The degenerate Arnold conjecture: critical-number and rational cup-length lower bounds for Hamiltonian fixed points
Arnold asked whether a Hamiltonian diffeomorphism of a closed symplectic manifold must have at least as many geometrically distinct fixed points as a smooth function on has critical points. Without nondegeneracy assumptions this is the critical-number bound , with the weaker rational cup-length form (unit included). These are known on tori and , and when and vanish on (Rudyak-Oprea). Do the critical-number and cup-length bounds hold for every 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, Arnold conjecture
- Posed by
- V. I. Arnold, First steps in symplectic topology, Russian Math. Surveys 41 (1986); recorded as Conjectures 1.4 and 1.6 in R. Golovko, On variants of Arnold conjecture, Arch. Math. (Brno) 56 (2020)
- Year posed
- 1986
- Years open
- 40y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Contested
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 45 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: on the complex quadric threefold there is a smooth Hamiltonian diffeomorphism with exactly three fixed points, at least one degenerate, while every smooth function has at least four critical points and the rational cup length is four; this disproves both unrestricted degenerate bounds, and contractibility conditions cannot rescue them. The construction composes a Hamiltonian involution fixing with a short invariant flow, using a three-critical-point function on . The count three is sharp by Gong (2025). It says nothing against the nondegenerate homological (Floer) bounds. Jiao (arXiv, 27 Sep 2026, after this paper) independently reports a two-fixed-point example on .
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 was read against the conjectures as the paper states them (Arnold 1986, Golovko 2020). It claims a smooth Hamiltonian diffeomorphism of the quadric with exactly 3 fixed points while . The proof was not refereed. Lean: ComparatorChallenges/ArnoldCounterexample.lean (OAI.ArnoldCounterexample.main, in lean/formalization.yaml) was read here; it states a Hamiltonian of the literal quadric with fixed set of size 3, the minimum critical number over all smooth functions, and a degenerate fixed point. It covers the critical-number form; the cup-length value is not formalized. Not rebuilt here. Graded contested because an earlier preprint (Ma) claims the opposite; see the claim note.
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.