Katok’s entropy rigidity conjecture

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves Katok’s entropy rigidity conjecture for closed connected Riemannian manifolds of dimension at least three with strictly negative sectional curvature: normalized Liouville measure maximizes entropy for the geodesic flow if and only if the metric is locally symmetric.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

The paper states

Theorem 1.1

In Entropy equality and local symmetry in negative curvature

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Suppose n\ge3 and every sectional curvature of g is strictly negative. Then

h_{m_L}(\phi_1)=\htop(\phi_1) \quad\Longleftrightarrow\quad \nabla^gR_g=0.

Equivalently, equality holds precisely when the universal Riemannian cover is a rank-one symmetric space of noncompact type, at an arbitrary overall scale.

Reported proof verification: The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations: One current paper, readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Entropy equality and local symmetry in negative curvature

Undated, current edition.

Read it here PDF

For every closed connected smooth Riemannian manifold of dimension at least three with strictly negative sectional curvature, we prove that normalized Liouville measure has maximal entropy for the unit-speed geodesic flow if and only if the metric is locally symmetric. This resolves Katok’s entropy rigidity conjecture positively in these dimensions, including all rank-one symmetric types at arbitrary scale.

Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

34 references in Entropy equality and local symmetry in negative curvature

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanation Suggest a source attribution Suggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
Author not stated. Source

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Identify the theorem the catalogue describes

Family 339 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

Family 339 The release has no Lean statement for this family. Open the form