The Morse-number Arnold conjecture for nondegenerate Hamiltonian diffeomorphisms

For a closed connected symplectic manifold $(M,\omega)$ let $\mathrm{Morse}(M)$ be the least number of critical points of a Morse function on $M$ . Floer theory and its virtual extensions show that a nondegenerate Hamiltonian diffeomorphism has at least the sum of the Betti numbers many fixed points, but the Morse number can be larger. The Morse-number form of the Arnold conjecture asks: if $H$ is a one-periodic Hamiltonian whose time-one map is nondegenerate, is the number of fixed points with contractible orbits at least $\mathrm{Morse}(M)$ ?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Symplectic topology; Hamiltonian fixed points, Morse theory
Posed by: V. I. Arnold, First steps in symplectic topology (1986); recorded as Conjecture 1.1 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: 35 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: there are a closed symplectic 12-manifold and a one-periodic Hamiltonian with all fixed points nondegenerate and contractible and $\#\mathrm{Fix}=\chi(M)+128<\mathrm{Morse}(M)$ ; Corollary 1.2 makes $\mathrm{Morse}(M)-\#\mathrm{Fix}$ arbitrarily large (the manifold depends on the deficit). The mechanism: the augmentation ideal of $A_5^k$ needs two generators while $d(A_5^k)$ is unbounded, realized by relative handle cancellation on $X\times\mathbb R^8$ . The fixed-point count still exceeds the homological (Floer) bounds. The examples are not simply connected; the October 5 companions give simply connected Kahler examples below the stable Morse number.

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 and Corollary 1.2 were read against Golovko's Conjecture 1.1 as the paper states it. The paper claims twelve-dimensional examples $M=X\times(S^2)^4$ , $\pi_1(X)=A_5^k$ , with all fixed points nondegenerate and contractible and $\#\mathrm{Fix}=\chi(M)+128<\mathrm{Morse}(M)$ , with arbitrarily large deficit. The proof was not refereed and is not formalized. Graded contested because Ma's 2013 preprint 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.

Source

CodeOpenAI math release: A nondegenerate counterexample to the Morse-number Arnold bound