Weak MTW curvature and the regularity of optimal transport on manifolds
From OpenAI Aftermath
Weak MTW curvature and the regularity of optimal transport on manifolds concerns the weak Ma–Trudinger–Wang (MTW) curvature condition for the squared-distance cost on a compact Riemannian manifold: whether it forces every tangent injectivity domain to be convex, as Villani conjectured, and whether it gives continuous optimal transport maps between densities bounded above and below. Two manuscripts in OpenAI's public mathematics collection (github.com/openai/math), listed together as its result 360 and dated 25 September 2026, claim both: convexity of every tangent injectivity domain under weak MTW alone, and Hölder estimates for the optimal map and its inverse that are uniform over a density class. The collection includes Lean formalizations of both main statements. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introductions of the manuscripts:
- Ma, Trudinger and Wang introduced a strictly positive curvature condition in the regularity theory of optimal transport, and Trudinger and Wang treated its degenerate form, the weak MTW condition. Loeper showed that the condition is necessary for continuity and developed the supporting-mountain principle; Kim and McCann gave it a geometric formulation.
- Villani proposed that weak MTW should force convexity of every tangent injectivity domain. Figalli, Gallouët and Rifford proved this for nonfocal manifolds, in which minimizing geodesics reach the cut locus before their first conjugate point, and Loeper and Villani obtained convexity under strong MTW and nonfocality.
- Figalli, Rifford and Villani proved the global supporting principle from weak MTW together with convex injectivity domains, and showed that on compact surfaces transport continuity is equivalent to weak MTW together with convex injectivity domains; they asked whether weak MTW alone suffices. Uniform Hölder estimates over density classes were known under strict MTW and nonfocality, by Loeper and Villani, and on products of round spheres, by Figalli, Kim and McCann.
The original results
- Claim, convexity. Theorem 1.1 of Global Support and Convex Injectivity Domains under Weak MTW states that on every smooth, connected, compact Riemannian manifold without boundary of dimension at least two that satisfies weak MTW, every tangent injectivity domain is convex; conjugate cut points are allowed, and no nonfocality, strict MTW or prior convexity is assumed. Theorem 1.2 states that every ordinary subgradient of a squared-distance cost potential is a minimizing velocity with a global supporting function, and gives compact convex sections and intermediate-time regularity. No density hypothesis is used.
- Claim, transport. Theorem 1.1 of Uniform Bi-Hölder Transport from Weak MTW states that on such a manifold, for fixed density bounds 0 < λ ≤ Λ, there are an exponent α and a constant C, depending only on the manifold and the bounds, such that for every pair of measurable probability densities between λ and Λ the optimal map has a homeomorphic representative, and both it and its inverse are α-Hölder with constant C. The manuscript states that the exponent comes from a contradiction argument and is not explicit, and that no uniformity over varying metrics is asserted. Combined with the necessity result of Figalli, Rifford and Villani, it states the equivalence of weak MTW and transport continuity.
- Method, as described: the convexity manuscript continues minimizing velocities through the cut locus without nonfocality, using a minimizing-branch Hessian, radial shortening of distance supports and an endpoint-moving Jacobi trial at possible conjugate cut points, with weak MTW applied only to directions perpendicular to the velocity. The transport manuscript bounds the oscillation of the dual potential on convex lifted sections, takes a supremum over the whole density class, and uses spectral estimates near conjugacy to reach a contradiction from two separated scales.
- Formalization, as described by the repository. The scope note states that the formalized results prove convexity of every open tangent injectivity domain under weak MTW without prior convexity or nonfocality, with global support at ordinary subgradients and the section estimates, and uniform bi-Hölder optimal transport for densities between fixed bounds, with constants uniform over the density class on a fixed manifold but not over varying metrics. It notes that a finite-family local-injectivity result does not cover the paper's assertion for arbitrary potentials. The Comparator statements are
WeakMTWGlobalSupport.leanandBiholderTransport.lean. - Sources (openai/math repository, commit
7ef5c0c): Global Support and Convex Injectivity Domains under Weak MTW and Uniform Bi-Hölder Transport from Weak MTW, both dated 25 September 2026.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 30 October 2026) |
|---|---|---|---|---|
| 2026-10-07 | Weak MTW curvature: convex injectivity domains and regular optimal transport, an article of the Latent Seminar blog at lowrank.github.io | Reading (exposition) | Explains the two dependent manuscripts: supporting geometry from weak Ma–Trudinger–Wang curvature, including convexity of tangent injectivity domains, and the companion's uniform Hölder estimates for optimal maps and their inverses, covering the geometric continuation argument, the section-to-Hölder reduction, the spectral estimates and the determinant argument. | 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 | MathVet (GitHub; review of openai/math, tables at repository commit c76f1e4, 14 October 2026) |
Review (fidelity of the formal statements) | Rates both challenges of this family "full" and records a sandboxed Comparator pass for each. It lists the formal definitions of the exponential map, the injectivity domain, the MTW quantity and the volume measure among the definitions to check. | MathVet states that it does not claim any theorem false and reports what was machine-checked as stated. |
| 2026-10-10 | cert-machine, openai/math lane K: WeakMTWGlobalSupport, BiholderTransport | Verification (two kernels) | Records that Lean's default kernel and the nanoda kernel accept both solutions, in 885 s and 2,706 s. | Repository. |
| 2026-10-13 | M. Colbrook and contributors, AIM — Open Applied Problems (GitHub), problems 394 and 401 | Catalogue | The collection records its problem 401, continuity of manifold transport under MTW and convex injectivity domains, as solved by the transport manuscript, which does not need convex injectivity domains in advance. Its problem 394, Villani's convex-injectivity-domain conjecture, is recorded as a claimed solution by the convexity manuscript, in a status review of 15 October 2026. | The status reviews are described as AI statement-level comparisons with the published manuscripts, not independent audits of the complete proofs or Lean kernel replays; for problem 394 the review states that the complete argument has not been independently audited. The collection is not affiliated with the American Institute of Mathematics. |
| 2026-10-19 | AI Proved What?, episode 360 Curvature keeps transport tame?, remade version of 23 October 2026 | Reading (exposition) | Video explainer of two claims: on every closed connected manifold of dimension at least two, weak MTW curvature makes every tangent injectivity domain convex, with no assumption about conjugate points, answering a conjecture of Villani from a 2011 survey; and, building on this, optimal transport maps between densities bounded above and below, and their inverses, are Hölder continuous with constants uniform over the density class. The episode recalls that Figalli, Gallouët and Rifford proved the convexity under nonfocality and that Figalli, Rifford and Villani showed the equivalence on surfaces. It states that smoothness and an explicit exponent are not claimed, that weak MTW is itself restrictive, forcing nonnegative sectional curvature, and that the equivalence of transport continuity and weak MTW is a one-line remark that is not formalized; it found no competing claim. It also reports its own checks of matchings of seven points on a line and of the MTW quantity on flat space and the round sphere. | 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 preprints: OpenAI, 24 + 41 pages, dated September 25, 2026; under expert review: not yet; peer reviewed: not yet; checked in Lean: both main theorems. The page states that both papers are unrefereed and unchanged since release, that the site did not run the Lean formalizations of about 19,000 and 74,000 lines, that the transport formalization re-proves what it needs rather than importing the convexity formalization, and that it found no specialist comment and no reported error. |
Notes
- The background summarizes the manuscripts' 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.
