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Three fixed points on the symplectic quadric threefold

OpenAI · September 23, 2026

Lean

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Abstract

We construct a smooth Hamiltonian diffeomorphism of the complex quadric threefold with exactly three fixed points, at least one of which is degenerate. The critical number and the unit-inclusive rational cup length of this manifold are both four. Thus the example disproves the unrestricted critical-number and rational cup-length forms of the Arnold conjecture.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Formalization

  • OpenAI math release, Lean scope note for family 347 (lean/docs/347.md, 2026). The release's own Lean formalization covers this paper's three-fixed-point counterexample on $Q^3$ (critical-number form only); the nondegenerate Morse-number construction is excluded.
  • OpenAI math release, OAI/Geometry/Arnold/Main.lean (Lean 4 / Mathlib, 2026). Contains the theorem OAI.ArnoldCounterexample.main (a Hamiltonian $\phi$ of $Q^3$ with exactly 3 fixed points, critical number at least 4, and a degenerate fixed point), proved from lemmas in other files; the comparator statement is lean/ComparatorChallenges/ArnoldCounterexample.lean, and the rational cup-length value is not formalized.

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

  • W. Gong, Two degenerate intersections of Clifford tori in S²×S² and CP² (arXiv:2610.02965, 2 Oct 2026). Constructs smooth Hamiltonians for which the Clifford torus meets its image in exactly two degenerate points, a Lagrangian-intersection analogue of the degenerate smooth counterexamples discussed in this paper; its acknowledgments cite Jiao's $S^2\times S^2$ example, not this paper.

Digests and expositions

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026,
  author = {{OpenAI}},
  title = {{Three fixed points on the symplectic quadric threefold}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026/paper.pdf}{OAI:A-degenerate-counterexample-to-the-critical-number-Arnold-bound-September-23-2026}},
  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: