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Hamiltonian Fixed Points Below the Stable Morse Number

OpenAI · October 5, 2026

Lean

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Abstract

We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points by constructing a simply connected closed Kähler manifold with sixteen fewer fixed points than its stable Morse number. All the corresponding periodic orbits are contractible. The construction combines a Hamiltonian involution with integral homology torsion at two different primes; its proof uses finite-dimensional Morse theory and complex blowups.

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Possibly relevant literature (found by the agent, not checked)

  • S. Rezchikov, Integral Arnol'd Conjecture (arXiv:2209.11165, 2022). Proves an integral version of the Arnold conjecture using global Kuranishi charts; relevant to the paper's discussion of integral refinements of the Floer bound, which it cites via Abouzaid–Blumberg and Bai–Xu.

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Cite

@misc{OAI:Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026,
  author = {{OpenAI}},
  title = {{Hamiltonian Fixed Points Below the Stable Morse Number}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026/paper.pdf}{OAI:Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-October-5-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/347.md. The Lean statements have not been checked by people against the paper's claims.

Family 347: Counterexamples to stable-Morse and strong Arnold fixed-point bounds

Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function.

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