The nearby Lagrangian conjecture
From OpenAI Aftermath
The nearby Lagrangian conjecture asks whether every closed exact embedded Lagrangian submanifold of the cotangent bundle T*Q of a closed manifold Q is Hamiltonian isotopic to the zero section. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 23 September 2026, claims a counterexample. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- The conjecture goes back to Arnold's discussion of Lagrangian knots; the manuscript uses the formulation of Abouzaid and Kragh. Intersection theorems of Hofer and of Laudenbach and Sikorav, and the generating-function methods of Chaperon, Laudenbach, Sikorav and Chekanov, constrain Hamiltonian deformations of the zero section.
- Positive results in low dimension include Hind's theorem for Lagrangian spheres in T*S2 and the theorem of Dimitroglou Rizell, Goodman and Ivrii for the cotangent bundle of the torus.
- Fukaya, Seidel and Smith, and independently Nadler, established cohomological and categorical constraints under additional assumptions. Abouzaid proved that the projection of a closed exact Lagrangian to the base is a homotopy equivalence when the Maslov class vanishes, and Kragh, with an appendix by Abouzaid, removed that hypothesis. Abouzaid and Kragh proved that the projection is a simple homotopy equivalence.
- Finer invariants were constrained by Abouzaid, Courte, Guillermou and Kragh (the stable Lagrangian Gauss map), by Abouzaid, Álvarez-Gavela, Courte and Kragh and by Porcelli and Smith (the normal invariant of the projection), and by Courte and Porcelli (parametrized Whitehead torsion). Álvarez-Gavela, Igusa and Sullivan introduced tube torsion for Legendrians. The manuscript notes that these results concern the topology of the projection and do not identify the Hamiltonian isotopy class.
The original result
- Claim. Theorem 1.1 of A counterexample to the nearby Lagrangian conjecture states that for some sufficiently large even integer N, the manifold Q = S9 × SN−1 admits a closed exact smoothly embedded Lagrangian L in T*Q that is diffeomorphic to Q but is not carried to the zero section by any compactly supported Hamiltonian isotopy. The manuscript states that the base and the Lagrangian are connected and simply connected, and presents the theorem as resolving the unrestricted conjecture negatively.
- Method, as described: the Lagrangian is given by a generating family that agrees at infinity with degree-two homogeneous functions parametrized by S9. Their negative domains define a nonzero class in the ninth homotopy group of the stable smooth tube space whose associated stable spherical fibration is trivial; the manuscript obtains it from Waldhausen's tube fibration and Rognes's computation of low-degree smooth Whitehead groups. A family with exactly one nondegenerate critical point in each fibre cannot have such data at infinity, by Morse theory and an argument with families of h-cobordisms. The extra factor SN−1 and a separation of momenta make the generated Lagrangian embedded. A Hamiltonian isotopy to the zero section would produce a family with one critical point in each fibre, which gives the contradiction.
- Source (openai/math repository, commit
7ef5c0c): A counterexample to the nearby Lagrangian conjecture. - Formalization. The collection lists no Lean formalization for this result.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 27 October 2026; AI Proved What? row 27 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | A counterexample to the nearby Lagrangian conjecture, an article of the Latent Seminar blog at lowrank.github.io | Reading, exposition | Explains the manuscript's construction: generating families with homogeneous data at infinity, the coupling to an additional sphere, the momentum tests that exclude double points, the obstruction to a family with one critical point in each fibre, and the transport of the generating family along a hypothetical Hamiltonian isotopy. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-13 | F. Nie, H. Ye, S. Liu, Z. Wang, S. Ermon and J. Zou, OSVerify, Stanford University: referee report on A counterexample to the nearby Lagrangian conjecture | Review, referee report | Reviews the manuscript version of 23 September 2026 and reports no major issue and one minor issue with a proposed repair: a citation of Porcelli and Smith gives the wrong theorem number. | Web page at osverify.github.io, first posted 13 October 2026. The page states that its findings and severity labels come from OSVerify's internal review and do not necessarily establish that the main theorem is false; the site's home page states that every reported issue was verified by human experts. |
| 2026-10-15 | Wenyuan Li, Zhejiang University, Nearby Lagrangian conjecture, present and future, research talk at the ZUMA geometry and topology seminar, abstract on the seminar page | Reading, talk | The abstract announces an explanation of the relation between Lagrangian submanifolds, generating families and parametric h-cobordism spaces, and of how the proposed counterexample is constructed; assuming the counterexample is correct, the speaker would explain why many problems remain to be understood and tools to be developed, with a perspective from C0-limits. A pretalk the same day surveyed earlier work. | Talk abstract on the seminar web page. This page lists only the abstract. |
| 2026-10-19 | AI Proved What?, episode 340 Arnold’s guess fails?, remade version of 23 October 2026 | Reading (exposition) | Video explainer of the claim that, for some large even N, the cotangent bundle of the product of a 9-sphere and an N − 1 sphere contains a closed exact embedded Lagrangian, diffeomorphic to the base, that no compactly supported Hamiltonian motion carries the zero section onto: a counterexample to the nearby Lagrangian conjecture. The episode states that N is not computed and that the construction needs N at least 40, and that an exotic Lagrangian and counterexamples in low dimensions are not claimed. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 19 October 2026: posted as a preprint: OpenAI, 28 pages, dated September 23, 2026; reviewed by OSVerify: no major and one minor issue, a citation number, not a certification; under expert review: not yet; peer reviewed: not yet; formalized in Lean: partly, about 22,000 uncatalogued lines with two key topological steps assumed. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
