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Math-463-A2 · revision r1 · on Math-463

OpenAI claim summary for Math-463

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.

Geodesic entropy rigidity: The main theorem gives precisely the entropy equality/local-symmetry equivalence in all dimensions at least three, with every rank-one symmetric type and arbitrary scale. The introduction explicitly identifies Katok 1982 Theorem B as the already proved surface branch, completing the unrestricted closed negative-curvature target.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

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

Manuscript

Contents
  1. Katok entropy rigidity and an exact time-change obstruction
  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

Katok entropy rigidity and an exact time-change obstruction

OpenAI's claim

Geodesic entropy rigidity: In negative curvature and dimension at least three, the paper claims that Liouville measure maximizes geodesic-flow entropy exactly for locally symmetric metrics. All rank-one symmetric geometries and their rescalings fall within the conclusion.

Mathematical statement of the claimed result

Mathematical claim 1.
For a closed connected smooth Riemannian $(M^n,g)$, $n\ge3$, with all sectional curvatures strictly negative, let $\phi_t$ be unit-speed geodesic flow on $SM$ and $m_L$ its normalized Liouville probability measure. The entropy-rigidity claim is \[h_{m_L}(\phi_1)=h_{\mathrm{top}}(\phi_1)\quad\Longleftrightarrow\quad\nabla^gR_g=0.\] Equivalently after passage to its universal cover, the metric belongs to a rank-one noncompact symmetric model at arbitrary scale.

Outline of the argument

Geodesic entropy rigidity: Entropy equality and thermodynamic formalism turn unstable volume expansion into a flow coboundary. The transfer function supplies transported conormal volume data along stable and unstable leaves. Successive regularity arguments upgrade these transverse data to smooth distributions; the resulting rigidity structure forces the metric to be locally symmetric, with the converse verified on the symmetric models.

Formal verification

Geodesic entropy rigidity: 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