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Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number

OpenAI · October 5, 2026

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Abstract

We construct a simply connected closed Kähler manifold of real dimension 3332 and minimal Chern number one whose Hamiltonian fixed-point count attains the cyclic integral Floer bound while falling below the stable Morse number. The Hamiltonian diffeomorphism has exactly $1\,872\,232$ fixed points, all nondegenerate and with contractible orbit loops, whereas the stable Morse number is $1\,872\,264$. Thus the stronger stable Morse bound fails by exactly 32, even when the cyclic integral bound is sharp.

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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 independently of Bai–Xu; relevant to the paper's claim that its fixed-point count attains the cyclically graded integral Floer bound $\beta_{\mathbb Z}^{\mathrm{cyc}}$.

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Cite

@misc{OAI:Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026,
  author = {{OpenAI}},
  title = {{Sharpness of the Cyclic Integral Floer Bound Below the Stable Morse Number}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Sharpness-of-the-Cyclic-Integral-Floer-Bound-Below-the-Stable-Morse-Number-October-5-2026/sharp-cyclic-integral-floer-bound-below-stable-morse-number.pdf}{OAI:Sharpness-of-the-Cyclic-Integral-Floer-Bound-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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