{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-cannon-conjecture-proof-claim/","md":"https://postcutoff.com/e/2026-10-06-openai-cannon-conjecture-proof-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-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":[{"n":1,"title":"OpenAI (PDF): A Modulus Proof of Cannon's Conjecture (23 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI Lean scope note for family 246","url":"https://github.com/openai/math/blob/main/lean/docs/246.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"oai-math-revised comparator results (CannonGeometricAction 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":4,"title":"Quasiworld seminar (22 Oct): Mario Bonk, The AI Proof of Cannon's conjecture (YouTube)","url":"https://www.youtube.com/watch?v=L2DU0VupdlU","type":"video","group":"reactions","domain":"youtube.com"},{"n":5,"title":"Proofs and Prompts: 100+ reactions to 100+ solutions (Baik, Wilton, Agol)","url":"https://proofsandprompts.com/2026/10/08/100-reactions-to-100-solutions/","type":"discussion","group":"reactions","domain":"proofsandprompts.com"},{"n":6,"title":"oai-math-revised AGENT_NOTES for the Cannon manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's math release claims a 31-page proof of Cannon's conjecture (a hyperbolic group with 2-sphere boundary acts geometrically on hyperbolic 3-space), formalized in Lean; Mario Bonk presented 'the AI proof' at the Quasiworld seminar","summary":"\"A Modulus Proof of Cannon's Conjecture\" (OpenAI, 23 Sept 2026, 31 pages; family 246 of the 6 Oct release): \"We prove that every hyperbolic group whose boundary is homeomorphic to the two-sphere admits a proper cocompact isometric action on hyperbolic three-space with finite kernel. This resolves Cannon's conjecture positively.\" The proof reduces the conjecture, via a criterion of Bourdon and Kleiner, to an upper bound on the combinatorial 2-modulus of curve families at a fixed scale in the group's boundary. OpenAI's Lean challenge CannonGeometricAction states the geometric-action conclusion with hyperbolicity defined by thin triangles in a Cayley graph; the oai-math-revised comparator run (25–26 Oct) accepted it. Mario Bonk presented the proof at the online Quasiworld seminar on 22 Oct. No written expert assessment has been published.","key_facts":["Manuscript (abstract, verbatim): 'We prove that every hyperbolic group whose boundary is homeomorphic to the two-sphere admits a proper cocompact isometric action on hyperbolic three-space with finite kernel. This resolves Cannon's conjecture positively.' 31 pages, dated 23 Sept 2026. Theorem 1.1 gives a homomorphism ρ : G → Isom(H³) with proper, cocompact action and finite kernel; 'For torsion-free G, the quotient is a closed hyperbolic three-manifold, possibly nonorientable'","Route (paper): Cannon's combinatorial Riemann mapping theorem (1994), the Cannon–Swenson formulation (1998) and the Cannon–Floyd–Parry modulus criterion (1999) turned the problem into control of discrete moduli; the proof uses the Bourdon–Kleiner criterion that a uniform bound for the combinatorial 2-modulus of curves of a fixed small diameter on an approximately self-similar metric sphere yields a quasi-Möbius parametrization by the round sphere. Bonk and Kleiner had proved the conjecture when the boundary's Ahlfors-regular conformal dimension is attained","Lean (OpenAI scope note lean/docs/246.md): 'The formalization proves that such a group admits an isometric action on H³ that is proper and cocompact and has finite kernel. Hyperbolicity is expressed through uniformly thin geodesic triangles in a Cayley graph, and the boundary hypothesis is a homeomorphism with S²'. Challenge CannonGeometricAction: 'accepted' in the oai-math-revised comparator run of 25–26 Oct (statement-to-paper fidelity not checked by that run)","Expert talk: 'Oct 22nd 2026, Quasiworld / Mario Bonk, The AI Proof of Cannon's conjecture' (YouTube, Quasiworld channel); the oai-math-revised hub lists it as a follow-up to this paper. No written summary of what Bonk concluded has been posted","Reactions (Proofs and Prompts, 8 Oct): Harry Hyungryul Baik: 'several of the problems or their variations were ones I had been working on right up until the day before, or were closely related to them. The Cannon conjecture is one example.' Henry Wilton named 'the Cannon conjecture' first among problems he cares about and said 'a cursory glance suggests that the write-ups vary wildly in quality. It will take months of work to go through the proofs'. Ian Agol, on the release's geometric group theory and topology results in general: problems 'that I might not have expected to be resolved in my lifetime (assuming they get checked to be correct)'"],"key_numbers":[],"tags":["math","geometric-group-theory","low-dimensional-topology","hyperbolic-groups","cannon-conjecture","quasiconformal-geometry","openai-math-release","lean","formal-verification"],"science":{"field":"mathematics","subfield":"geometric group theory / low-dimensional topology","problem":"Cannon's conjecture: does every word-hyperbolic group whose Gromov boundary is a 2-sphere act geometrically (properly, cocompactly, isometrically) on hyperbolic 3-space?","result":"Claimed proof for all such groups (finite kernel allowed), via a fixed-scale combinatorial 2-modulus bound and the Bourdon–Kleiner criterion; Lean formalization of the conclusion","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; presented in a seminar talk by Mario Bonk; no written human check yet","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Lean formalization accepted by an independent comparator run; unrefereed 31-page manuscript; expert seminar presentation (22 Oct) without a published verdict"},"status":"pending","shock":"A central conjecture of geometric group theory, open since the 1990s, is claimed in a short machine-written paper that one of the main contributors to its analytic approach presented in a seminar two weeks later."},"body_md":"## What happened\n\nCannon's conjecture predicts that a hyperbolic group whose boundary at infinity is a 2-sphere is, up to finite kernel, a cocompact\ngroup of isometries of hyperbolic 3-space. It is the group-theoretic face of hyperbolic geometrization in dimension three: such a\ntorsion-free group would be the fundamental group of a closed hyperbolic 3-manifold. Analytic approaches through quasiconformal\ngeometry (Cannon, Floyd, Parry, Bonk, Kleiner) reduced it to estimates on discrete moduli of curve families. Bonk and Kleiner proved it\nwhen the boundary's conformal dimension is attained.\n\nOpenAI's 6 Oct release includes a 31-page manuscript, dated 23 Sept, claiming the missing modulus bound in general. It then applies a\ncriterion of Bourdon and Kleiner to obtain a quasi-Möbius parametrization of the boundary by the round sphere. The conclusion is\nformalized in Lean, and a community comparator run accepted it. On 22 Oct Mario Bonk gave a talk on \"the AI proof\" at Quasiworld, an\nonline seminar in quasiconformal geometry.\n\n## Why it matters\n\nIf correct, the result settles a central problem of geometric group theory, and by a short argument in the analytic programme rather\nthan through cubulation. A seminar talk by one of the main contributors to that programme is the strongest public engagement with the paper\nso far, but no written verdict exists. Our coverage of the release had listed the claim only in passing.","disputed":[{"label":"Verification","text":"Lean formalization accepted by an independent comparator run; unrefereed 31-page manuscript; expert seminar presentation (22 Oct) without a published verdict","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":"2026-10-17-non-sofic-hyperbolic-group-sandhu-thom","url":"https://postcutoff.com/e/2026-10-17-non-sofic-hyperbolic-group-sandhu-thom/","date":"2026-10-17","date_precision":"day","short_title":"A non-sofic word-hyperbolic group","deck":"Sandhu and Thom build one from OpenAI's non-residually-finite hyperbolic group, starting from an AI-generated construction; Douglas–Mghirbi give an independent AI-assisted proof with Lean","takeaway":"Whether every hyperbolic group is sofic was a well-known open question in group theory. Two groups now say no. Both build on the torsion-free, non-residually-finite hyperbolic group in OpenAI's 6 October release. One is co-written by Andreas Thom, an expert in sofic groups. The other, by Douglas and Mghirbi, comes with a Lean formalization.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":4,"official":4,"filed":"2026-10-20","updated":"2026-10-20","orgs":["TU Dresden","OpenAI"]},{"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"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (CONTENTS entry, PDF (31 pp), Lean scope note, comparator results, YouTube metadata of Bonk's talk, oai-math-revised notes, Proofs and Prompts reactions)"}],"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/cannon-conjecture-openai-lean"}