{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-finitely-presented-burnside-periodic-group-claim/","md":"https://postcutoff.com/e/2026-10-06-openai-finitely-presented-burnside-periodic-group-claim/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":"2026-10-06-openai-finitely-presented-burnside-periodic-group-claim","date":"2026-10-06","date_precision":"day","short_title":"Finitely presented Burnside problem answered negatively? (OpenAI, Lean)","deck":null,"takeaway":"Two manuscripts credited to 'OpenAI' claim an infinite group with a finite presentation in which every element has finite order. Whether such a group exists, the finitely presented form of Burnside's 1902 question, had stayed open even after the infinite finitely generated examples of Golod–Shafarevich and Novikov–Adian. The claimed group is the Steinberg group St₁₂(R) over a characteristic-two algebra and also has Kazhdan's property (T). A companion dated 5 Oct makes it residually finite, with all element orders powers of 2. The existence statement is formalized in Lean and was accepted in a community comparator run; no group theorist has published an assessment.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI (PDF): An infinite finitely presented periodic group (23 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI (PDF): An infinite finitely presented residually finite 2-group (5 Oct 2026)","url":"https://github.com/openai/math/blob/main/preprints/An-infinite-finitely-presented-residually-finite-2-group-October-5-2026/residually-finite-torsion.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"OpenAI Lean scope note for family 247","url":"https://github.com/openai/math/blob/main/lean/docs/247.md","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"oai-math-revised comparator results (PeriodicGroup accepted)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv","type":"code","group":"primary","domain":"github.com"},{"n":5,"title":"GitHub: KitaKen1/all-prime-residually-finite-p-groups (Lean follow-up for every prime)","url":"https://github.com/KitaKen1/all-prime-residually-finite-p-groups","type":"code","group":"primary","domain":"github.com"},{"n":6,"title":"formal-conjectures PR #7412: Formalize the finitely presented Burnside problem and its variants","url":"https://github.com/google-deepmind/formal-conjectures/pull/7412","type":"code","group":"primary","domain":"github.com"},{"n":7,"title":"Wikipedia: Burnside problem (background)","url":"https://en.wikipedia.org/wiki/Burnside_problem","type":"docs","group":"primary","domain":"en.wikipedia.org"},{"n":8,"title":"Proofs and Prompts: 100+ reactions to 100+ solutions (8 Oct)","url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/","type":"discussion","group":"reactions","domain":"proofsandprompts.com"},{"n":9,"title":"oai-math-revised AGENT_NOTES for the periodic-group manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":7,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's math release claims an infinite finitely presented periodic group, answering the finitely presented Burnside problem negatively (a Steinberg group over an F₂-algebra, with property (T)); a residually finite 2-group version followed on 5 Oct; Lean covers the first","summary":"Family 247 of OpenAI's 6 Oct 2026 math release contains \"An infinite finitely presented periodic group\" (OpenAI, 23 Sept, 48 pages): \"We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively.\" The group is St₁₂(R) for a unital associative F₂-algebra R; the paper adds that it has property (T) and is nonamenable, and that the same algebra gives a finitely presented nil algebra that is not nilpotent, answering the finite-presentation version of Kurosh's question. The companion \"An infinite finitely presented residually finite 2-group\" (5 Oct, 25 pages) proves the group residually finite and extracts a finite-index subgroup in which every element has 2-power order. Lean (challenge PeriodicGroup) covers the existence of an infinite finitely presented periodic group and was accepted in the oai-math-revised comparator run (25–26 Oct); the residual-finiteness and algebra results are not formalized.","key_facts":["Manuscript 1 (abstract, verbatim): 'We construct an infinite group with an ordinary finite presentation in which every element has finite order, answering the finitely presented Burnside question negatively. We also construct an infinite-dimensional finitely presented nonunital nil associative algebra over F₂ that is Jacobson radical but not nilpotent.' Theorem 1.1: 'the construction below gives a unital associative F₂-algebra R for which St₁₂(R) is infinite, finitely presented, and periodic.' Corollary 1.2: the group has Kazhdan's property (T), hence is nonamenable (via Ershov and Jaikin-Zapirain's theorem on Steinberg groups). 48 pages, dated 23 Sept 2026","Manuscript 2 (5 Oct 2026, 25 pages): 'We prove that the infinite, ordinarily finitely presented periodic Steinberg group Γ = St₁₂(R) of a companion paper is residually finite. Its finite-index subgroup G = ker(Γ → St₁₂(F₂)) is infinite, ordinarily finitely presented, and residually finite, and every element of G has finite 2-power order. The orders of its elements are unbounded'","Background as the paper recounts it: Burnside's questions started the study of finiteness under restrictions on element orders; Golod built infinite finitely generated periodic groups from nil algebras via the Golod–Shafarevich inequality; Novikov and Adian built infinite finitely generated groups of bounded exponent; Zelmanov solved the restricted Burnside problem; Ol'shanskii and Sapir 'explicitly distinguish the finitely presented periodic-group question'. Infinite finitely generated periodic groups with property (T) were known (Ershov); the new claim is the combination with an ordinary finite presentation","Lean (OpenAI scope note lean/docs/247.md): 'The formalization gives a negative answer by constructing an infinite finitely presented periodic group, including a witness realized as a Steinberg group over an algebra of characteristic two. Periodicity means that each element has some finite order; no common exponent is asserted. The paper's separate nil-algebra and radical-algebra conclusions are outside these selected statements.' Challenge PeriodicGroup: 'accepted' in the oai-math-revised comparator run of 25–26 Oct. The residually finite 2-group is not in the scope note","Follow-up: Kenta Kitamura's GitHub repository claims infinite finitely presented residually finite p-groups for every prime p, with a Lean proof; its README opens 'OpenAI solved the following finitely presented Burnside problem'. His formal-conjectures PR #7412 (opened 16 Oct, still open) would add the problem with 'research solved' tags, reusing OpenAI's theorem for the main statement","Reactions: none from group theorists on this paper found (web, Hacker News, oai-math-revised notes) as of 29 Oct. In general reactions, Xiaolei Wu wrote that the release's group-theory section 'looked unreal. These were almost all the major open problems in my field!' (Proofs and Prompts, 8 Oct)"],"key_numbers":[],"tags":["math","group-theory","burnside-problem","periodic-groups","steinberg-groups","nil-algebras","openai-math-release","lean","formal-verification"],"science":{"field":"mathematics","subfield":"group theory (Burnside-type problems, finitely presented groups)","problem":"Finitely presented Burnside problem: must a finitely presented group in which every element has finite order be finite?","result":"Claimed infinite finitely presented periodic group St₁₂(R) over an F₂-algebra, with property (T); claimed residually finite version with a finite-index 2-power-torsion subgroup; claimed finitely presented nil algebra that is not nilpotent","open_since":"","ai_system":["OpenAI internal model (math release)"],"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; no human exposition or check yet","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Lean formalization of the existence of an infinite finitely presented periodic group, accepted by an independent comparator run; residual finiteness and the algebra results are only in unrefereed manuscripts"},"status":"pending","shock":"A finiteness question that Golod, Adian and Ol'shanskii's methods left open is claimed in two short machine-written papers, with an explicit Steinberg-group witness rather than a small-cancellation construction."},"body_md":"## What happened\n\nBurnside asked in 1902 whether a finitely generated group in which every element has finite order must be finite. Golod and Shafarevich\nshowed in 1964 that it need not be, and Novikov and Adian (1968) found infinite examples of bounded exponent. Whether a group given by finitely many\ngenerators and finitely many defining relations can be infinite and periodic remained open; Ol'shanskii and Sapir single it out as a\nseparate question.\n\nOpenAI's 6 Oct release claims such a group: the Steinberg group St₁₂(R) over a specially built algebra R of characteristic two. The\nsame construction is said to give a finitely presented nil algebra that is not nilpotent. A second paper, dated 5 Oct, claims the group\nis residually finite and contains a finite-index subgroup whose elements all have order a power of 2. The existence statement is\nformalized in Lean, and a community comparator run accepted it. A volunteer has since claimed, with Lean, residually finite examples for\nevery prime.\n\n## Why it matters\n\nIf the construction holds, one of the classical finiteness questions of group theory has a negative answer with an algebraic, explicit\nwitness. Because the Lean statement is short, specialists can check whether it says what group theorists mean: an ordinary finite\npresentation and elements of finite order. That check, and any expert reading of the 48-page argument, has not been published. Our\ncoverage of the release had not listed this claim at all.","disputed":[{"label":"Verification","text":"Lean formalization of the existence of an infinite finitely presented periodic group, accepted by an independent comparator run; residual finiteness and the algebra results are only in unrefereed manuscripts","sources":[]}],"related":[{"id":"2026-10-24-oai-math-revised-community-hub","url":"https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/","date":"2026-10-24","date_precision":"day","short_title":"oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)","deck":null,"takeaway":"This is the first organized attempt by research mathematicians to digest and credit the OpenAI catalogue.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":6,"official":5,"filed":"2026-10-25","updated":"2026-10-27","orgs":["OpenAI","mathcommunity"]},{"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":"2026-10-06-openai-cannon-conjecture-proof-claim","url":"https://postcutoff.com/e/2026-10-06-openai-cannon-conjecture-proof-claim/","date":"2026-10-06","date_precision":"day","short_title":"Cannon's conjecture claimed (OpenAI, Lean); Bonk's seminar talk","deck":null,"takeaway":"A 31-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims Cannon's conjecture: every word-hyperbolic group whose boundary is a 2-sphere acts properly and cocompactly by isometries on hyperbolic 3-space. For torsion-free groups this makes them fundamental groups of closed hyperbolic 3-manifolds, a group-theoretic counterpart of geometrization. The proof bounds combinatorial modulus and then applies a Bourdon–Kleiner criterion. The Lean statement was accepted in a community comparator run. On 22 Oct Mario Bonk, who with Bruce Kleiner proved a key special case, gave a seminar talk titled 'The AI Proof of Cannon's conjecture'.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-09-07-koethe-conjecture-disproved","url":"https://postcutoff.com/e/2026-09-07-koethe-conjecture-disproved/","date":"2026-09-07","date_precision":"day","short_title":"Pre-release GPT-6 Astra disproves the Köthe conjecture with a Lean-verified counterexample","deck":null,"takeaway":"If it survives review, it resolves one of the most famous open problems in ring theory, found autonomously and verified formally.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-10-19","orgs":["OpenAI","Epoch AI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (CONTENTS entry, both PDFs (48 and 25 pp), Lean scope note, comparator results, Kitamura's repository and formal-conjectures PR, Proofs and Prompts reactions; web and HN searches found no specialist comment)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":null,"updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/finitely-presented-burnside"}