--- id: "2026-10-06-openai-nearby-lagrangian-conjecture-counterexample" url: "https://postcutoff.com/e/2026-10-06-openai-nearby-lagrangian-conjecture-counterexample/" as_of: "2026-10-30T13:42:00+01:00" date: "2026-10-06" date_precision: day category: science importance: 4 confidence: medium status: [Awaiting review] verification: Preprint only sources: 4 editor: human editor human_review: null version: "2026-10-29" --- As of: 2026-10-30 13:42 CET. Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/ Canonical page: https://postcutoff.com/e/2026-10-06-openai-nearby-lagrangian-conjecture-counterexample/ # OpenAI claims a counterexample to the nearby Lagrangian conjecture Full 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 "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 ## What happened In a cotangent bundle, the zero section is the model example of an exact Lagrangian submanifold. The nearby Lagrangian conjecture, which grew out of Arnold's questions on Lagrangian knots and was posed by Arnold, Lalonde and Sikorav, says every closed exact Lagrangian there is the zero section in disguise, up to Hamiltonian isotopy. Over the 2-sphere and the 2-torus it is a theorem. In general a long line of work showed that such a Lagrangian is indistinguishable from the base by homotopy theory: the projection is a homotopy equivalence (Abouzaid, Kragh), even a simple one (Abouzaid–Kragh). OpenAI's manuscript claims it fails. For some large even N it constructs an exact Lagrangian in the cotangent bundle of S⁹ × S^{N−1} that is even diffeomorphic to the base but cannot be isotoped onto the zero section. The obstruction comes from algebraic K-theory of spaces, through generating families. On 15 Oct the symplectic topologist Wenyuan Li devoted a seminar talk to how the construction works, stating he would discuss consequences "assuming the counterexample is correct". ## Why it matters The nearby Lagrangian conjecture is one of the guiding problems of symplectic topology. A counterexample would show that homotopy-level rigidity, proved over the past two decades, does not extend to Hamiltonian isotopy in high dimensions. The claim is unrefereed and unformalized. As of 29 Oct the only expert engagement we found treats its correctness as an open assumption. ## What is disputed or not yet verified - Verification: Unrefereed manuscript without formalization; presented, conditionally on correctness, in an expert seminar talk ## Your AI and this story - GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff - Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff - Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff - Grok 4.7 (training cutoff May 2026): 128 days after its cutoff ## Sources 1. [OpenAI: A counterexample to the nearby Lagrangian conjecture (23 Sept 2026, PDF)](https://github.com/openai/math/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/paper.pdf) (github.com, paper) 2. [ZUMA-GT seminar: Wenyuan Li, Nearby Lagrangian conjecture, past / present and future (15 Oct 2026)](https://zuma-talks.github.io/gt/index.html) (zuma-talks.github.io, discussion) 3. [oai-math-revised: community notes on the nearby Lagrangian manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-counterexample-to-the-nearby-Lagrangian-conjecture-September-23-2026/AGENT_NOTES.md) (github.com, discussion) 4. [Latent Seminar: A counterexample to the nearby Lagrangian conjecture (exposition, 7 Oct)](https://lowrank.github.io/latent-seminar/OpenAI-Math/340-nearby-lagrangian-counterexample) (lowrank.github.io, discussion) ## Changes - 2026-10-29 (filed): Created (families 251–378; the manuscript's introduction read, ZUMA-GT seminar page, oai-math-revised notes) ## Related - 2026-10-24: [oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)](https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/index.md) - 2026-10-06: [OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems](https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/index.md)