{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-nearby-lagrangian-conjecture-counterexample/","md":"https://postcutoff.com/e/2026-10-06-openai-nearby-lagrangian-conjecture-counterexample/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-nearby-lagrangian-conjecture-counterexample","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims a counterexample to the nearby Lagrangian conjecture","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release (family 340, dated 23 Sept) claims to disprove the nearby Lagrangian conjecture, a central conjecture of symplectic topology posed by Arnold, Lalonde and Sikorav. For some large even N, it builds a closed exact Lagrangian in the cotangent bundle of S⁹×S^{N−1} that is diffeomorphic to the base but cannot be moved onto the zero section by a Hamiltonian isotopy. Earlier work had shown such Lagrangians are always simple homotopy equivalent to the base (Abouzaid–Kragh). On 15 Oct Wenyuan Li gave a seminar talk on how the construction works, 'assuming the counterexample is correct'. No Lean proof; no expert verdict.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: A counterexample to the nearby Lagrangian conjecture (23 Sept 2026, PDF)","url":"https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"ZUMA-GT seminar: Wenyuan Li, Nearby Lagrangian conjecture, past / present and future (15 Oct 2026)","url":"https://zuma-talks.github.io/gt/index.html","type":"discussion","group":"reactions","domain":"zuma-talks.github.io"},{"n":3,"title":"oai-math-revised: community notes on the nearby Lagrangian manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"},{"n":4,"title":"Latent Seminar: A counterexample to the nearby Lagrangian conjecture (exposition, 7 Oct)","url":"https://lowrank.github.io/latent-seminar/OpenAI-Math/340-nearby-lagrangian-counterexample","type":"discussion","group":"reactions","domain":"lowrank.github.io"}],"official":1,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's release claims a counterexample to Arnold's nearby Lagrangian conjecture: an exact Lagrangian in T*(S⁹×S^{N−1}) not Hamiltonian isotopic to the zero section","summary":"\"A counterexample to the nearby Lagrangian conjecture\" (OpenAI, 23 Sept 2026, released 6 Oct, family 340): \"For some sufficiently large even integer N, we construct a closed exact smoothly embedded Lagrangian in T*(S⁹ × S^{N−1}) that is diffeomorphic to the base but is not Hamiltonian isotopic to the zero section.\" The conjecture predicts every closed exact Lagrangian in a cotangent bundle is Hamiltonian isotopic to the zero section. It holds over S² and T² (Hind; Dimitroglou Rizell–Goodman–Ivrii), and in general the projection was known to be a simple homotopy equivalence (Abouzaid, Kragh, Abouzaid–Kragh). The construction uses generating families and tube spaces (algebraic K-theory of spaces). Wenyuan Li (Zhejiang University) presented it at the ZUMA-GT seminar on 15 Oct. No formalization; no published verification.","key_facts":["Claim (Theorem 1.1, verbatim): 'For some sufficiently large even integer N, the manifold Q = S⁹ × S^{N−1} admits a closed exact smoothly embedded Lagrangian L ⊂ T*Q such that L is diffeomorphic to Q and no compactly supported Hamiltonian isotopy carries the zero section to L.' The paper adds: 'Both the base and the Lagrangian in this example are connected and simply connected.'","Prior state (as the paper recounts it): Hind (T*S²) and Dimitroglou Rizell–Goodman–Ivrii (cotangent torus) proved positive cases; Fukaya–Seidel–Smith and Nadler gave cohomological constraints; Abouzaid proved the projection is a homotopy equivalence when the Maslov class vanishes, Kragh removed that hypothesis, and Abouzaid–Kragh proved simple homotopy equivalence; later work constrained the Gauss map and the normal invariant. None of these identifies the Hamiltonian isotopy class","Method: generating families with a nonzero stable 'tube' class (smooth tube spaces link generating functions to Waldhausen's algebraic K-theory of spaces), with a proof that coincident cotangent branches are always separated, so the Lagrangian is embedded rather than immersed","Expert engagement: Wenyuan Li (Zhejiang University), ZUMA-GT seminar, 15 Oct, research talk 'Nearby Lagrangian conjecture, present and future': 'Then we discuss how the recently proposed counterexample (of embedded nearby Lagrangian) by OpenAI is constructed. Finally, assuming the counterexample is correct, I will explain why I think there are still many problems to be understood…'","Status: no Lean scope note for family 340; no revision or withdrawal in history.md; Latent Seminar's exposition (7 Oct) states it does not verify the complete proof"],"key_numbers":[],"tags":["math","symplectic-topology","lagrangian-submanifolds","nearby-lagrangian-conjecture","generating-functions","counterexample","openai-math-release"],"science":{"field":"mathematics","subfield":"symplectic topology (exact Lagrangian submanifolds of cotangent bundles)","problem":"Nearby Lagrangian conjecture (Arnold): every closed exact Lagrangian in T*Q is Hamiltonian isotopic to the zero section","result":"Claimed counterexample in T*(S⁹ × S^{N−1}) for some large even N: an embedded exact Lagrangian diffeomorphic to the base, not Hamiltonian isotopic to the zero section","open_since":"","ai_system":["OpenAI internal model (math release)"],"human_role":"Autonomous AI output released by OpenAI","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed manuscript without formalization; presented, conditionally on correctness, in an expert seminar talk"},"status":"pending","shock":"After two decades of results pushing toward the conjecture (homotopy, then simple homotopy equivalence), the claim is that it is false in high dimensions."},"body_md":"## What happened\n\nIn a cotangent bundle, the zero section is the model example of an exact Lagrangian submanifold. The nearby Lagrangian conjecture,\nwhich grew out of Arnold's questions on Lagrangian knots and was posed by Arnold, Lalonde and Sikorav, says every closed exact\nLagrangian there is the zero section in disguise, up to Hamiltonian isotopy. Over the 2-sphere and the 2-torus it is a theorem. In\ngeneral a long line of work showed that such a Lagrangian is indistinguishable from the base by homotopy theory: the projection is a\nhomotopy equivalence (Abouzaid, Kragh), even a simple one (Abouzaid–Kragh).\n\nOpenAI's manuscript claims it fails. For some large even N it constructs an exact Lagrangian in the cotangent bundle of S⁹ × S^{N−1}\nthat is even diffeomorphic to the base but cannot be isotoped onto the zero section. The obstruction comes from algebraic K-theory of\nspaces, through generating families. On 15 Oct the symplectic topologist Wenyuan Li devoted a seminar talk to how the construction\nworks, stating he would discuss consequences \"assuming the counterexample is correct\".\n\n## Why it matters\n\nThe nearby Lagrangian conjecture is one of the guiding problems of symplectic topology. A counterexample would show that homotopy-level\nrigidity, proved over the past two decades, does not extend to Hamiltonian isotopy in high dimensions. The claim is unrefereed and\nunformalized. As of 29 Oct the only expert engagement we found treats its correctness as an open assumption.","disputed":[{"label":"Verification","text":"Unrefereed manuscript without formalization; presented, conditionally on correctness, in an expert seminar talk","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"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (families 251–378; the manuscript's introduction read, ZUMA-GT seminar page, oai-math-revised notes)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Families 251–378; the manuscript's introduction read, ZUMA-GT seminar page, oai-math-revised notes","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/openai-counterexample-nearby"}