The stable Morse number as a lower bound for nondegenerate Hamiltonian fixed points
The stable Morse number of a closed manifold is the least number of critical points of a Morse function equal to a nondegenerate quadratic form outside a compact set ( varying); . Dimitroglou Rizell and Golovko proved for nondegenerate Hamiltonians when and vanish on . Does the stable Morse bound hold for every nondegenerate Hamiltonian diffeomorphism of every closed symplectic manifold?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Symplectic topology; Hamiltonian fixed points, stable Morse theory
- Posed by
- Recorded as Conjecture 1.2 in R. Golovko, On variants of Arnold conjecture, Arch. Math. (Brno) 56 (2020); background in Arnold (1986)
- Year posed
- 2020
- Years open
- 6y
- Solved
- 2026-10-05
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Contested
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Principal Theorem 1.1: a simply connected closed Kahler (, ) and a Hamiltonian with all 318952 fixed points nondegenerate and contractible. Mechanism: a Hamiltonian involution on whose fixed components carry Smale-minimal Morse functions, with mixed 2- and 3-primary torsion making ; no Floer theory. Companions: in real dimension 22 the deficit is unbounded with ratio ; in dimension 3332 the count attains the cyclic integral Floer bound of Bai-Xu yet is 32 below . Since these also refute the Morse-number form in the simply connected setting. The homological bounds remain intact.
What the AI did
The release README says all results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal paper was read against Golovko's Conjecture 1.2; it claims a simply connected closed Kahler manifold of real dimension 1412 with a nondegenerate Hamiltonian having fixed points. The companions claim a dimension-22 family with and a dimension-3332 example attaining the cyclic integral Floer bound. Proofs not refereed; not formalized. Graded contested because Ma's 2013 preprint claims nondegenerate fixed points are always at least the ordinary Morse number, which is at least the stable Morse number; the stable-Morse papers themselves do not cite Ma.
Claim issue
The release says: 'Ma [Theorem 1.5] asserted that the minimum number of fixed points over all Hamiltonian diffeomorphisms always equals Crit(M). Theorem 1.1 contradicts that assertion' (degenerate paper) and 'Ma's preprint asserts that the minimum number of fixed points of a nondegenerate Hamiltonian diffeomorphism always equals the ordinary Morse number ... Theorem 1.1 contradicts that unrestricted assertion' (Morse-number paper). The other side: R. Ma, Proofs on Arnold conjectures, arXiv 0808.0613v7 (2013), Theorem 1.5, claims these lower bounds hold in general. Ma's preprint is unpublished; no referee has ruled between them. The stable-Morse papers do not cite Ma; the conflict is by implication, since Morse(M) >= SM(M).