we-solve (beta)

Math-183-A2 · revision r1 · on Math-183

OpenAI claim summary for Math-183

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.

Degenerate Arnold fixed-point obstruction: The quadric is a closed connected symplectic six-manifold and has exactly three fixed points while Crit equals four for arbitrary smooth functions. Degenerate fixed points are permitted by the precise site definition, so this is a direct counterexample.

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. Arnold Conjecture on 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

Arnold Conjecture on Hamiltonian Fixed Points

OpenAI's claim

Degenerate Arnold fixed-point obstruction: A Hamiltonian map of the complex quadric threefold is claimed to have exactly three fixed points although its critical number and rational cup-length lower bound are four. At least one fixed point is degenerate.

Mathematical statement of the claimed result

Mathematical claim 1.
For the complex quadric $Q^3=\{[z]\in\mathbb{CP}^4:\sum_{i=0}^4z_i^2=0\}$ with restricted Fubini--Study symplectic form, the claim constructs a smooth Hamiltonian diffeomorphism $\phi$ with \[\#\operatorname{Fix}(\phi)=3<4=\operatorname{Crit}(Q^3)=\operatorname{cuplength}(Q^3;\mathbb Q).\] At least one fixed point is degenerate. Here $\operatorname{Crit}$ minimizes over all smooth functions, including degenerate ones; the cup-length convention is the longest nonvanishing product of positive-degree classes, with one added to that count.

Outline of the argument

Degenerate Arnold fixed-point obstruction: A finite-order Hamiltonian map is perturbed by the flow of an invariant extension of a function on its fixed locus. A power identity excludes new fixed points elsewhere, and symmetric criticality identifies the remaining fixed points with critical points on that locus. A suitably chosen function on the quadric surface then supplies the three-point count.

Formal verification

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

Degenerate Arnold fixed-point obstruction: The manuscript links ArnoldCounterexample.lean (declaration OAI.ArnoldCounterexample.main).

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

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