{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev/","md":"https://postcutoff.com/e/2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev/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":"2026-10-29-kervaire-conjecture-openai-kourovka-confirmed-bogachev","date":"2026-10-29","date_precision":"day","short_title":"Kervaire's conjecture: OpenAI proof confirmed in the Kourovka Notebook","deck":null,"takeaway":"Kervaire's conjecture says that adding one generator and one relation to a nontrivial group never gives the trivial group. It was known for torsion-free groups (Klyachko 1993) and residually finite groups (Gerstenhaber–Rothaus 1962). OpenAI's 23–24 Sept release (family 256) claimed it in general, and also the Kervaire–Laudenbach conjecture on independent systems of equations (Howie's form). The Kourovka Notebook's update file now marks problems 17.94 and 18.87(a) solved: 'AI solution confirmed by N. V. Bogachev'. The same file lists 48 AI solutions confirmed by named mathematicians and 35 still unverified, including OpenAI's claim of infinite finitely presented periodic groups (8.52).","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":[{"n":1,"title":"Kourovka Notebook, 21st edition update file (November 2026 update)","url":"https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/21upd-3.pdf","type":"official","group":"primary","domain":"kourovkanotebookorg.wordpress.com"},{"n":2,"title":"OpenAI preprint hosted by Kourovka: Problem 17.94","url":"https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/17-94.pdf","type":"paper","group":"primary","domain":"kourovkanotebookorg.wordpress.com"},{"n":3,"title":"OpenAI preprint hosted by Kourovka: Problem 18.87","url":"https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/10/18-87.pdf","type":"paper","group":"primary","domain":"kourovkanotebookorg.wordpress.com"},{"n":4,"title":"openai/math: The Kervaire Theorem for Groups (24 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026/The-Kervaire-Theorem-for-Groups-September-24-2026.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":5,"title":"openai/math: Nonsingular systems of equations over arbitrary groups (5 Oct 2026)","url":"https://github.com/openai/math/blob/main/preprints/Nonsingular-systems-of-equations-over-arbitrary-groups-October-5-2026/nonsingular-systems-over-arbitrary-groups.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":6,"title":"openai/math Lean scope for family 256","url":"https://github.com/openai/math/blob/main/lean/docs/256.md","type":"code","group":"primary","domain":"github.com"}],"official":6,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Kourovka Notebook"],"title":"Kourovka Notebook marks OpenAI's proofs of Kervaire's conjecture and the Kervaire–Laudenbach conjecture as solved, 'AI solution confirmed by N. V. Bogachev'; its update lists 48 confirmed and 35 unverified AI solutions","summary":"The update file of the 21st edition of the Kourovka Notebook (headed \"November 2026 update\", at kourovkanotebookorg.wordpress.com, read 29–30 Oct) marks Problem 17.94 (Mycielski: can ℤ ∗ G be the normal closure of one element? — the negation of Kervaire's conjecture) answered \"No, it cannot\", and Problem 18.87(a) (Roman'kov: the Kervaire–Laudenbach conjecture) \"proved\", both with \"AI solution confirmed by N. V. Bogachev\" and OpenAI preprints hosted by the Notebook. These are the companion papers of OpenAI family 256 ('The Kervaire Theorem for Groups', 24 Sept; 'Nonsingular systems of equations over arbitrary groups', 5 Oct), whose Lean development covers the single-relator case. The Notebook's update marks unconfirmed AI solutions with ♯ and asks readers to check them.","key_facts":["Problem 17.94 (Kourovka, 'Well-known problem', J. Mycielski): 'Can the free product ℤ ∗ G of the infinite cyclic group ℤ and a nontrivial group G be the normal closure of a single element? Editors' comments: the negation is also known as Kervaire's conjecture. It was proved for torsion-free groups (A. Klyachko … 1993) and for residually finite groups (M. Gerstenhaber, O. S. Rothaus … 1962).' Entry now: '∗ No, it cannot (AI solution confirmed by N. V. Bogachev: OpenAI, Preprint, 2026, …/2026/10/17-94.pdf)'","Problem 18.87(a) (V. A. Roman'kov), the Kervaire–Laudenbach conjecture: 'every independent system of equations with coefficients in an arbitrary group G has a solution in some overgroup'; entry: '∗ a) The conjecture is proved (AI solution confirmed by N. V. Bogachev: OpenAI, Preprint, 2026, …/2026/10/18-87.pdf)'. Parts b) (nilpotent) and c) (solvable) are not marked","OpenAI family 256 (openai/math): 'Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture … A companion proves Kervaire's conjecture'; Lean scope lean/docs/256.md covers the single-relator case","Policy of the Notebook (update file, verbatim): 'Problems are marked with ♯ in those cases where some Artificial Intelligence (AI) system proposed a solution that has not yet been confirmed by mathematicians. Such a problem should be regarded as unsolved … in the first case a solution will be marked as a proper solution with ∗, and in the second altogether deleted'","Tally in the same file (our count of the phrases): 48 'AI solution confirmed by …' and 35 'unverified AI solution'. Confirmed contributors: Achyuth Jayadevan 23 (Zenodo preprints, confirmed by e.g. A. A. Buturlakin, O. G. Kharlampovich, M. Dixon, J. S. Wilson), P. Monticone 9, Nilradical 5, OpenAI 3 (17.94, 18.87(a), and 12.20 Thompson's group F nonamenable, confirmed by S.-H. Kim and Y. Lodha), others 1–2 each. Unverified include OpenAI on 8.52 (Olshanskii's 'well-known problem': infinite finitely presented periodic groups exist) and Jayadevan on 8.54(b)","Verification status: Kervaire and Kervaire–Laudenbach confirmed by a named group theorist and accepted by the Notebook's editors; OpenAI preprints not refereed; Lean only for the single-relator case"],"key_numbers":[],"tags":["math","group-theory","combinatorial-group-theory","kourovka-notebook","openai-math-release","lean","expert-confirmed"],"science":{"field":"mathematics","subfield":"group theory (combinatorial group theory, equations over groups)","problem":"Kervaire's conjecture (1960s; Kourovka 17.94) and the Kervaire–Laudenbach conjecture (Kourovka 18.87(a))","result":"OpenAI's proofs accepted as solutions in the Kourovka Notebook after confirmation by N. V. Bogachev","open_since":"","ai_system":["OpenAI internal model (unreleased, unnamed)"],"human_role":"Proofs from OpenAI's internal model run; human confirmation by N. V. Bogachev for the Notebook","verification":{"key":"lean","label":"Lean-verified","text":"Expert-confirmed (Kourovka Notebook); Lean for the single-relator case; preprints not refereed"},"status":"confirmed","shock":"A conjecture open since the 1960s, with only special cases known, is now listed as solved in the standard problem book of group theory on the strength of an AI proof."},"body_md":"## What happened\n\nKervaire's conjecture asks whether a nontrivial group can be killed by adding one new generator and one relation. It was proved in the\n1960s for residually finite groups and in 1993 for torsion-free groups, but stayed open in general. OpenAI's September release claimed it,\ntogether with a stronger statement about systems of equations over groups.\n\nThe Kourovka Notebook, the standard collection of open problems in group theory, now lists both as solved. Its update file credits OpenAI\npreprints that the Notebook hosts and says the AI solutions were confirmed by N. V. Bogachev. The same file marks 35 other AI claims as\nunverified and invites readers to check them.\n\n## Why it matters\n\nIt is one of the few famous claims from OpenAI's release that an outside expert has confirmed in public, and the Notebook gives the\nconfirmation an institutional form. The update also shows how much AI work the problem book now processes: 83 AI solutions in one file,\nmost of them from individual users rather than labs.","disputed":[],"related":[{"id":"2026-10-30-ai-assisted-results-october-30-listing","url":"https://postcutoff.com/e/2026-10-30-ai-assisted-results-october-30-listing/","date":"2026-10-30","date_precision":"day","short_title":"AI-assisted results, 30 Oct: arXiv announcement of 29–30 Oct","deck":null,"takeaway":"The 30 Oct sweep listed 374 arXiv papers that mention AI. We skimmed all their AI statements and read about 35 PDFs. Six results got their own entries: Das–Fox's Murty–Simon proof (GPT-5.6 Sol found the bound), Colbrook–Stepaniants's spectral isometry proof (found by GPT-6 Astra, Lean), the Θ(log k log log k) ratio of greedy for online Steiner forest (GPT Astra, Lean), Gáspár's Laczkovich solution (key proposition by ChatGPT), Ren's pure-semisimplicity counterexample (GPT-6 Astra, Lean) and Petty's conjecture in dimension 5 (Tencent's hyprover agent). PNNL's OpenConjecture study also has one. This listing has the rest, among them a single-author ChatGPT 6 Pro proof of the Alon–Seymour–Thomas separator conjecture, b₄ ≤ 7 for Borsuk's problem and Fiedorowicz's conjecture from a Claude idea.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":42,"official":42,"filed":"2026-10-30","updated":"2026-10-30","orgs":["OpenAI","Anthropic","Google","Tencent","various researchers"]},{"id":"2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition","url":"https://postcutoff.com/e/2026-10-13-thompson-group-f-nonamenable-kim-lodha-exposition/","date":"2026-10-13","date_precision":"day","short_title":"Thompson's group F is nonamenable","deck":"Specialists Sang-hyun Kim and Yash Lodha publish a human-written version of OpenAI's proof","takeaway":"The amenability of Thompson's group F, open since the 1970s, was one of the headline claims of OpenAI's 6 Oct math release. Two experts on groups of piecewise linear maps have now written the proof out themselves and generalized it, without AI.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":3,"filed":"2026-10-14","updated":"2026-10-19","orgs":["OpenAI","KIAS","Purdue University"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-05-09-deepmind-ai-co-mathematician","url":"https://postcutoff.com/e/2026-05-09-deepmind-ai-co-mathematician/","date":"2026-05-09","date_precision":"day","short_title":"Google DeepMind's 'AI co-mathematician' sets FrontierMath Tier 4 record and helps resolve a Kourovka Notebook problem","deck":null,"takeaway":"FrontierMath Tier 4, built to resist AI for years, was nearly half solved 18 months after launch.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["Google DeepMind","University of Oxford"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-30","type":"filed","text":"Created from lead; Kourovka update PDF read and AI-solution phrases counted (science agent)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-30","run":null,"sources_read":"Lead; Kourovka update PDF read and AI-solution phrases counted (science agent)","updated":"2026-10-30","human_review":null,"version":{"date":"2026-10-30"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":182,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":121,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":212,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":151,"in_training_data":false}],"short_url":"https://postcutoff.com/s/kervaire-conjecture-openai-proof"}