AI-assisted results in the 22 Oct 2026 arXiv listing
The gradient conjecture at infinity with GPT-6 Astra, Claude-built zero-knowledge PCPs, nucleon algorithms with GPT Sol 6.1 and Fable 5.1, Kelly’s Condorcet conjectures
Awaiting review
The takeaway
Thursday’s arXiv announcement brought experts’ write-ups of two claims from OpenAI’s 6 Oct release (Nagata’s conjecture, and a Hochster counterexample surveyed by Anghel), a human exposition of the Kozma–Nitzan proof, and a 3D counterexample to Gale’s conjecture found with GPT-6 Astra. This page lists the next tier.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 22 October 2026
Your AI and this story
- GPT-6 Astra175 days after its cutoff
- Claude Opus 5.5114 days after its cutoff
- Gemini 4 Argon205 days after its cutoff
- Grok 4.7144 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 114 days before it.
Key facts
- Own entries: Nagata (2610.30050), Hochster small CM (2610.30027), Gale in R³ (2610.28828). Enriched: dying percolation (Gouéré, 2610.30063)
- Gradient conjecture at infinity and the definable Thom gradient conjecture (Thang Do, Arnulf Jentzen, Benno Kuckuck; 2610.29211): every gradient trajectory of a definable C¹ potential has finite spherical length. ‘The authors gratefully acknowledge extensive assistance from OpenAI’s GPT-5.6 Sol and GPT-6 Astra/Sol … used via ChatGPT/Codex as research collaborators for developing mathematical arguments and ideas, exploring proof strategies, testing arguments, identifying gaps and possible counterexamples’; all proofs ‘independently checked by the authors’
- Quasilinear-size constant-query zero-knowledge PCPs (Inhyuk Park, Chenglin Fan; 2610.29066), improving the Õ(T+b²) length of Zeilberger–O’Connor–Gur–Spooner (2026): ‘We used Anthropic’s Claude extensively, under the direction of the authors, to assist with developing the constructions and the proofs, with checking the proofs, with the computations reported in Appendix B, and with drafting this paper. Its use materially affected every section of the paper’
- Nucleon of matching games (Haris Aziz, Gergely Csáji; 2610.29731), resolving three open problems from a recent survey (polynomial algorithm for matching games; NP-hardness for b-matching and partitioned games): ‘The authors acknowledge the assistance of OpenAI’s GPT Sol 6.1 and Anthropic’s Fable 5.1 in working out the proof details of the combinatorial algorithm.’
- Kelly’s two monotonicity conjectures on Condorcet winner probabilities for odd electorates (Seick Kim, Hocheol Shin; 2610.29904): ‘Large language models (LLMs) were used to explore proof ideas and assist with computational checks, literature searches, and language polishing.’
- Functional limit theorems for self-interacting random walks (Kosygina, Marêché, Mountford, Peterson; 2610.28935), solving a problem open since Tóth’s mid-90s work: ‘The literature search for these results with large language models (Gemini, Google; Claude, Anthropic) produced their initial proof drafts instead of references. We verified, corrected, and rewrote these proofs’ (appendix lemmas only)
- Rayleigh’s conjecture for clamped plates, a simpler planar proof of Nadirashvili’s theorem plus nonlinear extensions (Dorin Bucur, Richard Laugesen; 2610.29840): ‘The authors used OpenAI’s ChatGPT through Codex to explore the theorems and conjectures, check various endpoint arguments, locate and verify references, and prepare this manuscript and its figures.’
- Supporting roles (checking, exploring, drafting): conjugacy problem for virtually soluble groups of finite Prüfer rank (2610.29563, ChatGPT ‘for checking mathematical arguments, exploring proof strategies’); Hong–Krahn–Szegő sharp stability (2610.28757); gradient absorption prevents blow-up (2610.28913, ChatGPT); low-degree filtrations in Grassmannian cohomology (2610.28780, Codex for ‘exploration of some of the proof strategies and calculations’); pivot-based distance oracles in sparse graphs (2610.29624, ChatGPT and Claude); local Gan–Gross–Prasad epsilon dichotomy (2610.29483: ‘The draft of this paper was produced from handwritten notes with GPT 5.6, and GPT Astra was used for proofreading’)
Show 1 more
- No substantive AI role: Velani’s conjecture on dyadic approximation in the Cantor set (Sam Chow, Han Yu; 2610.28839: ‘This work was completed entirely without AI’, apart from a figure and search results); Ward’s conjecture on Griesmer codes (2610.29275), Kemper–Malle conjecture (2610.29849), Schäffer’s conjecture for even k ≤ 42 (2610.28768), eventual stability counterexample (2610.28688): no AI statement in the sweep’s scan
What happened
Three results from Thursday’s announcement have their own entries:
- Jenia Tevelev’s re-proof of OpenAI’s Nagata proof, which he calls “a stunning achievement”;
- Cristian Anghel’s survey of the counterexamples to Hochster’s small Cohen–Macaulay conjecture, which gives a complete proof of OpenAI’s theorem and extends it to every dimension;
- Vsevolod Voronov’s 101-point counterexample to Gale’s conjecture, found with algorithms developed with GPT-6 Astra.
Jean-Baptiste Gouéré’s exposition of OpenAI’s Kozma–Nitzan proof was added to the dying-percolation entry.
In the papers listed here, AI plays two kinds of role:
- AI as an extensive collaborator: the gradient conjecture at infinity (GPT-5.6 Sol and GPT-6 Astra/Sol), quasilinear zero-knowledge PCPs (Claude “materially affected every section”), the nucleon algorithm (GPT Sol 6.1 and Fable 5.1 worked out proof details), and appendix lemmas that Gemini and Claude drafted when asked for references.
- Support: exploring ideas, checking, drafting and editing (Kelly’s conjectures, Rayleigh’s conjecture, Gan–Gross–Prasad and others).
Older items still listed in the sweep (14–19 Oct ML, physics and materials papers that only mention LLMs) are out of scope here.
Why it matters
Mathematicians in the fields concerned are now writing up OpenAI’s 6 Oct claims (Nagata, Hochster, Kozma–Nitzan), and they speak openly about the editorial burden this places on them. Meanwhile, disclosures of AI as a central collaborator on long-standing conjectures have become routine.
What is disputed or not yet verified
| Verification | Unrefereed arXiv preprints, proofs checked by their authors |
|---|
Sources
13 sources from 1 site. Numbers match the chips in the text.
13 sources: 13 primary
Primary
- arXiv 2610.29211: On the gradient conjecture at infinity (Do, Jentzen, Kuckuck)arxiv.org, paper
- arXiv 2610.29066: Quasilinear-size constant-query zero-knowledge PCPs (Park, Fan)arxiv.org, paper
- arXiv 2610.29731: Computing the nucleon of matching games (Aziz, Csáji)arxiv.org, paper
- arXiv 2610.29904: Kelly’s conjectures on Condorcet winner probabilities (Kim, Shin)arxiv.org, paper
- arXiv 2610.28935: Functional limit theorems for self-interacting random walksarxiv.org, paper
- arXiv 2610.29840: Rayleigh’s conjecture for vibrating plates (Bucur, Laugesen)arxiv.org, paper
- arXiv 2610.29563: Conjugacy problem for virtually soluble groups of finite Prüfer rankarxiv.org, paper
- arXiv 2610.28757: Sharp quantitative stability of the Hong–Krahn–Szegő inequalityarxiv.org, paper
- arXiv 2610.28913: Strong gradient absorption and initial decay prevent blow-uparxiv.org, paper
- arXiv 2610.28780: Low-degree filtrations in Grassmannian cohomologyarxiv.org, paper
- arXiv 2610.29624: Tight space bounds for pivot-based distance oraclesarxiv.org, paper
- arXiv 2610.29483: Local Gan–Gross–Prasad conjecture: epsilon dichotomyarxiv.org, paper
Show all 13 sources
- arXiv 2610.28839: On a conjecture of Velani (Chow, Yu)arxiv.org, paper
Changes
- Filed (17 PDFs read in full text)