Lean statement

Differentiability of the exponential first-passage limit shape

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.PlanarFPP.manuscriptMain

Lean statement file: PlanarFirstPassage.leanShow on the mapFamily 212. 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 manuscriptMain (Ω : Type) [MeasurableSpace Ω] (P : Measure Ω) (τ : Edge → Ω → ℝ)    (henv : ExponentialEnvironment P τ) :    ∃ μ : Plane → ℝ, TimeConstantNorm μ ∧ IsTimeConstant P τ μ ∧      DifferentiableAwayFromOrigin μ ∧ UniqueNormalizedSupports μ ∧ C1UnitSphere μ ∧      (∀ v ∈ frontier {z | μ z ≤ 1}, ∃! L : Set Plane,        ∃ g : Plane →L[ℝ] ℝ, g ≠ 0 ∧ (∀ z, μ z ≤ 1 → g z ≤ g v) ∧ L={z | g z=g v}) ∧      (∀ v ∈ frontier {z | μ z ≤ 1}, HasC1CurveChart (frontier {z | μ z ≤ 1}) v) := 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.PlanarFPP.Vertex line 18
```
abbrev Vertex := ℤ × ℤ
```
abbreviation OAI.PlanarFPP.Edge line 20
```
abbrev Edge := Vertex × Bool
```

inductive type OAI.PlanarFPP.Step line 22

inductive Step where  | east | west | north | south  deriving DecidableEq

definition OAI.PlanarFPP.Step.delta line 26

def Step.delta : Step → Vertex  | .east => (1, 0)  | .west => (-1, 0)  | .north => (0, 1)  | .south => (0, -1)

definition OAI.PlanarFPP.Step.edge line 32

def Step.edge (p : Vertex) : Step → Edge  | .east => (p, false)  | .west => (p + (-1, 0), false)  | .north => (p, true)  | .south => (p + (0, -1), true)

definition OAI.PlanarFPP.displacement line 38

def displacement (ds : List Step) : Vertex := (ds.map Step.delta).sum

definition OAI.PlanarFPP.pathCost line 40

def pathCost (τ : Edge → ℝ) (p : Vertex) : List Step → ℝ  | [] => 0  | d :: ds => τ (d.edge p) + pathCost τ (p + d.delta) ds

abbreviation OAI.PlanarFPP.Plane line 44

abbrev Plane := EuclideanSpace ℝ (Fin 2)

definition OAI.PlanarFPP.latticeFloor line 46

noncomputable def latticeFloor (v : Plane) : Vertex := (⌊v 0⌋, ⌊v 1⌋)

definition OAI.PlanarFPP.passageTime line 48

noncomputable def passageTime (τ : Edge → ℝ) (p q : Vertex) : ℝ :=  sInf {t | ∃ ds : List Step, p + displacement ds = q ∧ pathCost τ p ds = t}

structure OAI.PlanarFPP.TimeConstantNorm line 51

structure TimeConstantNorm (μ : Plane → ℝ) : Prop where  nonneg : ∀ v, 0 ≤ μ v  zero_iff : ∀ v, μ v = 0 ↔ v = 0  triangle : ∀ v w, μ (v + w) ≤ μ v + μ w  homogeneous : ∀ (a : ℝ) v, μ (a • v) = |a| * μ v

structure OAI.PlanarFPP.ExponentialEnvironment line 57

structure ExponentialEnvironment {Ω : Type*} [MeasurableSpace Ω]    (P : Measure Ω) (τ : Edge → Ω → ℝ) : Prop where  probability : IsProbabilityMeasure P  measurable : ∀ e, Measurable (τ e)  independent : iIndepFun τ P  marginal : ∀ e, Measure.map (τ e) P = expMeasure 1

definition OAI.PlanarFPP.IsTimeConstant line 64

def IsTimeConstant {Ω : Type*} [MeasurableSpace Ω]    (P : Measure Ω) (τ : Edge → Ω → ℝ) (μ : Plane → ℝ) : Prop :=  ∀ v : Plane, ∀ᵐ ω ∂P,    Tendsto (fun t : ℝ => passageTime (fun e => τ e ω) (0, 0)      (latticeFloor (t • v)) / t) atTop (𝓝 (μ v))

definition OAI.PlanarFPP.DifferentiableAwayFromOrigin line 70

def DifferentiableAwayFromOrigin (μ : Plane → ℝ) : Prop :=  ∀ v : Plane, v ≠ 0 → ∃ ℓ : Plane →L[ℝ] ℝ, HasFDerivAt μ ℓ v

definition OAI.PlanarFPP.UniqueNormalizedSupports line 73

def UniqueNormalizedSupports (μ : Plane → ℝ) : Prop :=  ∀ v : Plane, μ v = 1 → ∃! ℓ : Plane →L[ℝ] ℝ,    ℓ v = 1 ∧ ∀ w, ℓ w ≤ μ w

definition OAI.PlanarFPP.C1UnitSphere line 77

def C1UnitSphere (μ : Plane → ℝ) : Prop :=  ∀ v : Plane, μ v = 1 → ∃ U : Set Plane, IsOpen U ∧ v ∈ U ∧    ∃ g : Plane → ℝ, ContDiffOn ℝ 1 g U ∧      (∀ w ∈ U, μ w = 1 ↔ g w = 0) ∧      ∀ w ∈ U, g w = 0 → ∃ ℓ : Plane →L[ℝ] ℝ,        HasFDerivAt g ℓ w ∧ ℓ ≠ 0

definition OAI.PlanarFPP.HasC1CurveChart line 84

def HasC1CurveChart (S : Set Plane) (v : Plane) : Prop :=  ∃ U : Set Plane, IsOpen U ∧ v ∈ U ∧ v ∈ S ∧ ∃ γ : ℝ → Plane, ∃ θ : Plane → ℝ,    ContDiff ℝ 1 γ ∧ ContinuousOn θ U ∧    (∀ t, γ t ∈ S ∩ U) ∧ (∀ t, θ (γ t)=t) ∧    (∀ z ∈ S ∩ U, γ (θ z)=z) ∧    (∀ t, ∃ d : Plane, HasDerivAt γ d t ∧ d ≠ 0)

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: exponential, supporting, curve, differentiable, unique, norm, unit, edge and one more. Found in the paper’s title, which its main theorem states: passage. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: shape and limit.

Statements in Strict convexity and differentiability of the planar exponential first-passage limit shape come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Strict convexity and differentiability of the planar exponential first-passage limit shape

Theorem (The exponential limit shape)

For independent exponential edge weights of any rate \lambda>0, the norm \mu is Fréchet differentiable on \R^2\setminus\{0\}, and its unit ball is strictly convex. In particular,

\mu((1-t)x+ty)<1 \quad\text{if }x,y\in\partial\Bcal,\ x\ne y,\ 0<t<1,

and \partial\Bcal is a C^1 curve with a unique supporting line at every point.

Section 1, Introduction Read it in the paper TeX source, line 21

Show the TeX

For independent exponential edge weights of any rate $\lambda>0$,
the norm $\mu$ is Fr\'echet differentiable on $\R^2\setminus\{0\}$,
and its unit ball is strictly convex. In particular,
\[
 \mu((1-t)x+ty)<1
 \quad\text{if }x,y\in\partial\Bcal,\ x\ne y,\ 0<t<1,
\]
and $\partial\Bcal$ is a $C^1$ curve with a unique supporting line
at every point.

Other possible paper matches, closest wording first

Theorem 1.2 of Strict convexity and differentiability of the planar exponential first-passage limit shape. Words of the Lean statement, its names split into words, that also appear in the theorem: curve, differentiable, norm, edge and independent. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: shape.
Proposition 14.3 of Strict convexity and differentiability of the planar exponential first-passage limit shape. Words of the Lean statement, its names split into words, that also appear in the theorem: displacement, delta, origin, list, zero, vertex and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: selected.
Corollary 17.4 of Strict convexity and differentiability of the planar exponential first-passage limit shape. Words of the Lean statement, its names split into words, that also appear in the theorem: time, euclidean, zero, plane, passage and unit. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: first.
Corollary 17.1 of Strict convexity and differentiability of the planar exponential first-passage limit shape. Words of the Lean statement, its names split into words, that also appear in the theorem: paths, unit, edge, probability and independent. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: limit.
Proposition 5.4 of No bigeodesics in planar first-passage percolation. Words of the Lean statement, its names split into words, that also appear in the theorem: lattice, constant, edge and chart.
Proposition 6.4 of Strict convexity and differentiability of the planar exponential first-passage limit shape. Words of the Lean statement, its names split into words, that also appear in the theorem: path and origin.

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 212’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 part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Stated: exponential case manuscriptMain (norm μ with IsTimeConstant, DifferentiableAwayFromOrigin, UniqueNormalizedSupports, C1UnitSphere, unique supporting line, HasC1CurveChart on the frontier) and GammaFPP.gamma_differentiability for every shape/rate > 0. Not stated: the first headline claim (no doubly infinite geodesic for iid nonatomic weights with finite second moment of the min of four) and strict convexity of the exponential limit shape (docs: 'Strict convexity is outside these selected differentiability statements'; bigeodesics are not mentioned in the docs scope).
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: The no-bigeodesics theorem (the headline's first sentence) is not stated; strict convexity of the exponential limit shape is not stated (declared).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The paper also claims strict convexity for the exponential limit shape. Strict convexity is outside these selected differentiability statements.”
Quoted from OpenAI’s Lean note for this family

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

Supporting material

The complete Lean statement file

102 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: 55 to its own declarations, 40 to Mathlib’s documentation. 18 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 manuscriptMain 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 17 other declarations of the file; with the theorem itself they span 64 lines. 9 of them are named in the statement itself.

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

abbrev Vertexline 18library
abbrev Edgeline 20in the statementlibrary
inductive Stepline 22library
def Step.deltaline 26library
def Step.edgeline 32library
def displacementline 38library
def pathCostline 40library
abbrev Planeline 44in the statementlibrary
def latticeFloorline 46library
def passageTimeline 48library
structure TimeConstantNormline 51in the statementlibrary
structure ExponentialEnvironmentline 57in the statementlibrary
def IsTimeConstantline 64in the statementlibrary
def DifferentiableAwayFromOriginline 70in the statementlibrary
def UniqueNormalizedSupportsline 73in the statementlibrary
def C1UnitSphereline 77in the statementlibrary
def HasC1CurveChartline 84in the statementlibrary
theorem manuscriptMainline 91selected theoremlibrary

import Mathlib.Algebra.BigOperators.Group.List.Basicimport Mathlib.Algebra.Group.Prodimport Mathlib.Analysis.Calculus.ContDiff.Defsimport Mathlib.Analysis.Calculus.Deriv.Basicimport Mathlib.Analysis.Calculus.FDeriv.Basicimport Mathlib.Analysis.InnerProductSpace.PiL2import Mathlib.Analysis.LocallyConvex.WithSeminormsimport Mathlib.Probability.Distributions.Exponentialimport Mathlib.Probability.Independence.Basicnamespace OAInamespace PlanarFPPopen Set MeasureTheory ProbabilityTheory Filteropen scoped Topologyabbrev Vertex := ℤ × ℤabbrev Edge := Vertex × Boolinductive Step where  | east | west | north | south  deriving DecidableEqdef Step.delta : Step → Vertex  | .east => (1, 0)  | .west => (-1, 0)  | .north => (0, 1)  | .south => (0, -1)def Step.edge (p : Vertex) : Step → Edge  | .east => (p, false)  | .west => (p + (-1, 0), false)  | .north => (p, true)  | .south => (p + (0, -1), true)def displacement (ds : List Step) : Vertex := (ds.map Step.delta).sumdef pathCost (τ : Edge → ℝ) (p : Vertex) : List Step → ℝ  | [] => 0  | d :: ds => τ (d.edge p) + pathCost τ (p + d.delta) dsabbrev Plane := EuclideanSpace ℝ (Fin 2)noncomputable def latticeFloor (v : Plane) : Vertex := (⌊v 0⌋, ⌊v 1⌋)noncomputable def passageTime (τ : Edge → ℝ) (p q : Vertex) : ℝ :=  sInf {t | ∃ ds : List Step, p + displacement ds = q ∧ pathCost τ p ds = t}structure TimeConstantNorm (μ : Plane → ℝ) : Prop where  nonneg : ∀ v, 0 ≤ μ v  zero_iff : ∀ v, μ v = 0 ↔ v = 0  triangle : ∀ v w, μ (v + w) ≤ μ v + μ w  homogeneous : ∀ (a : ℝ) v, μ (a • v) = |a| * μ vstructure ExponentialEnvironment {Ω : Type*} [MeasurableSpace Ω]    (P : Measure Ω) (τ : Edge → Ω → ℝ) : Prop where  probability : IsProbabilityMeasure P  measurable : ∀ e, Measurable (τ e)  independent : iIndepFun τ P  marginal : ∀ e, Measure.map (τ e) P = expMeasure 1def IsTimeConstant {Ω : Type*} [MeasurableSpace Ω]    (P : Measure Ω) (τ : Edge → Ω → ℝ) (μ : Plane → ℝ) : Prop :=  ∀ v : Plane, ∀ᵐ ω ∂P,    Tendsto (fun t : ℝ => passageTime (fun e => τ e ω) (0, 0)      (latticeFloor (t • v)) / t) atTop (𝓝 (μ v))def DifferentiableAwayFromOrigin (μ : Plane → ℝ) : Prop :=  ∀ v : Plane, v ≠ 0 → ∃ ℓ : Plane →L[ℝ] ℝ, HasFDerivAt μ ℓ vdef UniqueNormalizedSupports (μ : Plane → ℝ) : Prop :=  ∀ v : Plane, μ v = 1 → ∃! ℓ : Plane →L[ℝ] ℝ,    ℓ v = 1 ∧ ∀ w, ℓ w ≤ μ wdef C1UnitSphere (μ : Plane → ℝ) : Prop :=  ∀ v : Plane, μ v = 1 → ∃ U : Set Plane, IsOpen U ∧ v ∈ U ∧    ∃ g : Plane → ℝ, ContDiffOn ℝ 1 g U ∧      (∀ w ∈ U, μ w = 1 ↔ g w = 0) ∧      ∀ w ∈ U, g w = 0 → ∃ ℓ : Plane →L[ℝ] ℝ,        HasFDerivAt g ℓ w ∧ ℓ ≠ 0def HasC1CurveChart (S : Set Plane) (v : Plane) : Prop :=  ∃ U : Set Plane, IsOpen U ∧ v ∈ U ∧ v ∈ S ∧ ∃ γ : ℝ → Plane, ∃ θ : Plane → ℝ,    ContDiff ℝ 1 γ ∧ ContinuousOn θ U ∧    (∀ t, γ t ∈ S ∩ U) ∧ (∀ t, θ (γ t)=t) ∧    (∀ z ∈ S ∩ U, γ (θ z)=z) ∧    (∀ t, ∃ d : Plane, HasDerivAt γ d t ∧ d ≠ 0)theorem manuscriptMain (Ω : Type) [MeasurableSpace Ω] (P : Measure Ω) (τ : Edge → Ω → ℝ)    (henv : ExponentialEnvironment P τ) :    ∃ μ : Plane → ℝ, TimeConstantNorm μ ∧ IsTimeConstant P τ μ ∧      DifferentiableAwayFromOrigin μ ∧ UniqueNormalizedSupports μ ∧ C1UnitSphere μ ∧      (∀ v ∈ frontier {z | μ z ≤ 1}, ∃! L : Set Plane,        ∃ g : Plane →L[ℝ] ℝ, g ≠ 0 ∧ (∀ z, μ z ≤ 1 → g z ≤ g v) ∧ L={z | g z=g v}) ∧      (∀ v ∈ frontier {z | μ z ≤ 1}, HasC1CurveChart (frontier {z | μ z ≤ 1}) v) := by  sorryend PlanarFPPend 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.

Strict convexity and differentiability of the planar exponential first-passage limit shape

Theorem (The exponential limit shape)

For independent exponential edge weights of any rate \lambda>0, the norm \mu is Fréchet differentiable on \R^2\setminus\{0\}, and its unit ball is strictly convex. In particular,

\mu((1-t)x+ty)<1 \quad\text{if }x,y\in\partial\Bcal,\ x\ne y,\ 0<t<1,

and \partial\Bcal is a C^1 curve with a unique supporting line at every point.

Section 1, Introduction Read it in the paper TeX source, line 21

Show the TeX

For independent exponential edge weights of any rate $\lambda>0$,
the norm $\mu$ is Fr\'echet differentiable on $\R^2\setminus\{0\}$,
and its unit ball is strictly convex. In particular,
\[
 \mu((1-t)x+ty)<1
 \quad\text{if }x,y\in\partial\Bcal,\ x\ne y,\ 0<t<1,
\]
and $\partial\Bcal$ is a $C^1$ curve with a unique supporting line
at every point.

Theorem (Differentiability for Gamma weights)

For independent Gamma edge weights with any fixed shape \kappa>0 and rate \lambda>0, the norm \mu is Fréchet differentiable at every nonzero vector, and \partial\Bcal is a C^1 curve.

Section 1, Introduction Read it in the paper TeX source, line 44

Show the TeX

For independent Gamma edge weights with any fixed shape $\kappa>0$
and rate $\lambda>0$, the norm $\mu$ is Fr\'echet differentiable at
every nonzero vector, and $\partial\Bcal$ is a $C^1$ curve.

Configuration and file facts

File: lean/ComparatorChallenges/PlanarFirstPassage.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Probability/FirstPassage/BoundaryCurve.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.Algebra.BigOperators.Group.List.Basic, Mathlib.Algebra.Group.Prod, Mathlib.Analysis.Calculus.ContDiff.Defs, Mathlib.Analysis.Calculus.Deriv.Basic, Mathlib.Analysis.Calculus.FDeriv.Basic, Mathlib.Analysis.InnerProductSpace.PiL2, Mathlib.Analysis.LocallyConvex.WithSeminorms, Mathlib.Probability.Distributions.Exponential, Mathlib.Probability.Independence.Basic
Imports from outside Mathlib: None
Local vocabulary: 17 local vocabulary declarations
Lines of Lean, without comments or blank lines: 64
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