

• Arad–Herzog conjecture (Arad & Herzog, Products of Conjugacy Classes in Groups, LNM 1112, 1985): Aran S. Ziegler, ‘A proof of the Arad–Herzog conjecture’, Zenodo 23450488 (Oct 19, 2026; manuscript dated 18 Oct, ‘Private review version 3 … unreleased’; ~66 pp. plus a review-support zip). The paper finishes the remaining families on top of Fisman–Arad (1987, alternating groups), Moori–Tong-Viet (2011), Guralnick–Malle–Tiep (2013, projective linear and more) and Guralnick–Navarro (2016, square case). It also identifies an exceptional family of products with exactly two classes in odd-dimensional orthogonal groups over odd fields
• Arad–Herzog AI statement: ‘The author used agentic AI tools developed by OpenAI and Anthropic to carry out their research strategy and assist with Lean formalisation and manuscript preparation. The author takes full responsibility for the content.’
• Arad–Herzog Lean caveat (§12): the compiled declaration AH.FinalLiterature.global_literature_relative states the conjecture for all finite simple nonabelian groups, but its dependency closure contains 859 axioms. Besides the 3 standard Lean axioms, these are ‘587 binary ordinary imports and 269 other named mathematical inputs’ (published classification and character-theoretic results taken as axioms). The ‘revised verification artifacts are currently held in the private project archive’. The author is not otherwise known in the field, and there was no expert reaction as of Oct 19
• Messerschmidt’s 2018 conjecture confirmed: Martino Lupini (Università di Bologna, ERC grant DAT), ‘Lipschitz positive decompositions of von Neumann preduals’, Zenodo 23440415 (Oct 17). The selfadjoint predual of a von Neumann algebra has a Lipschitz positive decomposition iff the algebra is subhomogeneous, so infinite-dimensional selfadjoint trace class answers the Lipschitz decomposition problem negatively. The finite-dimensional constants satisfy c·log log N ≤ c_N ≤ C·log N. Disclosure: ‘OpenAI Codex was used in October 2026 … for mathematical exploration, proof development, source retrieval, manuscript drafting … and to assist in checking proofs’
• Lupini, ‘Common null complements for polynomial images of finite sums of perfect sets’, Zenodo 23440154 (Oct 17): for sums of D+2 compact perfect sets, one null F_σ set B gives P(C)+B = ℝ for every nonconstant polynomial of degree ≤ D, so Jung and Lai’s Conjecture 4.8 holds for such sums (it stays open in general). Codex was used ‘for mathematical exploration’ and ‘Separate Codex collaborators developed and reviewed arguments’
• Matching complexes: Zihuan Xu, ‘A Planar Counterexample to the Sphere-Wedge Conjecture for Matching Complexes’, Zenodo 23437631 (Oct 17). The matching complex of the planar pentagonal bipyramid collapses to a torus, which disproves the conjecture that matching complexes of planar graphs are contractible or wedges of spheres; it is the smallest possible counterexample, unique among 7-vertex planar graphs. ‘An OpenAI Codex assistant was used in developing the arguments, drafting the manuscript, and writing the verification programs’
• Cubelike Graph Core Conjecture (Mančinska–Pivotto–Roberson–Royle), partial: Qihang Wang, ‘Integral Cayley graphs and abelian rigidity in cores of cubelike graphs’, Zenodo 23445385 (Oct 18). It proves the conjecture for cores with a normal abelian Cayley representation and classifies the abelian groups on which every integral Cayley graph is cubelike. Models named: ‘OpenAI GPT-5.6 Sol, Anthropic Claude Fable 5, Anthropic Claude Fable 5.1, OpenAI GPT-6.1 Sol, OpenAI GPT-6 Astra, and Anthropic Claude Opus 5.5’; acknowledges Gewu Intelligence Lab (the same author and lab as arXiv 2610.20649 in the 16 Oct catalogue)
• Fluid dynamics: Adilkhan Salkimbayev, ‘Linear stability of asymmetric Burgers vortices at large circulation beyond the half-strain threshold’, Zenodo 23448800 (Oct 18). Maekawa’s vortex is linearly stable for every asymmetry λ ≤ 0.9399 at large circulation (previously known only for λ < 1/2), with interval-arithmetic certificates (python-flint). ‘The work was carried out with substantial help from generative AI (Anthropic’s Claude)’