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Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two

OpenAI · October 5, 2026

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Abstract

We disprove the stable Morse lower bound for nondegenerate Hamiltonian fixed points with examples in fixed real dimension twenty-two. For every integer m ≥ 1, we construct a simply connected closed Kähler manifold with stable Morse number $80+1968m$ and a Hamiltonian diffeomorphism with exactly $80+1952m$ fixed points, all nondegenerate and with contractible orbit loops. The deficit is therefore unbounded, and the fixed-point count is at most 127/128 of the stable Morse number.

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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 comparison with the integral Floer bound, which the paper cites via Bai–Xu.

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Cite

@misc{OAI:Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026,
  author = {{OpenAI}},
  title = {{Hamiltonian Fixed Points Below the Stable Morse Number in Dimension Twenty-Two}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-October-5-2026/hamiltonian-fixed-points-below-stable-morse-number.pdf}{OAI:Hamiltonian-Fixed-Points-Below-the-Stable-Morse-Number-in-Dimension-Twenty-Two-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.

Other papers in this family: