Katok's entropy rigidity conjecture
From OpenAI Aftermath
Katok's entropy rigidity conjecture asks whether, for the geodesic flow of a closed Riemannian manifold of negative sectional curvature, the entropy of the normalized Liouville measure equals the topological entropy only when the metric is locally symmetric. The topological entropy measures the exponential growth of orbit complexity, and the variational principle places the Liouville entropy below it. Katok proved the conjecture for surfaces in 1982. A manuscript dated 23 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims it in every dimension at least three. The collection has no Lean formalization of this result. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- Katok formulated the conjecture in 1982 and proved the surface case, and he obtained the higher-dimensional statement within the conformal class of a locally symmetric metric.
- Flaminio proved that the entropy gap has positive second variation at real hyperbolic metrics, along fixed-volume paths transverse to the diffeomorphism orbit, and gave a real hyperbolic three-manifold at which the fixed-volume Liouville-entropy Hessian is indefinite. Humbert proved rigidity in a small neighbourhood of any real or complex hyperbolic metric.
- For flows on three-dimensional phase spaces, Foulon and De Simoi, Leguil, Vinhage and Yang characterized the flows whose measure of maximal entropy lies in the smooth volume class. The theorem of Besson, Courtois and Gallot on volume entropy concerns a different minimization problem.
The original result
- Claim. The main theorem of Entropy equality and local symmetry in negative curvature states that for a closed connected smooth Riemannian manifold of dimension n ≥ 3 with strictly negative sectional curvature, the normalized Liouville measure has maximal entropy for the geodesic flow if and only if the metric is locally symmetric, that is, if and only if its universal cover is a rank-one symmetric space of noncompact type at some scale. This includes real, complex, quaternionic and Cayley hyperbolic geometry. The manuscript notes that when the metric is not locally symmetric, the measure of maximal entropy and the Liouville measure are mutually singular.
- Method, as described: entropy equality, through thermodynamic formalism and Livšic cohomology, gives conormal volume forms that the flow transports by exponential factors. Jets of stable and unstable plaques in a polynomial normal form yield coherent formal affine fields, and an algebra of formal contact symmetries, analysed with Cartan–Guillemin structure theory, gives a dichotomy. The infinite-dimensional alternative is excluded with Busemann functions at infinity; in the finite-dimensional one the stable and unstable splitting is proved smooth, and theorems of Benoist–Foulon–Labourie and of Besson–Courtois–Gallot, with Ruan's correction for the Cayley case, then give local symmetry.
- Source (openai/math repository, commit
7ef5c0c): Entropy equality and local symmetry in negative curvature, dated 23 September 2026. - Formalization. The repository has no Lean scope note for this family.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | Entropy equality and local symmetry in negative curvature, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the two entropies, hyperbolicity and the missing transverse regularity, a periodic-orbit identity, the transported conormal volumes, finite jets and the formal alternative, the exclusion of the mixed-rectangle alternative, the recovery of smoothness and the final rigidity step, with proof outlines. It treats the algebraic classification as a stated input. | Blog article; it states that its review covers statements, introductions and selected proof mechanisms and is not an independent verification of the complete proofs. |
| 2026-10-08 | T. Humbert, reaction in 100+ reactions to 100+ solutions, a collected post on the Proofs and Prompts blog | Critique (comment) | Humbert, whose doctoral work was a local version of the conjecture near complex hyperbolic metrics, describes the paper on first reading as mostly unreadable. He criticizes OpenAI's standards for mathematical publication and the unpaid work of validation that this asks of mathematicians. He does not name a specific mathematical error. | Blog post dated 8 October 2026, with reactions added through 19 October 2026. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_107 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995) | Review (AI-assisted audit) | Reports no gap. The reviewer read every section and checked in detail the entropy forms and the Livšic step, the normal-form dynamics, the trace and dilation identities, the algebraic dichotomy, the Busemann contradiction, the smoothness argument and the final rigidity step with its scaling. It did not verify the measurable-selection details or every interpolation constant line by line, and it describes its verdict on these analytic parts as a structural check, not a full certification. | Part of an audit of the collection's manuscripts that the audit describes as complete. The reviews are published under CC BY 4.0. |
| 2026-10-19 | AI Proved What?, episode 339 Most chaotic only if symmetric?, remade version of 23 October 2026 | Reading (exposition) | Video explainer of the claim. The episode states that surfaces, which are Katok's own theorem, and manifolds of only nonpositive curvature are not claimed. It notes that the final step relies on classical rigidity theorems and, for the Cayley case, on a 2024 paper of Ruan, and that the paper does not cite the 1985 problem list of Burns and Katok or an unpublished 1990 preprint of Hamenstädt that claimed the three-dimensional case. | Web page and video; the site states that its scripts and animations are produced with AI assistance and that the narration is an AI-generated voice. Status record last checked 27 October 2026: posted as a preprint: OpenAI, 66 pages, dated September 23, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: not yet. The page states that it found no specialist assessment and no reported error. |
Notes
- The background summarizes the manuscript's own account of earlier work.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
