{"schema":"postcutoff/event@1","as_of":"2026-10-22T13:31:00+02:00","url":"https://postcutoff.com/e/2026-10-22-ai-assisted-results-october-22-listing/","md":"https://postcutoff.com/e/2026-10-22-ai-assisted-results-october-22-listing/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-22-ai-assisted-results-october-22-listing","date":"2026-10-22","date_precision":"day","short_title":"AI-assisted results in the 22 Oct 2026 arXiv listing","deck":"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","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.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.29211: On the gradient conjecture at infinity (Do, Jentzen, Kuckuck)","url":"https://arxiv.org/abs/2610.29211","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"arXiv 2610.29066: Quasilinear-size constant-query zero-knowledge PCPs (Park, Fan)","url":"https://arxiv.org/abs/2610.29066","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"arXiv 2610.29731: Computing the nucleon of matching games (Aziz, Csáji)","url":"https://arxiv.org/abs/2610.29731","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"arXiv 2610.29904: Kelly's conjectures on Condorcet winner probabilities (Kim, Shin)","url":"https://arxiv.org/abs/2610.29904","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"arXiv 2610.28935: Functional limit theorems for self-interacting random walks","url":"https://arxiv.org/abs/2610.28935","type":"paper","group":"primary","domain":"arxiv.org"},{"n":6,"title":"arXiv 2610.29840: Rayleigh's conjecture for vibrating plates (Bucur, Laugesen)","url":"https://arxiv.org/abs/2610.29840","type":"paper","group":"primary","domain":"arxiv.org"},{"n":7,"title":"arXiv 2610.29563: Conjugacy problem for virtually soluble groups of finite Prüfer rank","url":"https://arxiv.org/abs/2610.29563","type":"paper","group":"primary","domain":"arxiv.org"},{"n":8,"title":"arXiv 2610.28757: Sharp quantitative stability of the Hong–Krahn–Szegő inequality","url":"https://arxiv.org/abs/2610.28757","type":"paper","group":"primary","domain":"arxiv.org"},{"n":9,"title":"arXiv 2610.28913: Strong gradient absorption and initial decay prevent blow-up","url":"https://arxiv.org/abs/2610.28913","type":"paper","group":"primary","domain":"arxiv.org"},{"n":10,"title":"arXiv 2610.28780: Low-degree filtrations in Grassmannian cohomology","url":"https://arxiv.org/abs/2610.28780","type":"paper","group":"primary","domain":"arxiv.org"},{"n":11,"title":"arXiv 2610.29624: Tight space bounds for pivot-based distance oracles","url":"https://arxiv.org/abs/2610.29624","type":"paper","group":"primary","domain":"arxiv.org"},{"n":12,"title":"arXiv 2610.29483: Local Gan–Gross–Prasad conjecture: epsilon dichotomy","url":"https://arxiv.org/abs/2610.29483","type":"paper","group":"primary","domain":"arxiv.org"},{"n":13,"title":"arXiv 2610.28839: On a conjecture of Velani (Chow, Yu)","url":"https://arxiv.org/abs/2610.28839","type":"paper","group":"primary","domain":"arxiv.org"}],"official":13,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"],"title":"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","summary":"Catalogue of AI-disclosing papers in the 22 Oct 2026 arXiv announcement (submitted 20–21 Oct). Own entries: Tevelev's re-proof of OpenAI's Nagata proof, Anghel's survey and generalization of OpenAI's counterexample to Hochster's small Cohen–Macaulay conjecture, and Voronov's GPT-6 Astra-assisted counterexample to Gale's conjecture in R³. Gouéré's exposition of the Kozma–Nitzan proof was added to the dying-percolation entry. The rest is listed here.","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')","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"],"key_numbers":[],"tags":["math","computer-science","arxiv","catalogue","chatgpt","gpt-6-astra","gpt-6-1-sol","gpt-5-6-sol","codex","claude","claude-fable-5-1","gemini"],"science":{"field":"mathematics","subfield":"multiple (o-minimal dynamics, complexity theory, game theory, probability, spectral theory)","problem":"Gradient conjecture at infinity; zero-knowledge PCP length; nucleon computation; Kelly's Condorcet conjectures; see list","result":"Catalogue of AI-assisted results in the 22 Oct 2026 arXiv announcement beyond those with their own entries","open_since":"","ai_system":["GPT-6 Astra","GPT-6 Sol","GPT-5.6 Sol","GPT Sol 6.1","ChatGPT","Codex","Claude","Claude Fable 5.1","Gemini"],"human_role":"From AI as an extensive research collaborator (gradient conjecture, zero-knowledge PCPs) to checking and editing","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed arXiv preprints, proofs checked by their authors"},"status":"pending","shock":"","verified_by":["preprint"],"kind":"batch"},"body_md":"## What happened\n\nThree results from Thursday's announcement have their own entries:\n- Jenia Tevelev's re-proof of OpenAI's Nagata proof, which he calls \"a stunning achievement\";\n- 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;\n- Vsevolod Voronov's 101-point counterexample to Gale's conjecture, found with algorithms developed with GPT-6 Astra.\n\nJean-Baptiste Gouéré's exposition of OpenAI's Kozma–Nitzan proof was added to the dying-percolation entry.\n\nIn the papers listed here, AI plays two kinds of role:\n- **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.\n- **Support**: exploring ideas, checking, drafting and editing (Kelly's conjectures, Rayleigh's conjecture, Gan–Gross–Prasad and others).\n\nOlder items still listed in the sweep (14–19 Oct ML, physics and materials papers that only mention LLMs) are out of scope here.\n\n## Why it matters\n\nMathematicians 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.","disputed":[{"label":"Verification","text":"Unrefereed arXiv preprints, proofs checked by their authors","sources":[]}],"related":[{"id":"2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition","url":"https://postcutoff.com/e/2026-10-21-nagata-conjecture-openai-proof-tevelev-exposition/","date":"2026-10-21","date_precision":"day","short_title":"Nagata's 1959 conjecture","deck":"Algebraic geometer Jenia Tevelev writes out OpenAI's 'remarkably simple' proof and calls the release 'a stunning achievement'","takeaway":"Nagata's conjecture on plane curves through very general points, open since 1959, was one of the claims in OpenAI's 6 Oct maths release. A UMass Amherst specialist has now reorganized, simplified and published the full proof himself, while criticizing how the manuscripts were dumped on the community.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","UMass Amherst"]},{"id":"2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey","url":"https://postcutoff.com/e/2026-10-21-hochster-small-cohen-macaulay-counterexample-openai-anghel-survey/","date":"2026-10-21","date_precision":"day","short_title":"Hochster's 1973 small Cohen–Macaulay conjecture","deck":"An OpenAI manuscript's counterexample is surveyed, re-proved and extended to every dimension by Cristian Anghel","takeaway":"OpenAI's 6 Oct release included a model-written counterexample to the domain form of Hochster's small Cohen–Macaulay conjecture, a central question of commutative algebra since 1973. A survey by Cristian Anghel, whose September preprints started this line of work, gives a complete proof of it and generalizes it to all dimensions.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":5,"official":5,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","Simion Stoilow Institute of Mathematics of the Romanian Academy"]},{"id":"2026-10-21-ai-assisted-results-october-21-listing","url":"https://postcutoff.com/e/2026-10-21-ai-assisted-results-october-21-listing/","date":"2026-10-21","date_precision":"day","short_title":"AI-assisted results in the 21 Oct 2026 arXiv listing","deck":"A ChatGPT 6.1 Sol counterexample to Frankl–Ota–Tokushige, klt Fano proofs from ChatGPT, Vince's conjecture with GPT Astra, Pólya for spherical shells, a Claude-built 55-addition 3×3 scheme and a Claude Opus 5.5 lemma in reverse mathematics","takeaway":"Wednesday's arXiv announcement brought four results with their own entries: the MDS conjecture in even characteristic, the double-cap conjecture, the quasitrace/monotone completeness counterexamples and the Neumann–Zagier conjecture. An expert verification of Claude's Hadamard matrices was added to that entry. This page lists the next tier, about thirty papers.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":18,"official":18,"filed":"2026-10-21","updated":"2026-10-21","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-10-20-gale-conjecture-r3-counterexample-gpt-6-astra","url":"https://postcutoff.com/e/2026-10-20-gale-conjecture-r3-counterexample-gpt-6-astra/","date":"2026-10-20","date_precision":"day","short_title":"Gale's 1953 conjecture fails in 3D","deck":"A 101-point set found with LLM code evolution and GPT-6 Astra beats the sphere's Borsuk partition ratio","takeaway":"Vsevolod Voronov built a 101-point set in space whose every split into four parts leaves a part of diameter above 0.88928, more than the sphere's 0.88807. That refutes Gale's 1953 conjecture in dimension 3. The search algorithms were developed with GPT-6 Astra, some proposed by the AI, and the set is checkable in exact arithmetic.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":2,"official":2,"filed":"2026-10-22","updated":"2026-10-22","orgs":["OpenAI","Anthropic"]},{"id":"2026-09-03-dying-percolation-theta-pc-zero","url":"https://postcutoff.com/e/2026-09-03-dying-percolation-theta-pc-zero/","date":"2026-09-03","date_precision":"day","short_title":"Claude-written Lean proof claims the dying percolation conjecture θ(p_c)=0 in every dimension","deck":null,"takeaway":"If the formal statement matches the intended theorem, a famous problem in mathematical physics is settled by machine-written formal mathematics.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":12,"official":6,"filed":"2026-09-29","updated":"2026-10-22","orgs":["Anthropic","OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-22","type":"filed","text":"Created (17 PDFs read in full text)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-22","run":null,"sources_read":"17 PDFs read in full text","updated":"2026-10-22","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":175,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":114,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":205,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":144,"in_training_data":false}],"short_url":null}