Arnold's nearby Lagrangian conjecture
Let be a closed connected smooth manifold and its cotangent bundle with the canonical symplectic form. A closed embedded Lagrangian is exact if the canonical one-form restricts to an exact form on . Known constraints are strong: Abouzaid (2012) and Kragh (2013) proved the projection is a homotopy equivalence, Abouzaid and Kragh (2018) a simple homotopy equivalence, and the conjecture holds for (Hind) and (Dimitroglou Rizell, Goodman, Ivrii). Is every closed exact embedded Lagrangian in Hamiltonian isotopic to the zero section?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Symplectic topology; exact Lagrangians in cotangent bundles
- Posed by
- V. I. Arnold (historical setting in Arnold, First steps in symplectic topology, 1986); modern formulation as in Abouzaid (2012) and Kragh (2013)
- Year posed
- 1986
- Years open
- 40y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 62 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for some sufficiently large even , admits a closed exact smoothly embedded Lagrangian , diffeomorphic to and with and simply connected, that no compactly supported Hamiltonian isotopy carries to the zero section. is cut out by a generating family whose data at infinity form a nontrivial class of the stable smooth tube space with trivial associated spherical fibration; a Morse and h-cobordism argument shows such data cannot come from a family with one nondegenerate critical point per fibre, as an isotopy to the zero section would force. Not shown: an explicit , counterexamples in low dimension, or any failure of the homotopy-theoretic consequences already proved.
What the AI did
The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, using on average about three hours of ChatGPT Pro thinking compute per result, with outputs grouped into families and manuscripts. The README's two exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region write-up, which was human-edited for readability) do not concern this family. The manuscript is credited to OpenAI alone, names no human author and has no acknowledgements. This family has no Lean formalization (there is no lean/docs/340.md at the pinned commit). The release does not say how much human review happened before publication.
Verification
No independent mathematician has checked this yet. Checked here: the abstract and Theorem 1.1 of the TeX source against the conjecture as the manuscript states it (every closed exact embedded Lagrangian in T*Q is Hamiltonian isotopic to the zero section). The theorem claims the negation for one base, so it addresses the conjecture as posed for all closed Q. No formalization exists for this family and the argument was not checked. Scope the paper itself states: N is some sufficiently large even integer, not given explicitly, so the counterexample lives in very high dimension; it says nothing about low-dimensional bases or specific bases such as spheres or tori. The proof leans on Waldhausen's tube fibration and Rognes's computation of low-degree smooth Whitehead groups to produce a nonzero class in the ninth homotopy group of the stable tube space with trivial image in BG. Given that the field broadly expected the conjecture to hold, an expert reading is especially wanted.