Most chaotic only if symmetric?
An AI preprint claims Katok’s entropy rigidity conjecture in every dimension from three up
Covers result 339 in OpenAI’s catalog: its one paper. See its connections to other results and the work it cites.
3:55. English captions available. The video loads from YouTube when you press play.
Chapters
The claim, in plain words
On a negatively curved space, geodesics fan out exponentially, so the geodesic flow is chaotic. Two numbers measure how chaotic. Topological entropy counts the growth of distinct long orbits; Liouville entropy measures the chaos seen from a starting point and direction chosen uniformly at random. The second can never exceed the first. On a genus-2 surface of constant curvature minus one, both equal 1, and our computer counts orbit points growing at that rate. Bend the surface so its curvature varies, and our computer finds the Liouville entropy is about 0.96, below Katok’s bound of 0.975, which the topological entropy can never fall below.
In 1982 Katok proved that on surfaces the two entropies agree only in constant curvature, and wrote that it looks like a reasonable conjecture that locally symmetric spaces are the only negatively curved cases where the Liouville measure has maximal entropy. In higher dimensions these include complex, quaternionic and Cayley spaces, whose curvature is not constant; at a standard scale our computer reproduces Katok’s table of entropies, from 2 up to 22. Besson, Courtois and Gallot proved a related but different conjecture in 1995. Before this release, the main higher-dimensional result was Humbert’s: rigidity near real and complex hyperbolic metrics. A 1990 preprint by Hamenstädt claimed the three-dimensional case, but was never published.
Result 339 is one OpenAI paper, 66 pages, dated September 23 and made public October 6, 2026. It claims Katok’s conjecture in every dimension from 3 up: on a closed manifold with strictly negative curvature, the Liouville entropy equals the topological entropy exactly when the metric is locally symmetric, that is, when its universal cover is a rank-one symmetric space at some scale.
What it doesn’t claim
Not claimed: surfaces, which are Katok’s own theorem; manifolds whose curvature is only nonpositive; or a machine-checked proof, since there is no Lean.
Where it stands
- Posted as a preprint
- OpenAI, 66 pages, dated September 23, 2026
- Peer reviewed
- Not yet
- Formalized in Lean
- Not yet
Last checked October 19, 2026. The paper is unrefereed, unchanged since release, and has no Lean. Its final step relies on those classical rigidity theorems, and for the Cayley case on a 2024 paper of Ruan. We found no specialist assessment and no reported error. The paper does not cite the 1985 Burns and Katok problem list, or Hamenstädt’s unpublished preprint. OSVerify has not reviewed it.
How the proof goes
- From equal entropies to formal fields. Equal entropies mean that the expansion rate equals a constant up to a coboundary, by thermodynamic formalism and Livšic’s theorem; Katok suggested this route in 1985, and our symbolic toy shows the same equality case. The paper then records higher-order jets of the stable and unstable leaves, puts them in exact polynomial normal form, and builds two formal affine fields of covectors.
- A dichotomy, then classical rigidity. From these it builds a formal algebra of contact symmetries and proves a dichotomy: a finite-dimensional algebra yields formal stable and unstable fields, while an infinite-dimensional one yields special stable and unstable directions. The infinite case is ruled out with Busemann functions at infinity. In the finite case the stable and unstable splitting is smooth, and classical rigidity theorems of Benoist, Foulon and Labourie, and of Besson, Courtois and Gallot, then give local symmetry.
The numbers in the video
- Ours: on a genus-2 surface of constant curvature −1 both entropies are 1; bent to area 13.21, its Liouville entropy is about 0.961, below Katok’s bound 0.975.
- Ours: at a standard scale the rank-one symmetric spaces have entropies 2, 4, 10 and 22 (real 3-dim, complex 4-dim, quaternionic 8-dim, Cayley 16-dim), Katok’s table.
- Ours: orbit points of a genus-2 surface group within distance 9: 2,057, against 2,025 predicted.
- Ours: in a symbolic toy, Liouville entropy touches topological entropy (log 2) only when the two expansion rates are equal.
Read the source
- Entropy equality and local symmetry in negative curvature (PDF), 66 pages.
- The collection’s catalog, where this is result 339.
The paper
OpenAI groups related papers into one result; this one has a single paper.
Transcript
Read the full script
0:00The question
On a negatively curved space, geodesics fan out exponentially, so the geodesic flow is chaotic. Two numbers measure how chaotic. Topological entropy counts the growth of distinct long orbits; Liouville entropy measures the chaos seen from a starting point and direction chosen uniformly at random. The second can never exceed the first. On a genus-2 surface of constant curvature minus one, both equal 1, and our computer counts orbit points growing at that rate. Bend the surface so its curvature varies, and our computer finds the Liouville entropy is about 0.96, below Katok’s bound of 0.975, which the topological entropy can never fall below. In 1982 Katok proved that on surfaces the two entropies agree only in constant curvature, and wrote that it looks like a reasonable conjecture that locally symmetric spaces are the only negatively curved cases where the Liouville measure has maximal entropy. In higher dimensions these include complex, quaternionic and Cayley spaces, whose curvature is not constant; at a standard scale our computer reproduces Katok’s table of entropies, from 2 up to 22. Besson, Courtois and Gallot proved a related but different conjecture in 1995. Before this release, the main higher-dimensional result was Humbert’s: rigidity near real and complex hyperbolic metrics. A 1990 preprint by Hamenstädt claimed the three-dimensional case, but was never published.
1:38The claim
Result 339 is one OpenAI paper, 66 pages, dated September 23 and made public October 6, 2026. It claims Katok’s conjecture in every dimension from 3 up: on a closed manifold with strictly negative curvature, the Liouville entropy equals the topological entropy exactly when the metric is locally symmetric, that is, when its universal cover is a rank-one symmetric space at some scale. Not claimed: surfaces, which are Katok’s own theorem; manifolds whose curvature is only nonpositive; or a machine-checked proof, since there is no Lean.
2:17How the proof goes
Equal entropies mean that the expansion rate equals a constant up to a coboundary, by thermodynamic formalism and Livšic’s theorem; Katok suggested this route in 1985, and our symbolic toy shows the same equality case. The paper then records higher-order jets of the stable and unstable leaves, puts them in exact polynomial normal form, and builds two formal affine fields of covectors. From these it builds a formal algebra of contact symmetries and proves a dichotomy: a finite-dimensional algebra yields formal stable and unstable fields, while an infinite-dimensional one yields special stable and unstable directions. The infinite case is ruled out with Busemann functions at infinity. In the finite case the stable and unstable splitting is smooth, and classical rigidity theorems of Benoist, Foulon and Labourie, and of Besson, Courtois and Gallot, then give local symmetry.
3:16Where it stands
The paper is unrefereed, unchanged since release, and has no Lean. Its final step relies on those classical rigidity theorems, and for the Cayley case on a 2024 paper of Ruan. We found no specialist assessment and no reported error. The paper does not cite the 1985 Burns and Katok problem list, or Hamenstädt’s unpublished preprint. If it holds, among closed negatively curved manifolds of dimension 3 and up, the uniform measure is the most chaotic only for the locally symmetric ones, as Katok guessed.
Spot a mistake?
This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.