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Math-2653 · mathematics · Geometry

Morse lower bound for nondegenerate Hamiltonian fixed points

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every closed connected symplectic manifold M and every Hamiltonian diffeomorphism $\varphi $ with nondegenerate fixed points, must $\#Fix(\varphi )$ be at least the minimum number of critical points of a Morse function on M? The minimum is restricted to Morse functions, whereas the existing Crit record is not.

Short English statement

For every closed connected symplectic manifold M and every Hamiltonian diffeomorphism $\varphi $ with nondegenerate fixed points, must $\#Fix(\varphi )$ be at least the minimum number of critical points of a Morse function on M? The minimum is restricted to Morse functions, whereas the existing Crit record is not.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2653-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2653

Ordinary Morse-number obstruction: The paper claims nondegenerate Hamiltonian maps on closed twelve-dimensional symplectic manifolds with fewer fixed points than their ordinary Morse numbers. The shortfall can be made arbitrarily large and all associated orbit loops are contractible. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

Morse lower bound for nondegenerate Hamiltonian fixed points

OpenAI's claim

Ordinary Morse-number obstruction: The paper claims nondegenerate Hamiltonian maps on closed twelve-dimensional symplectic manifolds with fewer fixed points than their ordinary Morse numbers. The shortfall can be made arbitrarily large and all associated orbit loops are contractible.

Mathematical statement of the claimed result

Mathematical claim 1.
A claimed example uses a connected compact symplectic manifold $(M,\omega)$ without boundary, of real dimension twelve. A smooth Hamiltonian $H$ of time period one has only nondegenerate time-one fixed points, and \[\#\operatorname{Fix}_0(\phi_H^1;H)=\#\operatorname{Fix}(\phi_H^1)=\chi(M)+128<\operatorname{Morse}(M).\] The subscript $0$ counts contractible Hamiltonian orbits; $\operatorname{Morse}(M)$ is the minimum critical-point number among ordinary Morse functions. Nondegenerate means the derivative has no eigenvalue $1$.

Outline of the argument

Ordinary Morse-number obstruction: Finite groups with increasing group-generator number but bounded augmentation-ideal generation create a gap between ordinary and stabilized Morse counts. A relative twelve-dimensional handle construction realizes a small critical-point model while preserving quadratic collars. Local Hamiltonian realization on a closed product transfers that reduced count to nondegenerate fixed points.

Formal verification

Family 347 describes Lean coverage for the degenerate quadric critical-number obstruction; the nondegenerate ordinary and stable-Morse constructions are separate.

Ordinary Morse-number obstruction: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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