Lean statement

Endpoint gradient bound for the centered disk maximal operator

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.CenteredDiskEndpoint.main_endpoint

Lean statement file: DiskMaximal.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 085 link it.

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.

def EndpointStatement : Prop :=  ∃ C : ℝ, 0 ≤ C ∧    ∀ (f : Plane → ℝ) (g : Plane → Plane),      Integrable f → Integrable g → HasWeakGradient f g →      (∀ᵐ x, maximal f x ≠ ⊤) ∧      LocallyIntegrable (maximalReal f) ∧      InW11Loc (maximalReal f) ∧      ∃ G : Plane → Plane, Integrable G ∧        HasWeakGradient (maximalReal f) G ∧ gradientNorm G ≤ C * gradientNorm g⋮theorem main_endpoint : EndpointStatement := 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.CenteredDiskEndpoint.Plane line 12
```
abbrev Plane := EuclideanSpace ℝ (Fin 2)
```

definition OAI.CenteredDiskEndpoint.absoluteDiskAverage line 15

def absoluteDiskAverage (f : Plane → ℝ) (x : Plane) (r : ℝ) : ℝ≥0∞ :=  (∫⁻ y in Metric.ball x r, ENNReal.ofReal |f y|) /    ENNReal.ofReal (Real.pi * r ^ 2)

definition OAI.CenteredDiskEndpoint.maximal line 20

def maximal (f : Plane → ℝ) (x : Plane) : ℝ≥0∞ :=  ⨆ (r : ℝ) (_ : 0 < r), absoluteDiskAverage f x r

definition OAI.CenteredDiskEndpoint.maximalReal line 24

def maximalReal (f : Plane → ℝ) (x : Plane) : ℝ := (maximal f x).toReal

definition OAI.CenteredDiskEndpoint.testPartial line 27

def testPartial (φ : Plane → ℝ) (i : Fin 2) (x : Plane) : ℝ :=  fderiv ℝ φ x (EuclideanSpace.single i 1)

definition OAI.CenteredDiskEndpoint.HasWeakGradient line 31

def HasWeakGradient (f : Plane → ℝ) (g : Plane → Plane) : Prop :=  ∀ (φ : Plane → ℝ), ContDiff ℝ ∞ φ → HasCompactSupport φ →    ∀ i : Fin 2, (∫ x, f x * testPartial φ i x) = -∫ x, g x i * φ x

definition OAI.CenteredDiskEndpoint.InW11Loc line 36

def InW11Loc (f : Plane → ℝ) : Prop :=  LocallyIntegrable f ∧    ∃ g : Plane → Plane, LocallyIntegrable g ∧ HasWeakGradient f g

definition OAI.CenteredDiskEndpoint.gradientNorm line 41

def gradientNorm (g : Plane → Plane) : ℝ := ∫ x, ‖g x‖

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: represented, valued, locally, integrable, functions, distributional, loc, derivatives and two more. Found in the paper’s title, which its main theorem states: gradient, disk, maximal, endpoint, centered and bound. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: belongs, mf, globally, nabla and constant.

Theorem 1.1 of An Endpoint Gradient Bound for the Centered Disk Maximal Operator

Theorem

There is an absolute constant C<\infty such that, for every real-valued f\in W^{1,1}(\R^2), the function Mf is locally integrable, belongs to W^{1,1}_{\mathrm{loc}}(\R^2), and satisfies

\int_{\R^2}|\nabla Mf|\,\dd x \le C\int_{\R^2}|\nabla f|\,\dd x.

In particular, its distributional derivatives are represented by globally integrable functions.

Section 1, Introduction Read it in the paper TeX source, line 19

Show the TeX

There is an absolute constant $C<\infty$ such that, for every
real-valued $f\in W^{1,1}(\R^2)$, the function $Mf$ is locally
integrable, belongs to $W^{1,1}_{\mathrm{loc}}(\R^2)$, and satisfies
\begin{equation}
 \int_{\R^2}|\nabla Mf|\,\dd x
 \le C\int_{\R^2}|\nabla f|\,\dd x.
\end{equation}
In particular, its distributional derivatives are represented by
globally integrable functions.

Other possible paper matches, closest wording first

Corollary 6.3 of An Endpoint Gradient Bound for the Centered Disk Maximal Operator. Words of the Lean statement, its names split into words, that also appear in the theorem: inputs, bounded, integrable, functions, distributional, finite and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: almost, mf, globally and constant.
Proposition 1.2 of An Endpoint Gradient Bound for the Centered Disk Maximal Operator. Words of the Lean statement, its names split into words, that also appear in the theorem: including, radius, finite, independent, real and absolute. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one, nabla and constant.
Proposition 4.3 of An Endpoint Gradient Bound for the Centered Disk Maximal Operator. Words of the Lean statement, its names split into words, that also appear in the theorem: independent, derivative, real and absolute. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant.
Proposition 3.3 of An Endpoint Gradient Bound for the Centered Disk Maximal Operator. Words of the Lean statement, its names split into words, that also appear in the theorem: support, radius and absolute. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nabla and constants.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 277 s; peak memory 6.5 GiB).
Reported 11 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 085’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: None; the BV extension is not claimed (docs). Weak gradients are encoded by hand (HasWeakGradient against C_c^∞ test functions), which is the standard definition.
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 formalization proves that every real $f\in W^{1,1}(\mathbb R^2)$ has a maximal function that is finite almost everywhere, belongs to $W^{1,1}_{\mathrm{loc}}$, and has a globally integrable weak gradient satisfying $\|\nabla Mf\|_1\le C\|\nabla f\|_1$ for one absolute constant $C$.”
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

63 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: 36 to its own declarations, 18 to Mathlib’s documentation. 10 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 main_endpoint 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 9 other declarations of the file; with the theorem itself they span 27 lines. 1 of them is named in the statement itself.

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

abbrev Planeline 12library
def absoluteDiskAverageline 15library
def maximalline 20library
def maximalRealline 24library
def testPartialline 27library
def HasWeakGradientline 31library
def InW11Locline 36library
def gradientNormline 41library
def EndpointStatementline 46in the statementlibrary
theorem main_endpointline 56selected theoremlibrary

import Mathlibnamespace OAIopen MeasureTheory Setopen scoped ENNReal NNReal Topology ContDiffnoncomputable sectionnamespace CenteredDiskEndpoint/-- The Euclidean plane, not the product space with its max norm. -/abbrev Plane := EuclideanSpace ℝ (Fin 2)/-- Absolute disk average, retaining possible infinite values. -/def absoluteDiskAverage (f : Plane → ℝ) (x : Plane) (r : ℝ) : ℝ≥0∞ :=  (∫⁻ y in Metric.ball x r, ENNReal.ofReal |f y|) /    ENNReal.ofReal (Real.pi * r ^ 2)/-- The actual centered disk maximal function, over every positive real radius. -/def maximal (f : Plane → ℝ) (x : Plane) : ℝ≥0∞ :=  ⨆ (r : ℝ) (_ : 0 < r), absoluteDiskAverage f x r/-- Real representative; the endpoint also asserts that `maximal f` is finite a.e. -/def maximalReal (f : Plane → ℝ) (x : Plane) : ℝ := (maximal f x).toReal/-- The i-th coordinate derivative of a smooth test function. -/def testPartial (φ : Plane → ℝ) (i : Fin 2) (x : Plane) : ℝ :=  fderiv ℝ φ x (EuclideanSpace.single i 1)/-- Distributional first derivatives represented by a vector-valued function. -/def HasWeakGradient (f : Plane → ℝ) (g : Plane → Plane) : Prop :=  ∀ (φ : Plane → ℝ), ContDiff ℝ ∞ φ → HasCompactSupport φ →    ∀ i : Fin 2, (∫ x, f x * testPartial φ i x) = -∫ x, g x i * φ x/-- Local W^(1,1), expressed using the same distributional gradient convention. -/def InW11Loc (f : Plane → ℝ) : Prop :=  LocallyIntegrable f ∧    ∃ g : Plane → Plane, LocallyIntegrable g ∧ HasWeakGradient f g/-- The gradient L1 norm in Euclidean norm. -/def gradientNorm (g : Plane → Plane) : ℝ := ∫ x, ‖g x‖/-- Exact endpoint, including finiteness before passing to the real representative.Quantifying the input gradient makes the norm bound independent of a choice ofrepresentative of its weak derivatives. -/def EndpointStatement : Prop :=  ∃ C : ℝ, 0 ≤ C ∧    ∀ (f : Plane → ℝ) (g : Plane → Plane),      Integrable f → Integrable g → HasWeakGradient f g →      (∀ᵐ x, maximal f x ≠ ⊤) ∧      LocallyIntegrable (maximalReal f) ∧      InW11Loc (maximalReal f) ∧      ∃ G : Plane → Plane, Integrable G ∧        HasWeakGradient (maximalReal f) G ∧ gradientNorm G ≤ C * gradientNorm gtheorem main_endpoint : EndpointStatement := by  sorryend CenteredDiskEndpointendend 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.

An Endpoint Gradient Bound for the Centered Disk Maximal Operator

Theorem

There is an absolute constant C<\infty such that, for every real-valued f\in W^{1,1}(\R^2), the function Mf is locally integrable, belongs to W^{1,1}_{\mathrm{loc}}(\R^2), and satisfies

\int_{\R^2}|\nabla Mf|\,\dd x \le C\int_{\R^2}|\nabla f|\,\dd x.

In particular, its distributional derivatives are represented by globally integrable functions.

Section 1, Introduction Read it in the paper TeX source, line 19

Show the TeX

There is an absolute constant $C<\infty$ such that, for every
real-valued $f\in W^{1,1}(\R^2)$, the function $Mf$ is locally
integrable, belongs to $W^{1,1}_{\mathrm{loc}}(\R^2)$, and satisfies
\begin{equation}
 \int_{\R^2}|\nabla Mf|\,\dd x
 \le C\int_{\R^2}|\nabla f|\,\dd x.
\end{equation}
In particular, its distributional derivatives are represented by
globally integrable functions.

Proposition (Signed finite-band estimate)

There is an absolute constant C such that, for every real g\in C_c^\infty(\R^2) and 0<a\le b with b/a a power of two, including one,

\int_{\R^2}|\nabla S_{a,b}g|\,\dd x \le C\int_{\R^2}|\nabla g|\,\dd x.

The constant is independent of a, b, and the number of dyadic radius scales between them.

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

Show the TeX

There is an absolute constant $C$ such that, for every real
$g\in C_c^\infty(\R^2)$ and $0<a\le b$ with $b/a$ a power of two, including one,
\begin{equation}
 \int_{\R^2}|\nabla S_{a,b}g|\,\dd x
 \le C\int_{\R^2}|\nabla g|\,\dd x.
\end{equation}
The constant is independent of $a$, $b$, and the number of dyadic
radius scales between them.

Configuration and file facts

File: lean/ComparatorChallenges/DiskMaximal.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 085 link it.
Solution module: OAI.Analysis.DiskMaximal.Main
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: 9 local vocabulary declarations
Lines of Lean, without comments or blank lines: 27
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.

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