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Math-2653-A1 · revision r1 · on Math-2653

OpenAI claim summary for Math-2653

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Morse lower bound for nondegenerate Hamiltonian fixed points
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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