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Math-2300-A1 · revision r1 · on Math-2300

OpenAI claim summary for Math-2300

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: 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.

Infinitely many prime closed geodesics: The theorem provides infinitely many geodesic images for every Riemannian sphere, so covers the Riemannian S2 subcase. Long source question permits general Finsler metrics, including irreversible ones; that extra class is not treated.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

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

Manuscript

Contents
  1. A hyperbolic geodesic and the robust infinite branch
  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

A hyperbolic geodesic and the robust infinite branch

OpenAI's claim

Infinitely many prime closed geodesics: Every smooth Riemannian sphere of dimension at least two is claimed to have infinitely many prime closed geodesics with distinct images. The paper also obtains infinitude on every nonempty closed three-manifold.

Mathematical statement of the claimed result

Mathematical claim 1.
Whatever smooth metric is chosen on the standard sphere $S^n$, at each integer dimension $n\ge2$ the manuscript claims infinitely many prime closed geodesics having pairwise distinct images. Prime means not an iterate of a shorter closed geodesic. Its three-manifold corollary asserts the same conclusion for every nonempty closed smooth three-manifold and every smooth metric, allowing nonorientability.

Outline of the argument

Infinitely many prime closed geodesics: Assuming finitely many prime images gives a uniform budget for local loop-space crossings, including degenerate root types. Reflection-equivariant cohomology and half-period translation compare ordinary and root-degree filtrations. Extra classes would force a quadratic discrepancy, but the exact-sequence and filling estimates allow only a linear budget; covering and three-manifold results provide the final transfer.

Formal verification

Infinitely many prime closed geodesics: 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