Lean statement

Weak Hessian bounds along every geodesic

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.WeakHessian.every_geodesic

Lean statement file: WeakHessian.leanShow on the mapFamily 356. 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 every_geodesic [CompleteSpace X] [TopologicalSpace.SeparableSpace X]    (m : Measure X) (K N : ℝ) (hN : 1 < N)    (hfull : FullSupport m) (hfinite : FiniteOnBoundedSets m) (hrcd : RCD m K N)    (F G : X → ℝ) (hFbound : BoundedFunction F) (hFlip : GloballyLipschitz F)    (hGbound : BoundedFunction G) (hGcont : Continuous G)    (hHess : WeakHessianUpperBound m F G)    (σ : Curve X) (hσ : IsGeodesic σ) : CurveSecondDerivativeBound F G σ := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

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.WeakHessian.Time line 12
```
abbrev Time := Set.Icc (0 : ℝ) 1
```

definition OAI.WeakHessian.timeZero line 14

def timeZero : Time := ⟨0, le_rfl, zero_le_one⟩

definition OAI.WeakHessian.timeOne line 15

def timeOne : Time := ⟨1, zero_le_one, le_rfl⟩

definition OAI.WeakHessian.clampTime line 17

def clampTime (t : ℝ) : Time :=  ⟨max 0 (min 1 t), le_max_left _ _, max_le zero_le_one (min_le_left _ _)⟩

abbreviation OAI.WeakHessian.Curve line 22

abbrev Curve (X : Type*) [MetricSpace X] := C(Time, X)

definition OAI.WeakHessian.evalReal line 28

def evalReal (γ : Curve X) (t : ℝ) : X := γ (clampTime t)

definition OAI.WeakHessian.IsGeodesic line 31

def IsGeodesic (γ : Curve X) : Prop :=  ∀ s t : Time,    dist (γ s) (γ t) = |(s : ℝ) - (t : ℝ)| * dist (γ timeZero) (γ timeOne)

definition OAI.WeakHessian.metricSpeed line 36

def metricSpeed (γ : Curve X) (t : ℝ) : ℝ≥0∞ :=  Filter.limsup (fun s : ℝ => ENNReal.ofReal    (dist (evalReal γ s) (evalReal γ t) / |s - t|)) (𝓝[≠] t)

definition OAI.WeakHessian.IsAC2 line 41

def IsAC2 (γ : Curve X) : Prop :=  ∃ a : ℝ → ℝ≥0∞, Measurable a ∧    (∫⁻ t in Set.Icc (0 : ℝ) 1, (a t) ^ 2) < ∞ ∧    ∀ s t : Time, (s : ℝ) ≤ (t : ℝ) →      edist (γ s) (γ t) ≤ ∫⁻ r in Set.Icc (s : ℝ) (t : ℝ), a r

definition OAI.WeakHessian.IsTestPlan line 50

def IsTestPlan (m : Measure X) (π : Measure (Curve X)) : Prop :=  IsProbabilityMeasure π ∧  (∃ C : ℝ≥0, ∀ t : Time,    Measure.map (fun γ : Curve X => γ t) π ≤ (C : ℝ≥0∞) • m) ∧  (∀ᵐ γ ∂π, IsAC2 γ) ∧  (∫⁻ γ, ∫⁻ t in Set.Icc (0 : ℝ) 1, (metricSpeed γ t) ^ 2 ∂volume ∂π) < ∞

definition OAI.WeakHessian.BoundedTracks line 58

def BoundedTracks (π : Measure (Curve X)) : Prop :=  ∃ o : X, ∃ R : ℝ, 0 < R ∧ ∀ᵐ γ ∂π, ∀ t : Time, dist (γ t) o ≤ R

definition OAI.WeakHessian.LocallyL2 line 61

def LocallyL2 (m : Measure X) (f : X → ℝ) : Prop :=  ∀ o : X, ∀ R : ℝ, 0 < R → MemLp f 2 (m.restrict (Metric.ball o R))

definition OAI.WeakHessian.IsLocalWeakUpperGradient line 65

def IsLocalWeakUpperGradient (m : Measure X) (f D : X → ℝ) : Prop :=  LocallyL2 m D ∧ (∀ᵐ x ∂m, 0 ≤ D x) ∧  ∀ π : Measure (Curve X), IsTestPlan m π → BoundedTracks π →    (∫⁻ γ, ENNReal.ofReal |f (γ timeOne) - f (γ timeZero)| ∂π) ≤      ∫⁻ γ, ∫⁻ t in Set.Icc (0 : ℝ) 1,        ENNReal.ofReal (D (evalReal γ t)) * metricSpeed γ t ∂volume ∂π

definition OAI.WeakHessian.IsMinimalLocalWeakUpperGradient line 72

def IsMinimalLocalWeakUpperGradient (m : Measure X) (f D : X → ℝ) : Prop :=  IsLocalWeakUpperGradient m f D ∧  ∀ D' : X → ℝ, IsLocalWeakUpperGradient m f D' → ∀ᵐ x ∂m, D x ≤ D' x

definition OAI.WeakHessian.weakGradient line 77

def weakGradient (m : Measure X) (f : X → ℝ) : X → ℝ :=  if h : ∃ D, IsMinimalLocalWeakUpperGradient m f D then Classical.choose h  else 0

definition OAI.WeakHessian.LocalSobolev line 81

def LocalSobolev (m : Measure X) (f : X → ℝ) : Prop :=  LocallyL2 m f ∧ ∃ D, IsMinimalLocalWeakUpperGradient m f D

definition OAI.WeakHessian.Sobolev line 84

def Sobolev (m : Measure X) (f : X → ℝ) : Prop :=  MemLp f 2 m ∧ LocalSobolev m f ∧ MemLp (weakGradient m f) 2 m

definition OAI.WeakHessian.gamma line 88

def gamma (m : Measure X) (f g : X → ℝ) (x : X) : ℝ :=  ((weakGradient m (fun y => f y + g y) x) ^ 2 -    (weakGradient m (fun y => f y - g y) x) ^ 2) / 4

definition OAI.WeakHessian.IsLaplacian line 93

def IsLaplacian (m : Measure X) (u L : X → ℝ) : Prop :=  Sobolev m u ∧ MemLp L 2 m ∧    ∀ v : X → ℝ, Sobolev m v →      (∫ x, gamma m u v x ∂m) = -(∫ x, L x * v x ∂m)

definition OAI.WeakHessian.InLaplacianDomain line 98

def InLaplacianDomain (m : Measure X) (u : X → ℝ) : Prop :=  ∃ L, IsLaplacian m u L

definition OAI.WeakHessian.laplacian line 101

def laplacian (m : Measure X) (u : X → ℝ) : X → ℝ :=  if h : ∃ L, IsLaplacian m u L then Classical.choose h else 0

definition OAI.WeakHessian.BoundedFunction line 104

def BoundedFunction (f : X → ℝ) : Prop := ∃ B : ℝ, ∀ x, |f x| ≤ B

definition OAI.WeakHessian.GloballyLipschitz line 106

def GloballyLipschitz (f : X → ℝ) : Prop := ∃ L : ℝ≥0, LipschitzWith L f

definition OAI.WeakHessian.IsTestFunction line 109

def IsTestFunction (m : Measure X) (g : X → ℝ) : Prop :=  InLaplacianDomain m g ∧ BoundedFunction g ∧ GloballyLipschitz g ∧    Sobolev m (laplacian m g)

19 more definitions are reached the same way; the full file lists them.

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: minimizing, sigma, weak, geodesic, gamma, real, speed, constant and eleven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: circ.

Statements in Weak Hessian bounds along every geodesic in RCD spaces come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Weak Hessian bounds along every geodesic in RCD spaces

Theorem (Weak Hessian bounds along every geodesic)

Let (M,d,m) be a full-support \RCD(K,N) space, where K\in\mathbb R and 1<N<\infty. Let F\colon M\to\mathbb R be bounded and globally Lipschitz, and let G\colon M\to\mathbb R be bounded and continuous. Suppose that for every compactly supported g\in\Test(M) and every nonnegative h\in\Lip_c(M),

H_F(\nabla g,\nabla g)(h) \le\int_M hG\Gamma(g)\dd m.

Then every constant-speed minimizing geodesic \sigma\colon[0,1]\to M, of length \ell, satisfies

(F\circ\sigma)''\le\ell^2G\circ\sigma \qquad\text{in distributions on }(0,1).

Section 1, Introduction Read it in the paper TeX source, line 52

Show the TeX

Let \((M,d,m)\) be a full-support \(\RCD(K,N)\) space,
where \(K\in\mathbb R\) and \(1<N<\infty\). Let
\(F\colon M\to\mathbb R\) be bounded and globally Lipschitz,
and let \(G\colon M\to\mathbb R\) be bounded and continuous.
Suppose that for every compactly supported \(g\in\Test(M)\)
and every nonnegative \(h\in\Lip_c(M)\),
\begin{equation}

H_F(\nabla g,\nabla g)(h)
\le\int_M hG\Gamma(g)\dd m.
\end{equation}
Then every constant-speed minimizing geodesic
\(\sigma\colon[0,1]\to M\), of length \(\ell\), satisfies
\begin{equation}

(F\circ\sigma)''\le\ell^2G\circ\sigma
\qquad\text{in distributions on }(0,1).
\end{equation}

Other possible paper matches, closest wording first

Proposition 4.1 of Weak Hessian bounds along every geodesic in RCD spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: cd, metric, gamma, real, distributional, continuous and eleven more.
Proposition 5.6 of Gigli’s distributional curvature characterization of Alexandrov spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: time, derivative, agree, local, optimal, norm and 15 more.
Theorem 4.1 of Gigli’s distributional curvature characterization of Alexandrov spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: one, endpoint, plan, compression, pi, gamma and 21 more.
Theorem 6.1 of Gigli’s distributional curvature characterization of Alexandrov spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: minimizing, sigma, weak, geodesic, real, speed and twelve more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: circ.
Theorem 5.10 of Gigli’s distributional curvature characterization of Alexandrov spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt, ball, hess, nonpositive, singular, coefficient and 17 more.
Theorem 1.1 of Gigli’s distributional curvature characterization of Alexandrov spaces. Words of the Lean statement, its names split into words, that also appear in the theorem: unreduced, separable, complete, metric, integer, rcd and six more. Found in the paper’s title, which its main theorem states: distributional.

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.

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

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 356’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds a supporting or a different result stated Their term: “supporting-only”.
The challenge statements state a lemma or auxiliary result, not the headline. Their record
Their note: Only every_geodesic: in a complete separable RCD(K,N) space (N > 1, full-support reference measure finite on bounded sets), a bounded Lipschitz F whose weak Hessian is ≤ G·metric (G bounded continuous) satisfies the distributional bound (F∘σ)'' ≤ G(σ)·d(σ(0),σ(1))² along every constant-speed geodesic. Gigli's conjecture (Alexandrov curvature ≥ κ ⇔ RCD((n-1)κ,n) with H^n and distributional sectional curvature ≥ κ) is not stated.
Definitions the group asks readers to compare with the paper: Entire RCD/Sobolev calculus is hand-built: CurvatureDimension (unreduced CD(K,N) via optimal dynamical plans, tauCoeff/sigmaCoeff), QuadraticCheegerEnergy, IsTestPlan, IsLocalWeakUpperGradient, weakGradient/laplacian defined by Classical.choose with fallback 0, weakHessian; the cone is 65,542 OAI lines here (recomputed; tmp/import_cones.json says 64,010)
Carlos Toledo’s cert-machine
Finds a supporting or a different result stated Their term: “DIFFERENT”.
The challenge statements state a different theorem from the headline. Their record
Their note: The challenge states a supporting result (on a full-support RCD(K,N) space with 1 < N < ∞, a bounded Lipschitz F with weak Hessian ≤ G·g has (F∘γ)'' ≤ G(γ)·d(γ₀,γ₁)² distributionally along every constant-speed geodesic). It is an ingredient of the characterization, but neither implies it nor is implied by it. Gigli's characterization (the headline) is not stated.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization transfers a weak Hessian upper bound to every prescribed minimizing geodesic in an $\mathrm{RCD}(K,N)$ space with finite $N>1$. If a bounded globally Lipschitz function $F$ has weak Hessian bounded above by $G$ times the metric, where $G$ is bounded and continuous, then every constant-speed geodesic $\gamma:[0,1]\to X$ satisfies $(F\circ\gamma)''\le G(\gamma)\,d(\gamma(0),\gamma(1))^2$ in the distributional sense.”
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

233 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: 115 to its own declarations, 141 to Mathlib’s documentation. 46 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. 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 every_geodesic 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 43 of the file’s 45 other declarations; with the theorem itself they span 146 lines. 9 of them are named in the statement itself.

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

abbrev Timeline 12library
def timeZeroline 14library
def timeOneline 15library
def clampTimeline 17library
abbrev Curveline 22in the statementlibrary
def evalRealline 28library
def IsGeodesicline 31in the statementlibrary
def metricSpeedline 36library
def IsAC2line 41library
def IsTestPlanline 50library
def BoundedTracksline 58library
def LocallyL2line 61library
def IsLocalWeakUpperGradientline 65library
def IsMinimalLocalWeakUpperGradientline 72library
def weakGradientline 77library
def LocalSobolevline 81library
def Sobolevline 84library
def gammaline 88library
def IsLaplacianline 93library
def InLaplacianDomainline 98library
def laplacianline 101library
def BoundedFunctionline 104in the statementlibrary
def GloballyLipschitzline 106in the statementlibrary
def IsTestFunctionline 109library
def weakHessianline 114library
def WeakHessianUpperBoundline 118in the statementlibrary
def localSlopeline 125library
def cheegerEnergyline 129library
def QuadraticCheegerEnergyline 138library
def sigmaCoeffline 144library
def tauCoeffline 157library
def HasSecondMomentline 160library
def IsCouplingline 163library
def QuadraticTransportCostline 166library
def IsOptimalCouplingline 169library
def endpointCouplingline 173library
def interpolationline 176library
def renyiIntegralline 180library
def CurvatureDimensionline 184library
def RCDline 200in the statementlibrary
def FullSupportline 203in the statementlibrary
def FiniteOnBoundedSetsline 206in the statementlibrary
def CurveSecondDerivativeBoundline 210in the statementlibrary
theorem every_geodesicline 220selected theoremlibrary

Other Lean rules that may affect the statement 3

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance curveMeasurableSpace line 24
instance curveBorelSpace line 26
attribute [local instance] Classical.propDecidable line 9

import Mathlibnamespace OAIopen Set Filter MeasureTheoryopen scoped ENNReal NNReal Topologynoncomputable sectionattribute [local instance] Classical.propDecidablenamespace WeakHessianabbrev Time := Set.Icc (0 : ℝ) 1def timeZero : Time := ⟨0, le_rfl, zero_le_one⟩def timeOne : Time := ⟨1, zero_le_one, le_rfl⟩def clampTime (t : ℝ) : Time :=  ⟨max 0 (min 1 t), le_max_left _ _, max_le zero_le_one (min_le_left _ _)⟩variable {X : Type*} [MetricSpace X]abbrev Curve (X : Type*) [MetricSpace X] := C(Time, X)instance curveMeasurableSpace : MeasurableSpace (Curve X) := borel (Curve X)instance curveBorelSpace : BorelSpace (Curve X) := ⟨rfl⟩def evalReal (γ : Curve X) (t : ℝ) : X := γ (clampTime t)/-- The constant-speed minimizing parametrization, including constant curves. -/def IsGeodesic (γ : Curve X) : Prop :=  ∀ s t : Time,    dist (γ s) (γ t) = |(s : ℝ) - (t : ℝ)| * dist (γ timeZero) (γ timeOne)/-- Metric speed. Limsup agrees a.e. with the metric derivative on AC² curves. -/def metricSpeed (γ : Curve X) (t : ℝ) : ℝ≥0∞ :=  Filter.limsup (fun s : ℝ => ENNReal.ofReal    (dist (evalReal γ s) (evalReal γ t) / |s - t|)) (𝓝[≠] t)/-- Absolute continuity with a square-integrable metric control. -/def IsAC2 (γ : Curve X) : Prop :=  ∃ a : ℝ → ℝ≥0∞, Measurable a ∧    (∫⁻ t in Set.Icc (0 : ℝ) 1, (a t) ^ 2) < ∞ ∧    ∀ s t : Time, (s : ℝ) ≤ (t : ℝ) →      edist (γ s) (γ t) ≤ ∫⁻ r in Set.Icc (s : ℝ) (t : ℝ), a rvariable [MeasurableSpace X] [BorelSpace X]/-- A 2-test plan: bounded compression and finite average kinetic energy. -/def IsTestPlan (m : Measure X) (π : Measure (Curve X)) : Prop :=  IsProbabilityMeasure π ∧  (∃ C : ℝ≥0, ∀ t : Time,    Measure.map (fun γ : Curve X => γ t) π ≤ (C : ℝ≥0∞) • m) ∧  (∀ᵐ γ ∂π, IsAC2 γ) ∧  (∫⁻ γ, ∫⁻ t in Set.Icc (0 : ℝ) 1, (metricSpeed γ t) ^ 2 ∂volume ∂π) < ∞/-- Uniform bounded spatial support; used only to define the local Sobolev class. -/def BoundedTracks (π : Measure (Curve X)) : Prop :=  ∃ o : X, ∃ R : ℝ, 0 < R ∧ ∀ᵐ γ ∂π, ∀ t : Time, dist (γ t) o ≤ Rdef LocallyL2 (m : Measure X) (f : X → ℝ) : Prop :=  ∀ o : X, ∀ R : ℝ, 0 < R → MemLp f 2 (m.restrict (Metric.ball o R))/-- Local weak upper gradients, in the integrated test-plan definition. -/def IsLocalWeakUpperGradient (m : Measure X) (f D : X → ℝ) : Prop :=  LocallyL2 m D ∧ (∀ᵐ x ∂m, 0 ≤ D x) ∧  ∀ π : Measure (Curve X), IsTestPlan m π → BoundedTracks π →    (∫⁻ γ, ENNReal.ofReal |f (γ timeOne) - f (γ timeZero)| ∂π) ≤      ∫⁻ γ, ∫⁻ t in Set.Icc (0 : ℝ) 1,        ENNReal.ofReal (D (evalReal γ t)) * metricSpeed γ t ∂volume ∂πdef IsMinimalLocalWeakUpperGradient (m : Measure X) (f D : X → ℝ) : Prop :=  IsLocalWeakUpperGradient m f D ∧  ∀ D' : X → ℝ, IsLocalWeakUpperGradient m f D' → ∀ᵐ x ∂m, D x ≤ D' x/-- A representative of the minimal weak upper gradient (zero off its domain). -/def weakGradient (m : Measure X) (f : X → ℝ) : X → ℝ :=  if h : ∃ D, IsMinimalLocalWeakUpperGradient m f D then Classical.choose h  else 0def LocalSobolev (m : Measure X) (f : X → ℝ) : Prop :=  LocallyL2 m f ∧ ∃ D, IsMinimalLocalWeakUpperGradient m f Ddef Sobolev (m : Measure X) (f : X → ℝ) : Prop :=  MemLp f 2 m ∧ LocalSobolev m f ∧ MemLp (weakGradient m f) 2 m/-- Polarization of the minimal weak gradient on an infinitesimally Hilbertian space. -/def gamma (m : Measure X) (f g : X → ℝ) (x : X) : ℝ :=  ((weakGradient m (fun y => f y + g y) x) ^ 2 -    (weakGradient m (fun y => f y - g y) x) ^ 2) / 4/-- Weak nonpositive generator convention, without a global L² assumption on F. -/def IsLaplacian (m : Measure X) (u L : X → ℝ) : Prop :=  Sobolev m u ∧ MemLp L 2 m ∧    ∀ v : X → ℝ, Sobolev m v →      (∫ x, gamma m u v x ∂m) = -(∫ x, L x * v x ∂m)def InLaplacianDomain (m : Measure X) (u : X → ℝ) : Prop :=  ∃ L, IsLaplacian m u Ldef laplacian (m : Measure X) (u : X → ℝ) : X → ℝ :=  if h : ∃ L, IsLaplacian m u L then Classical.choose h else 0def BoundedFunction (f : X → ℝ) : Prop := ∃ B : ℝ, ∀ x, |f x| ≤ Bdef GloballyLipschitz (f : X → ℝ) : Prop := ∃ L : ℝ≥0, LipschitzWith L f/-- Bounded globally Lipschitz Laplacian-domain functions with Sobolev Laplacian. -/def IsTestFunction (m : Measure X) (g : X → ℝ) : Prop :=  InLaplacianDomain m g ∧ BoundedFunction g ∧ GloballyLipschitz g ∧    Sobolev m (laplacian m g)/-- The weak Hessian, with ∇_(∇g)∇g = ∇(Γ(g)/2). -/def weakHessian (m : Measure X) (F g h : X → ℝ) : ℝ :=  -(∫ x, (gamma m h g x + h x * laplacian m g x) * gamma m F g x ∂m) -    ∫ x, h x * gamma m F (fun y => gamma m g g y / 2) x ∂mdef WeakHessianUpperBound (m : Measure X) (F G : X → ℝ) : Prop :=  ∀ g : X → ℝ, IsTestFunction m g → HasCompactSupport g →    ∀ h : X → ℝ, GloballyLipschitz h → HasCompactSupport h →      (∀ x, 0 ≤ h x) →      weakHessian m F g h ≤ ∫ x, h x * G x * gamma m g g x ∂m/-- Asymptotic Lipschitz slope; zero at an isolated point. -/def localSlope (f : X → ℝ) (x : X) : ℝ≥0∞ :=  Filter.limsup (fun y => ENNReal.ofReal (|f y - f x| / dist y x)) (𝓝[≠] x)/-- Cheeger energy is the L² relaxation of one half the squared local slope. -/def cheegerEnergy (m : Measure X) (f : X → ℝ) : ℝ≥0∞ :=  if MemLp f 2 m then    sInf {a | ∃ fs : ℕ → X → ℝ,      (∀ n, GloballyLipschitz (fs n) ∧ MemLp (fs n) 2 m) ∧      Tendsto (fun n => eLpNorm (fun x => fs n x - f x) 2 m) atTop (𝓝 0) ∧      a = Filter.liminf (fun n => (2 : ℝ≥0∞)⁻¹ *        ∫⁻ x, (localSlope (fs n) x) ^ 2 ∂m) atTop}  else ∞def QuadraticCheegerEnergy (m : Measure X) : Prop :=  ∀ f g : X → ℝ, cheegerEnergy m f < ∞ → cheegerEnergy m g < ∞ →    cheegerEnergy m (fun x => f x + g x) + cheegerEnergy m (fun x => f x - g x) =      2 * cheegerEnergy m f + 2 * cheegerEnergy m g/-- The unreduced Lott–Sturm–Villani distortion coefficient σ. -/def sigmaCoeff (K N t θ : ℝ) : ℝ≥0∞ :=  if t ≤ 0 then 0 else if 1 ≤ t then 1 else  if θ = 0 then ENNReal.ofReal t else  if 0 < K then    if K * θ ^ 2 < N * Real.pi ^ 2 then      ENNReal.ofReal (Real.sin (t * θ * Real.sqrt (K / N)) /        Real.sin (θ * Real.sqrt (K / N)))    else ∞  else if K = 0 then ENNReal.ofReal t else    ENNReal.ofReal (Real.sinh (t * θ * Real.sqrt (-K / N)) /      Real.sinh (θ * Real.sqrt (-K / N)))/-- The τ coefficient (not the reduced σ coefficient) in CD(K,N). -/def tauCoeff (K N t θ : ℝ) : ℝ≥0∞ :=  (ENNReal.ofReal t) ^ (1 / N) * (sigmaCoeff K (N - 1) t θ) ^ ((N - 1) / N)def HasSecondMoment (μ : Measure X) : Prop :=  ∃ o : X, (∫⁻ x, (edist x o) ^ 2 ∂μ) < ∞def IsCoupling (q : Measure (X × X)) (μ₀ μ₁ : Measure X) : Prop :=  Measure.map Prod.fst q = μ₀ ∧ Measure.map Prod.snd q = μ₁def QuadraticTransportCost (q : Measure (X × X)) : ℝ≥0∞ :=  ∫⁻ p, (edist p.1 p.2) ^ 2 ∂qdef IsOptimalCoupling (q : Measure (X × X)) (μ₀ μ₁ : Measure X) : Prop :=  IsCoupling q μ₀ μ₁ ∧ ∀ q' : Measure (X × X), IsCoupling q' μ₀ μ₁ →    QuadraticTransportCost q ≤ QuadraticTransportCost q'def endpointCoupling (π : Measure (Curve X)) : Measure (X × X) :=  Measure.map (fun γ : Curve X => (γ timeZero, γ timeOne)) πdef interpolation (π : Measure (Curve X)) (t : Time) : Measure X :=  Measure.map (fun γ : Curve X => γ t) π/-- The positive of minus the N-Rényi entropy; singular mass contributes zero. -/def renyiIntegral (m μ : Measure X) (N : ℝ) : ℝ≥0∞ :=  ∫⁻ x, (μ.rnDeriv m x) ^ (1 - 1 / N) ∂m/-- Unreduced CD(K,N), written with the standard dynamical optimal-plan formula. -/def CurvatureDimension (m : Measure X) (K N : ℝ) : Prop :=  ∀ μ₀ μ₁ : Measure X,    IsProbabilityMeasure μ₀ → IsProbabilityMeasure μ₁ →    HasSecondMoment μ₀ → HasSecondMoment μ₁ →    μ₀ ≪ m → μ₁ ≪ m →    ∃ π : Measure (Curve X), IsProbabilityMeasure π ∧      (∀ᵐ γ ∂π, IsGeodesic γ) ∧      IsOptimalCoupling (endpointCoupling π) μ₀ μ₁ ∧      ∀ N' : ℝ, N ≤ N' → ∀ t : Time,        (∫⁻ γ,          (tauCoeff K N' (1 - (t : ℝ)) (dist (γ timeZero) (γ timeOne)) *            (μ₀.rnDeriv m (γ timeZero)) ^ (-1 / N') +          tauCoeff K N' (t : ℝ) (dist (γ timeZero) (γ timeOne)) *            (μ₁.rnDeriv m (γ timeOne)) ^ (-1 / N')) ∂π) ≤        renyiIntegral m (interpolation π t) N'def RCD (m : Measure X) (K N : ℝ) : Prop :=  CurvatureDimension m K N ∧ QuadraticCheegerEnergy mdef FullSupport (m : Measure X) : Prop :=  ∀ U : Set X, IsOpen U → U.Nonempty → 0 < m Udef FiniteOnBoundedSets (m : Measure X) : Prop :=  ∀ B : Set X, Bornology.IsBounded B → m B < ∞/-- The literal second-distributional-derivative conclusion on (0,1). -/def CurveSecondDerivativeBound (F G : X → ℝ) (σ : Curve X) : Prop :=  ∀ φ : ℝ → ℝ, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) φ → HasCompactSupport φ →    tsupport φ ⊆ Set.Ioo (0 : ℝ) 1 → (∀ t, 0 ≤ φ t) →    (∫ t in Set.Icc (0 : ℝ) 1,      F (evalReal σ t) * deriv (deriv φ) t) ≤    (dist (σ timeZero) (σ timeOne)) ^ 2 *      ∫ t in Set.Icc (0 : ℝ) 1, G (evalReal σ t) * φ t/-- A weak upper Hessian bound controls the second distributional derivativealong every constant-speed minimizing geodesic. -/theorem every_geodesic [CompleteSpace X] [TopologicalSpace.SeparableSpace X]    (m : Measure X) (K N : ℝ) (hN : 1 < N)    (hfull : FullSupport m) (hfinite : FiniteOnBoundedSets m) (hrcd : RCD m K N)    (F G : X → ℝ) (hFbound : BoundedFunction F) (hFlip : GloballyLipschitz F)    (hGbound : BoundedFunction G) (hGcont : Continuous G)    (hHess : WeakHessianUpperBound m F G)    (σ : Curve X) (hσ : IsGeodesic σ) : CurveSecondDerivativeBound F G σ := by  sorryend WeakHessianendend OAI

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