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Math-274-A2 · revision r1 · on Math-274

OpenAI claim summary for Math-274

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 at unsolvedmath.com: partially solved. 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.

Nearby Lagrangian counterexample: The constructed Lagrangian is closed, exact and smoothly embedded in a cotangent bundle, but is not Hamiltonian isotopic to its zero section; one permitted base disproves this universal assertion. Compact-support failure suffices because a Hamiltonian isotopy of compact Lagrangians has compact image and can be cut off without altering that image.

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. Nearby Lagrangian conjecture
  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

Nearby Lagrangian conjecture

OpenAI's claim

Nearby Lagrangian counterexample: For some sufficiently large even $N$, the paper claims a closed embedded exact Lagrangian in $T^*(S^9\times S^{N-1})$ that cannot be taken to the zero section by a Hamiltonian isotopy, despite sharing the base’s diffeomorphism type. Thus the unrestricted nearby-Lagrangian assertion is contradicted.

Mathematical statement of the claimed result

Mathematical claim 1.
For some sufficiently large even $N$, let $Q=S^9\times S^{N-1}$. The manuscript claims a closed exact smoothly embedded Lagrangian $L\subset T^*Q$ with $L\cong Q$ as smooth manifolds, but with no compactly supported Hamiltonian isotopy sending the zero section onto $L$. Exactness means that the canonical cotangent one-form restricts to $dF$ on $L$ for some function $F$.

Outline of the argument

Nearby Lagrangian counterexample: Generating families are organized by their homogeneous quadratic behavior at large vertical radius. Stable tube data determine spherical fibrations, and a nontrivial obstruction is selected from the tube-to-fibration comparison. A regular vertical-gradient construction produces an exact immersion, followed by a separate injectivity argument; the retained stable obstruction prevents a Hamiltonian isotopy to the zero section.

Formal verification

Nearby Lagrangian counterexample: The pinned catalogue lists no Lean formalization for this result.

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