Plectis Atlas
Search

↑ and ↓ move, Enter opens, Esc closes.

Show on the map

Weak MTW curvature gives convexity and regular optimal transport

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

On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

The paper states

Theorem 1.1 (Uniform bi-Hölder transport)

In Uniform Bi-Holder Transport from Weak MTW, 25 September 2026

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.

The other possible matches are Theorem 1.2 (Global support and intermediate geometry) in Global Support and Convex Injectivity Domains under Weak MTW () and Theorem 1.1 (Convexity of injectivity domains) in Global Support and Convex Injectivity Domains under Weak MTW ().

Fix (M,g) as above and 0<\lambda\leq\Lambda<\infty for which the following density class is nonempty. There are \alpha\in(0,1] and C<\infty, depending only on (M,g),\lambda,\Lambda, such that the following holds for every pair of measurable densities satisfying

\int_M\rho_i\,d\vol=1,\qquad \lambda\leq\rho_i\leq\Lambda\quad\vol\text{-almost everywhere}, \qquad i=0,1.

The almost-everywhere unique optimal map from \rho_0\vol to \rho_1\vol for the cost c has a homeomorphic representative \widetilde T:M\to M, and

d(\widetilde T(x),\widetilde T(x'))\leq C d(x,x')^\alpha, \qquad d(\widetilde T^{-1}(y),\widetilde T^{-1}(y')) \leq C d(y,y')^\alpha

for all x,x',y,y'\in M.

Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
Two current papers, each 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.

Global Support and Convex Injectivity Domains under Weak MTW

25 September 2026, current edition.

Read it herePDF

We prove that weak Ma–Trudinger–Wang curvature on a smooth, connected, compact Riemannian manifold of dimension at least two without boundary implies convexity of every tangent injectivity domain, resolving Villani’s conjecture in this setting. Conjugate cut points are allowed. More generally, every ordinary subgradient of a squared-distance cost potential is a minimizing velocity with a global supporting mountain. No density hypothesis is used.

Uniform Bi-Holder Transport from Weak MTW

25 September 2026, current edition.

Read it herePDF

We prove that the weak Ma–Trudinger–Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities with fixed positive upper and lower bounds. The maps have homeomorphic representatives, conjugate cut points are allowed, and no density regularity is assumed.

Two Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

BiholderTransport

The paper states

Theorem 1.1 (Uniform bi-Hölder transport)

In Uniform Bi-Holder Transport from Weak MTW, 25 September 2026

Fix (M,g) as above and 0<\lambda\leq\Lambda<\infty for which the following density class is nonempty. There are \alpha\in(0,1] and C<\infty, depending only on (M,g),\lambda,\Lambda, such that the following holds for every pair of measurable densities satisfying

\int_M\rho_i\,d\vol=1,\qquad \lambda\leq\rho_i\leq\Lambda\quad\vol\text{-almost everywhere}, \qquad i=0,1.

The almost-everywhere unique optimal map from \rho_0\vol to \rho_1\vol for the cost c has a homeomorphic representative \widetilde T:M\to M, and

d(\widetilde T(x),\widetilde T(x'))\leq C d(x,x')^\alpha, \qquad d(\widetilde T^{-1}(y),\widetilde T^{-1}(y')) \leq C d(y,y')^\alpha

for all x,x',y,y'\in M.

The words they share: measurable, optimal, transport, cost, bi, almost, everywhere, uniform, density, holder, weak, mtw, map, unique, pair.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Uniform bi-Hölder optimal transport

BiholderTransport.lean, the release lists it as a main result.

theorem uniform_biHolder_transport    {n : ℕ} (hn : 2 ≤ n) {M : Type*}    [MetricSpace M] [CompactSpace M] [ConnectedSpace M]    [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ, Model n) ∞ M]    [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ, Model n) x)]    [IsContMDiffRiemannianBundle 𝓘(ℝ, Model n) ∞ (Model n)      (fun x : M => TangentSpace 𝓘(ℝ, Model n) x)]    [IsRiemannianManifold 𝓘(ℝ, Model n) M]    [MeasurableSpace M] [BorelSpace M]    (hmtw : WeakMTW (n := n) (M := M))    (lam cap : ℝ) (hlam : 0 < lam) (hcap : lam ≤ cap)    (hclass : ∃ rho : M → ℝ, AdmissibleDensity (metricVolume n) lam cap rho) :    ∃ alpha C : ℝ, 0 < alpha ∧ alpha ≤ 1 ∧ 0 ≤ C ∧      ∀ rho0 rho1 : M → ℝ,        AdmissibleDensity (metricVolume n) lam cap rho0 →        AdmissibleDensity (metricVolume n) lam cap rho1 →

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

WeakMTWGlobalSupport

The paper states

The words they share: convexity, injectivity, domains, weak, mtw, convex.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Convex injectivity domains under weak MTW

WeakMTWGlobalSupport.lean, the release lists it as a main result.

theorem current_main_convexity    {n : ℕ} (hn : 2 ≤ n) {M : Type*}    [MetricSpace M] [ChartedSpace (WeakMTW.Model n) M]    [IsManifold (WeakMTW.model n) ∞ M]    [CompactSpace M] [ConnectedSpace M]    [RiemannianBundle (fun x : M => TangentSpace (WeakMTW.model n) x)]    [IsContMDiffRiemannianBundle (WeakMTW.model n) ∞ (WeakMTW.Model n)      (fun x : M => TangentSpace (WeakMTW.model n) x)]    [IsRiemannianManifold (WeakMTW.model n) M]    (hmtw : WeakMTW.HasWeakMTW (n := n) (M := M)) :    WeakMTW.MainConvexity (n := n) (M := M) :=

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

Their note: current_main_convexity: on a compact connected smooth Riemannian manifold of dimension n ≥ 2, weak MTW (-3/2·∂⁴c ≥ 0 for orthogonal ξ, η at interior velocities) implies every open tangent injectivity domain is convex. uniform_biHolder_transport: for squared-distance cost and densities between fixed bounds there is one (α, C) such that for all pairs of densities the optimal map exists, is a homeomorphism, is a.e. unique, and T and T⁻¹ are α-Hölder with constant C.

Definitions the group asks readers to compare with the paper: Riemannian exponential is defined by if h : ∃ γ, complete geodesic with initial data then (Classical.choose h) 1 else x (geodesics characterized metrically as locally constant-speed distance realizers), injectivityDomain x := {v | ∃ a > 1, dist x (exp x (a•v)) = a‖v‖}, and mtw is an iterated ordinary derivative of the cost along exp-curves; calibrated Hausdorff measure metricVolume as volume

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

On a compact connected smooth Riemannian manifold of dimension at least two, the formalized result proves that weak MTW curvature implies convexity of every open tangent injectivity domain. Prior convexity or nonfocality is not assumed. Additional formalized results give global support at ordinary subgradients, compact convex lifted gap sections with the stated diameter bound, and scale and local-injectivity estimates for finite target families. The latter finite-family result does not cover the paper’s arbitrary-potential local-injectivity assertion.

The formalized result gives uniform bi-Hölder optimal transport for squared-distance cost on each fixed compact connected smooth Riemannian manifold of dimension at least two satisfying weak MTW. For probability densities between fixed positive lower and finite upper bounds, one exponent and constant work for every pair of densities. The optimal map is a homeomorphism, is unique almost everywhere, and both it and its inverse satisfy the global Hölder bound. The constants are uniform over the density class on the fixed manifold, not over varying metrics.

AI-drafted comparisons, with no recorded human review

  • Uniform bi-Hölder optimal transport BiholderTransport

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the transport paper: uniform bi-Hölder optimal maps for squared-distance cost between densities with fixed bounds under weak MTW.

  • Convex injectivity domains under weak MTW WeakMTWGlobalSupport

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: weak MTW curvature implies convexity of every open tangent injectivity domain on a compact manifold of dimension at least two. The paper’s arbitrary-potential local-injectivity assertion is not covered.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Uniform bi-Hölder optimal transport BiholderTransport

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Convex injectivity domains under weak MTW WeakMTWGlobalSupport

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Missing or incomplete source records

The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.

  • WeakMTWGlobalSupport: Tracked source lean/OAI/Geometry/WeakMTW/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb

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.

Within this family

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

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 explanationCompare one Lean statement with its paper: WeakMTWGlobalSupport, BiholderTransportSuggest a source attributionSuggest 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 “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

  • Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source

    The doc lists further results as formalised that no Comparator theorem name covers.

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.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form
  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 360 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

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 360 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form