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The degenerate Arnold conjecture: critical-number and rational cup-length lower bounds for Hamiltonian fixed points

Theorem 1.1: on the complex quadric threefold there is a smooth Hamiltonian diffeomorphism with exactly three fixed points, at least one degenerate, while every smooth function has at least four critical points and the rational cup length is four; this disproves both unrestricted degenerate bounds, and contractibility conditions cannot rescue them. The construction composes a Hamiltonian involution fixing Q2≅S2×S2Q^2\cong S^2\times S^2 with a short invariant flow, using a three-critical-point function on S2×S2S^2\times S^2. The count three is sharp by Gong (2025). It says nothing against the nondegenerate homological (Floer) bounds. Jiao (arXiv, 27 Sep 2026, after this paper) independently reports a two-fixed-point example on S2×S2S^2\times S^2.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: contested. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The degenerate Arnold conjecture: critical-number and rational cup-length lower bounds for Hamiltonian fixed points · Degenerate Arnold conjecture

Confidence: Not scored

Registry verification: contested · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Artifacts and verifiers

Lean proof (OAI.ArnoldCounterexample.main)

lean artifact · pending

Artifact ↗
Comparator statement: ArnoldCounterexample.lean

lean artifact · pending

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Compute record

No linked compute attempts recorded.

Lineage and corrections

The degenerate Arnold conjecture: critical-number and rational cup-length lower bounds for Hamiltonian fixed points parent of this event

This event attributed to Unreleased internal OpenAI model

VibeMathed record: The degenerate Arnold conjecture: critical-number and rational cup-length lower bounds for Hamiltonian fixed points evidence for this event

Comparator statement: ArnoldCounterexample.lean evidence for this event

Theorem 1.1: on the complex quadric threefold there is a smooth Hamiltonian diffeomorphism with exactly three fixed points, at least one degenerate, while every smooth function has at least four critical points and the rational cup length is four; this disproves both unrestricted degenerate bounds, and contractibility conditions cannot rescue them. The construction composes a Hamiltonian involution fixing Q2≅S2×S2Q^2\cong S^2\times S^2 with a short invariant flow, using a three-critical-point function on S2×S2S^2\times S^2. The count three is sharp by Gong (2025). It says nothing against the nondegenerate homological (Floer) bounds. Jiao (arXiv, 27 Sep 2026, after this paper) independently reports a two-fixed-point example on S2×S2S^2\times S^2. parent of this event

Lean proof (OAI.ArnoldCounterexample.main) evidence for this event

The degenerate Arnold conjecture: critical-number and rational cup-length lower bounds for Hamiltonian fixed points evidence for this event

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The degenerate Arnold conjecture: critical-number and rational cup-length lower bounds for Hamiltonian fixed points — Mathematical Frontier Network