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A nondegenerate counterexample to the Morse-number Arnold bound

OpenAI · September 23, 2026

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Abstract

We construct a smooth one-periodic Hamiltonian on a closed symplectic twelve-manifold whose time-one map has fewer fixed points than the manifold's ordinary Morse number. Every fixed point is nondegenerate and has a contractible Hamiltonian trajectory. This disproves the Morse-number form of the Arnold conjecture. The deficit can be arbitrarily large among twelve-dimensional examples.

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Cite

@misc{OAI:A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026,
  author = {{OpenAI}},
  title = {{A nondegenerate counterexample to the Morse-number Arnold bound}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026/paper.pdf}{OAI:A-nondegenerate-counterexample-to-the-Morse-number-Arnold-bound-September-23-2026}},
  year = {2026}
}

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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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