OpenAI claims a counterexample to the nearby Lagrangian conjecture
Awaiting review
Importance: major (4 of 5)The 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.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 29 October 2026
Your AI and this story
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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 |
|---|
Sources
4 sources from 3 sites. Numbers match the chips in the text.
4 sources: 1 primary, 3 reactions
Primary
Reactions
- ZUMA-GT seminar: Wenyuan Li, Nearby Lagrangian conjecture, past / present and future (15 Oct 2026)zuma-talks.github.io, discussion
- oai-math-revised: community notes on the nearby Lagrangian manuscriptgithub.com, discussion
- Latent Seminar: A counterexample to the nearby Lagrangian conjecture (exposition, 7 Oct)lowrank.github.io, discussion
Changes
- Filed (families 251–378; the manuscript’s introduction read, ZUMA-GT seminar page, oai-math-revised notes)