Lean statement

Uniform bi-Hölder optimal transport

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Theorem selected for verification: OAI.WeakMTWTransport.uniform_biHolder_transport

Lean statement file: BiholderTransport.leanShow on the mapFamily 360. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

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 →        ∃ T : M ≃ₜ M,          IsOptimalMap (metricVolume n) rho0 rho1 T ∧          (∀ S : M → M, IsOptimalMap (metricVolume n) rho0 rho1 S →            S =ᵐ[densityMeasure (metricVolume n) rho0] T) ∧          BiHolderEstimate alpha C T := by  sorry

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Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.WeakMTWTransport.Model line 12

abbrev Model (n : ℕ) := EuclideanSpace ℝ (Fin n)

definition OAI.WeakMTWTransport.cost line 19

def cost (x y : M) : ℝ := dist x y ^ 2 / 2

definition OAI.WeakMTWTransport.IsGeodesicWithInitialData line 65

def IsGeodesicWithInitialData (x : M) (v : TangentSpace 𝓘(ℝ, Model n) x)    (gamma : ℝ → M) : Prop :=  ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, Model n) ∞ gamma ∧  (⟨gamma 0, mfderiv 𝓘(ℝ, ℝ) 𝓘(ℝ, Model n) gamma 0 (1 : ℝ)⟩ :    TangentBundle 𝓘(ℝ, Model n) M) = ⟨x, v⟩ ∧  ∀ t : ℝ, ∃ eps : ℝ, 0 < eps ∧    ∀ s u : ℝ, |s - t| < eps → |u - t| < eps →      dist (gamma s) (gamma u) = |s - u| * ‖v‖

definition OAI.WeakMTWTransport.riemannianExp line 75

def riemannianExp (x : M) (v : TangentSpace 𝓘(ℝ, Model n) x) : M := by  classical  exact if h : ∃ gamma, IsGeodesicWithInitialData x v gamma then (Classical.choose h) 1 else x

definition OAI.WeakMTWTransport.injectivityDomain line 80

def injectivityDomain (x : M) : Set (TangentSpace 𝓘(ℝ, Model n) x) :=  {p | ∃ a : ℝ, 1 < a ∧ dist x (riemannianExp x (a • p)) = a * ‖p‖}

definition OAI.WeakMTWTransport.mtw line 84

def mtw (x : M) (p xi eta : TangentSpace 𝓘(ℝ, Model n) x) : ℝ :=  -(3 / 2 : ℝ) *    iteratedDeriv 2 (fun s : ℝ =>      iteratedDeriv 2 (fun t : ℝ =>        cost (riemannianExp x (t • xi)) (riemannianExp x (p + s • eta))) 0) 0

definition OAI.WeakMTWTransport.WeakMTW line 91

def WeakMTW : Prop :=  ∀ (x : M) (p : TangentSpace 𝓘(ℝ, Model n) x), p ∈ injectivityDomain x →    ∀ xi eta : TangentSpace 𝓘(ℝ, Model n) x,      inner ℝ xi eta = 0 → 0 ≤ mtw x p xi eta

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: measurable, optimal, transport, cost, bi, almost, everywhere, uniform and one more. Found in the paper’s title, which its main theorem states: holder, weak and mtw. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: map, unique and pair.

Statements in Uniform Bi-Holder Transport from Weak MTW come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Uniform Bi-Holder Transport from Weak MTW

Theorem (Uniform bi-Hölder transport)

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.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

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 $x,x',y,y'\in M$.

Other possible paper matches, closest wording first

Proposition 2.8 of Uniform Bi-Holder Transport from Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: almost, geodesic, everywhere, uniform, bounds, compact and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: maps and pair.
Corollary 3.6 of Uniform Bi-Holder Transport from Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: vector, weak, mtw, one and constants. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.
Theorem 1.1 of Global Support and Convex Injectivity Domains under Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: injectivity, domains, weak and mtw. Found in the paper’s title, which its main theorem states: global.
Theorem 1.2 of Global Support and Convex Injectivity Domains under Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: minimizing, 2/2, density, gamma, global, weak and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: map, homeomorphism, unique and pair.
Theorem 2.5 of Uniform Bi-Holder Transport from Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: gamma, global, one and exp. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: homeomorphism and unique.
Corollary 2.6 of Uniform Bi-Holder Transport from Weak MTW. Words of the Lean statement, its names split into words, that also appear in the theorem: injectivity and domains.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

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Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 360’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

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

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“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.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

127 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 50 to its own declarations, 61 to Mathlib’s documentation. 14 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves uniform_biHolder_transport exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 13 other declarations of the file; with the theorem itself they span 66 lines. 7 of them are named in the statement itself.

Names in the namespace OAI.WeakMTWTransport are listed without it.

abbrev Modelline 12in the statementlibrary
def costline 19library
def euclideanVolumeFactorline 22library
def metricVolumeline 30in the statementlibrary
def AdmissibleDensityline 34in the statementlibrary
def densityMeasureline 39in the statementlibrary
def IsOptimalMapline 43in the statementlibrary
def BiHolderEstimateline 51in the statementlibrary
def IsGeodesicWithInitialDataline 65library
def riemannianExpline 75library
def injectivityDomainline 80library
def mtwline 84library
def WeakMTWline 91in the statementlibrary
theorem uniform_biHolder_transportline 100selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen Set MeasureTheory Manifold Bundleopen scoped ContDiff Manifold ENNReal NNReal Topologynamespace WeakMTWTransportabbrev Model (n : ℕ) := EuclideanSpace ℝ (Fin n)section MetricDefinitionsvariable {M : Type*} [MetricSpace M]/-- Squared-distance cost with factor one half. -/def cost (x y : M) : ℝ := dist x y ^ 2 / 2/-- Calibration of Hausdorff measure to Euclidean volume. -/def euclideanVolumeFactor (n : ℕ) : ℝ≥0∞ :=  volume (Metric.ball (0 : Model n) 1) /    (Measure.hausdorffMeasure (n : ℝ) : Measure (Model n))      (Metric.ball (0 : Model n) 1)variable [MeasurableSpace M]/-- Riemannian volume calibrated to Lebesgue measure in Euclidean space. -/def metricVolume [BorelSpace M] (n : ℕ) : Measure M :=  euclideanVolumeFactor n • Measure.hausdorffMeasure (n : ℝ)/-- Almost-everywhere measurable probability densities between fixed bounds. -/def AdmissibleDensity (vol : Measure M) (lam cap : ℝ) (rho : M → ℝ) : Prop :=  AEMeasurable rho vol ∧ Integrable rho vol ∧  (∫ x, rho x ∂vol) = 1 ∧ (∀ᵐ x ∂vol, lam ≤ rho x ∧ rho x ≤ cap)/-- Multiplication by a density, ignoring exceptional negative values on null sets. -/def densityMeasure (vol : Measure M) (rho : M → ℝ) : Measure M :=  vol.withDensity (fun x => ENNReal.ofReal (rho x))/-- Optimality for the Monge minimization problem. -/def IsOptimalMap (vol : Measure M) (rho0 rho1 : M → ℝ) (T : M → M) : Prop :=  Measurable T ∧ Measure.map T (densityMeasure vol rho0) = densityMeasure vol rho1 ∧  ∀ S : M → M, Measurable S →    Measure.map S (densityMeasure vol rho0) = densityMeasure vol rho1 →    (∫ x, cost x (T x) * rho0 x ∂vol) ≤      (∫ x, cost x (S x) * rho0 x ∂vol)/-- The two global estimates have the very same exponent and constant. -/def BiHolderEstimate (alpha C : ℝ) (T : M ≃ₜ M) : Prop :=  (∀ x x', dist (T x) (T x') ≤ C * (dist x x') ^ alpha) ∧  (∀ y y', dist (T.symm y) (T.symm y') ≤ C * (dist y y') ^ alpha)end MetricDefinitionssection RiemannianDefinitionsvariable {n : ℕ} {M : Type*} [MetricSpace M] [ChartedSpace (Model n) M]  [IsManifold 𝓘(ℝ, Model n) ∞ M]  [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ, Model n) x)]/-- Complete affinely parametrized geodesic with prescribed initial tangent.Local metric minimization is an intrinsic definition of Riemannian geodesics. -/def IsGeodesicWithInitialData (x : M) (v : TangentSpace 𝓘(ℝ, Model n) x)    (gamma : ℝ → M) : Prop :=  ContMDiff 𝓘(ℝ, ℝ) 𝓘(ℝ, Model n) ∞ gamma ∧  (⟨gamma 0, mfderiv 𝓘(ℝ, ℝ) 𝓘(ℝ, Model n) gamma 0 (1 : ℝ)⟩ :    TangentBundle 𝓘(ℝ, Model n) M) = ⟨x, v⟩ ∧  ∀ t : ℝ, ∃ eps : ℝ, 0 < eps ∧    ∀ s u : ℝ, |s - t| < eps → |u - t| < eps →      dist (gamma s) (gamma u) = |s - u| * ‖v‖/-- Evaluation at time one of the geodesic determined by initial data. -/def riemannianExp (x : M) (v : TangentSpace 𝓘(ℝ, Model n) x) : M := by  classical  exact if h : ∃ gamma, IsGeodesicWithInitialData x v gamma then (Classical.choose h) 1 else x/-- The open injectivity domain, defined by extendible minimizing vectors. -/def injectivityDomain (x : M) : Set (TangentSpace 𝓘(ℝ, Model n) x) :=  {p | ∃ a : ℝ, 1 < a ∧ dist x (riemannianExp x (a • p)) = a * ‖p‖}/-- The Ma–Trudinger–Wang fourth derivative with factor -3/2. -/def mtw (x : M) (p xi eta : TangentSpace 𝓘(ℝ, Model n) x) : ℝ :=  -(3 / 2 : ℝ) *    iteratedDeriv 2 (fun s : ℝ =>      iteratedDeriv 2 (fun t : ℝ =>        cost (riemannianExp x (t • xi)) (riemannianExp x (p + s • eta))) 0) 0/-- Weak MTW only at interior velocities and only for orthogonal test vectors. -/def WeakMTW : Prop :=  ∀ (x : M) (p : TangentSpace 𝓘(ℝ, Model n) x), p ∈ injectivityDomain x →    ∀ xi eta : TangentSpace 𝓘(ℝ, Model n) x,      inner ℝ xi eta = 0 → 0 ≤ mtw x p xi etaend RiemannianDefinitions/-- Uniform bi-Hölder optimal transport for all densities between fixed boundson a compact connected smooth Riemannian manifold satisfying weak MTW. -/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 →        ∃ T : M ≃ₜ M,          IsOptimalMap (metricVolume n) rho0 rho1 T ∧          (∀ S : M → M, IsOptimalMap (metricVolume n) rho0 rho1 S →            S =ᵐ[densityMeasure (metricVolume n) rho0] T) ∧          BiHolderEstimate alpha C T := by  sorryend WeakMTWTransportendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Global Support and Convex Injectivity Domains under Weak MTW

Theorem (Convexity of injectivity domains)

If (M,g) satisfies (eq:weak-mtw), then I(x) is convex for every x\in M. More precisely, for every v_0,v_1\in I(x) and \theta\in[0,1],

(1-\theta)v_0+\theta v_1\in I(x).

Section 1, Introduction Read it in the paper TeX source, line 47

Show the TeX

If $(M,g)$ satisfies \eqref{eq:weak-mtw}, then $I(x)$ is convex for
every $x\in M$. More precisely, for every $v_0,v_1\in I(x)$ and
$\theta\in[0,1]$,
\[
 (1-\theta)v_0+\theta v_1\in I(x).
\]

Theorem (Global support and intermediate geometry)

Under (eq:weak-mtw), every potential u, with its dual potential v, has the following properties.

(i)
Every p\in\partial u(x) belongs to G_x and satisfies
u(x')\geq u(x)+c(x,\exp_xp)-c(x',\exp_xp) \qquad(x'\in M).
For every 0<t<1, the map F_t:\Gamma_u\to M, F_t(x,p)=\exp_x(tp), is a homeomorphism; tp\in I(x) and x is the unique global minimizing pole of Q_tu at F_t(x,p).
(ii)
For every dual pair (u,v), every x and every r, the lifted section
\mathcal S_x(r)=\left\{p\in G_x: u(x)+\frac{|p|^2}{2}+v(\exp_xp)\leq r\right\}
is compact and convex. If it is nonempty and r\geq0, then
\operatorname{diam}\mathcal S_x(r)^2 \leq8\left(r+\osc_{p\in\mathcal S_x(r)}v(\exp_xp)\right).
• For 0<t<1, one has Q_tu=-Q_{1-t}v\in C^{1,1}(M). Its C^{1,1} seminorm and the Lipschitz constants of the pole and momentum maps F_t^{-1} depend only on M,g,t. There are also r_t,b_t>0, depending only on M,g,t, such that, for z=F_t(x,p),
u(x')+\frac{c(x',z)}t \geq u(x)+\frac{c(x,z)}t+b_t d(x,x')^2 \quad\text{if }d(x,x')<r_t.
These constants can be chosen uniformly on compact subintervals of (0,1).

No density hypothesis is required.

Section 1, Introduction Read it in the paper TeX source, line 94

Show the TeX

Under \eqref{eq:weak-mtw}, every potential $u$, with its dual potential $v$,
has the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every $p\in\partial u(x)$ belongs to $G_x$ and satisfies
\[
 u(x')\geq u(x)+c(x,\exp_xp)-c(x',\exp_xp)
 \qquad(x'\in M).
\]
For every $0<t<1$, the map $F_t:\Gamma_u\to M$,
$F_t(x,p)=\exp_x(tp)$, is a homeomorphism; $tp\in I(x)$ and $x$
is the unique global minimizing pole of $Q_tu$ at $F_t(x,p)$.
\item For every dual pair $(u,v)$, every $x$ and every $r$, the lifted section
\[
 \mathcal S_x(r)=\left\{p\in G_x:
 u(x)+\frac{|p|^2}{2}+v(\exp_xp)\leq r\right\}
\]
is compact and convex. If it is nonempty and $r\geq0$, then
\[
 \operatorname{diam}\mathcal S_x(r)^2
 \leq8\left(r+\osc_{p\in\mathcal S_x(r)}v(\exp_xp)\right).
\]
\item For $0<t<1$, one has $Q_tu=-Q_{1-t}v\in C^{1,1}(M)$.
Its $C^{1,1}$ seminorm and the Lipschitz constants of the pole and momentum
maps $F_t^{-1}$ depend only on $M,g,t$. There are also $r_t,b_t>0$,
depending only on $M,g,t$, such that, for $z=F_t(x,p)$,
\[
 u(x')+\frac{c(x',z)}t
 \geq u(x)+\frac{c(x,z)}t+b_t d(x,x')^2
 \quad\text{if }d(x,x')<r_t.
\]
These constants can be chosen uniformly on compact subintervals of $(0,1)$.
\end{enumerate}
No density hypothesis is required.

Uniform Bi-Holder Transport from Weak MTW

Theorem (Uniform bi-Hölder transport)

\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.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

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 $x,x',y,y'\in M$.

Configuration and file facts

File: lean/ComparatorChallenges/BiholderTransport.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/BiholderTransport/Main.lean
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 13 local vocabulary declarations
Lines of Lean, without comments or blank lines: 68
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas