Lean statement

Quantitative higher-codimension Riesz rectifiability

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.RieszRectifiability.quantitative_higher_codimension_riesz_rectifiability

Lean statement file: RieszQuantitative.leanShow on the mapFamily 081. 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.

def QuantitativeFullStatement : Prop :=  ∀ d n : ℕ, 4 ≤ d → 2 ≤ n → n + 2 ≤ d →    ∀ C_AD : ℝ, 1 ≤ C_AD → ∀ C_R : ℝ≥0,      ∃ θ : ℝ, 0 < θ ∧ ∃ M : ℝ≥0,        ∀ μ : Measure (Ambient d), μ.Regular →          ADRegularWithConstant n C_AD μ →          RieszL2BoundedWithConstant n C_R μ →          BallImageConclusion n μ θ M⋮theorem quantitative_higher_codimension_riesz_rectifiability : QuantitativeFullStatement := 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.RieszRectifiability.Ambient line 10

abbrev Ambient (d : ℕ) := EuclideanSpace ℝ (Fin d)

definition OAI.RieszRectifiability.AdmissibleRadius line 12

def AdmissibleRadius {d : ℕ} (μ : Measure (Ambient d)) (r : ℝ) : Prop :=  0 < r ∧ ENNReal.ofReal r ≤ Metric.ediam μ.support

definition OAI.RieszRectifiability.kernel line 15

def kernel {d : ℕ} (n : ℕ) (x y : Ambient d) : Ambient d :=  (‖x - y‖ ^ (n + 1))⁻¹ • (x - y)

definition OAI.RieszRectifiability.truncated line 18

def truncated {d : ℕ} (n : ℕ) (μ : Measure (Ambient d))    (ε : ℝ) (f : Ambient d → ℝ) (x : Ambient d) : Ambient d :=  ∫ y in {y | ε < dist x y}, f y • kernel n x y ∂μ

definition OAI.RieszRectifiability.ADRegularWithConstant line 22

def ADRegularWithConstant {d : ℕ} (n : ℕ) (C : ℝ)    (μ : Measure (Ambient d)) : Prop :=  1 ≤ C ∧ ∀ x ∈ μ.support, ∀ r : ℝ, AdmissibleRadius μ r →    ENNReal.ofReal (r ^ n / C) ≤ μ (ball x r) ∧    μ (ball x r) ≤ ENNReal.ofReal (C * r ^ n)

definition OAI.RieszRectifiability.RieszL2BoundedWithConstant line 28

def RieszL2BoundedWithConstant {d : ℕ} (n : ℕ) (C : ℝ≥0)    (μ : Measure (Ambient d)) : Prop :=  ∀ ε : ℝ, 0 < ε → ∀ f : Ambient d → ℝ,    MemLp f 2 μ → MemLp (truncated n μ ε f) 2 μ ∧      eLpNorm (truncated n μ ε f) 2 μ ≤ (C : ℝ≥0∞) * eLpNorm f 2 μ

definition OAI.RieszRectifiability.BallImageConclusion line 34

def BallImageConclusion {d : ℕ} (n : ℕ) (μ : Measure (Ambient d))    (θ : ℝ) (M : ℝ≥0) : Prop :=  ∀ x ∈ μ.support, ∀ r : ℝ, AdmissibleRadius μ r →    ∃ g : (ball (0 : Ambient n) r) → Ambient d,      LipschitzWith M g ∧      ENNReal.ofReal (θ * r ^ n) ≤ μ (ball x r ∩ range g)

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: rectifiable, bound, ad, regular, riesz, measure, integers and constants. Found in the paper’s title, which its main theorem states: higher, codimension and uniform. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: ahlfors and david.

Theorem 1.3 of Riesz transforms and uniform rectifiability in higher codimension

Theorem

Let d\ge4 and 2\le n\le d-2 be integers. Let \mu be an n-Ahlfors–David regular Radon measure on \R^d satisfying (eq:riesz-bound). Then \mu is uniformly n-rectifiable in the sense of Definition def:ur. Its constants \theta and M can be chosen in terms of d,n,C_{\rm AD} and C_{\rm R} alone.

Section 1, The quantitative theorem and normalized pairings Read it in the paper TeX source, line 61

Show the TeX

Let $d\ge4$ and $2\le n\le d-2$ be integers. Let $\mu$ be an
$n$-Ahlfors--David regular Radon measure on $\R^d$ satisfying
\eqref{eq:riesz-bound}. Then $\mu$ is uniformly $n$-rectifiable in the
sense of Definition~\ref{def:ur}. Its constants $\theta$ and $M$ can
be chosen in terms of $d,n,C_{\rm AD}$ and $C_{\rm R}$ alone.

Other possible paper matches, closest wording first

Corollary 8.1 of Riesz transforms and uniform rectifiability in higher codimension. Words of the Lean statement, its names split into words, that also appear in the theorem: real, kernel, hard, rectifiability, norm, bound and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: operator, ahlfors and david.
Proposition 3.4 of Riesz transforms and uniform rectifiability in higher codimension. Words of the Lean statement, its names split into words, that also appear in the theorem: single, ball, truncated, hard, norm, bounded and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: operator and transform.
Theorem 7.3 of Riesz transforms and uniform rectifiability in higher codimension. Words of the Lean statement, its names split into words, that also appear in the theorem: images, lipschitz, ad, regular and constants. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive, pieces and david.
Theorem 6.4 of Riesz transforms and uniform rectifiability in higher codimension. Words of the Lean statement, its names split into words, that also appear in the theorem: radius, admissible, supported, uniform, measure, riesz and two more.
Proposition 6.3 of Riesz transforms and uniform rectifiability in higher codimension. Words of the Lean statement, its names split into words, that also appear in the theorem: natural, lipschitz, supported, riesz, integer and constants. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Proposition 5.1 of Riesz transforms and uniform rectifiability in higher codimension. Words of the Lean statement, its names split into words, that also appear in the theorem: conclusions, euclidean, lipschitz, supported, measure and constant.

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

One independent group reports that Comparator accepted a run for this Lean statement file: Carlos Toledo’s cert-machine. 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

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 081’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 for the headline; uniform n-rectifiability is encoded as big pieces of Lipschitz images (standard equivalent form). The variation / principal-value corollary is not stated.
Definitions the group asks readers to compare with the paper: ADRegularWithConstant, RieszL2BoundedWithConstant (real-valued f, vector-valued hard truncation truncated), BallImageConclusion are custom but standard in form; AdmissibleRadius uses Metric.ediam μ.support
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 proves that bounded Riesz transforms force quantitative uniform rectifiability in higher codimension. For $d\ge4$ and $2\le n\le d-2$, an $n$-Ahlfors–David regular measure whose positive hard truncations have one uniform $L^2$ operator bound has big pieces of Lipschitz images.”
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

62 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: 23 to its own declarations, 24 to Mathlib’s documentation. 9 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 quantitative_higher_codimension_riesz_rectifiability 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 8 other declarations of the file; with the theorem itself they span 34 lines. 1 of them is named in the statement itself.

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

abbrev Ambientline 10library
def AdmissibleRadiusline 12library
def kernelline 15library
def truncatedline 18library
def ADRegularWithConstantline 22library
def RieszL2BoundedWithConstantline 28library
def BallImageConclusionline 34library
def QuantitativeFullStatementline 42in the statementlibrary
theorem quantitative_higher_codimension_riesz_rectifiabilityline 57selected theoremlibrary

import Mathlibnamespace OAInamespace RieszRectifiabilitynoncomputable sectionopen MeasureTheory Set Metricopen scoped NNReal ENNRealabbrev Ambient (d : ℕ) := EuclideanSpace ℝ (Fin d)def AdmissibleRadius {d : ℕ} (μ : Measure (Ambient d)) (r : ℝ) : Prop :=  0 < r ∧ ENNReal.ofReal r ≤ Metric.ediam μ.supportdef kernel {d : ℕ} (n : ℕ) (x y : Ambient d) : Ambient d :=  (‖x - y‖ ^ (n + 1))⁻¹ • (x - y)def truncated {d : ℕ} (n : ℕ) (μ : Measure (Ambient d))    (ε : ℝ) (f : Ambient d → ℝ) (x : Ambient d) : Ambient d :=  ∫ y in {y | ε < dist x y}, f y • kernel n x y ∂μdef ADRegularWithConstant {d : ℕ} (n : ℕ) (C : ℝ)    (μ : Measure (Ambient d)) : Prop :=  1 ≤ C ∧ ∀ x ∈ μ.support, ∀ r : ℝ, AdmissibleRadius μ r →    ENNReal.ofReal (r ^ n / C) ≤ μ (ball x r) ∧    μ (ball x r) ≤ ENNReal.ofReal (C * r ^ n)def RieszL2BoundedWithConstant {d : ℕ} (n : ℕ) (C : ℝ≥0)    (μ : Measure (Ambient d)) : Prop :=  ∀ ε : ℝ, 0 < ε → ∀ f : Ambient d → ℝ,    MemLp f 2 μ → MemLp (truncated n μ ε f) 2 μ ∧      eLpNorm (truncated n μ ε f) 2 μ ≤ (C : ℝ≥0∞) * eLpNorm f 2 μdef BallImageConclusion {d : ℕ} (n : ℕ) (μ : Measure (Ambient d))    (θ : ℝ) (M : ℝ≥0) : Prop :=  ∀ x ∈ μ.support, ∀ r : ℝ, AdmissibleRadius μ r →    ∃ g : (ball (0 : Ambient n) r) → Ambient d,      LipschitzWith M g ∧      ENNReal.ofReal (θ * r ^ n) ≤ μ (ball x r ∩ range g)/-- A single mass fraction and Lipschitz bound work for all measures with fixed AD and Riesz bounds. -/def QuantitativeFullStatement : Prop :=  ∀ d n : ℕ, 4 ≤ d → 2 ≤ n → n + 2 ≤ d →    ∀ C_AD : ℝ, 1 ≤ C_AD → ∀ C_R : ℝ≥0,      ∃ θ : ℝ, 0 < θ ∧ ∃ M : ℝ≥0,        ∀ μ : Measure (Ambient d), μ.Regular →          ADRegularWithConstant n C_AD μ →          RieszL2BoundedWithConstant n C_R μ →          BallImageConclusion n μ θ Mendend RieszRectifiabilitynamespace RieszRectifiability/-- Class-uniform Lipschitz ball images from fixed AD and hard-truncation L² bounds. -/theorem quantitative_higher_codimension_riesz_rectifiability : QuantitativeFullStatement := by  sorryend RieszRectifiabilityend 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.

Riesz transforms and uniform rectifiability in higher codimension

Theorem

Section 1, The quantitative theorem and normalized pairings Read it in the paper TeX source, line 61

Show the TeX

Let $d\ge4$ and $2\le n\le d-2$ be integers. Let $\mu$ be an
$n$-Ahlfors--David regular Radon measure on $\R^d$ satisfying
\eqref{eq:riesz-bound}. Then $\mu$ is uniformly $n$-rectifiable in the
sense of Definition~\ref{def:ur}. Its constants $\theta$ and $M$ can
be chosen in terms of $d,n,C_{\rm AD}$ and $C_{\rm R}$ alone.

Proposition (Regular dyadic cells)

If D_E>0, there are a common \mu-null set N\subset E and families \mathcal D_k of Borel cells in E\setminus N such that:

(i)
each \mathcal D_k partitions E\setminus N, and cells of different generations are disjoint or nested;
(ii)
writing \ell(Q)=2^{-k} for Q\in\mathcal D_k, there are centers z_Q\in Q and constants 0<c_0\le1/2, C_0\ge2 for which
(E\setminus N)\cap B(z_Q,c_0\ell(Q))\subset Q \subset E\cap B(z_Q,C_0\ell(Q)),\qquad c_1\ell(Q)^n\le\mu(Q)\le C_1\ell(Q)^n;
(iii)
every non-top cell has one parent, and every cell has finitely many children. When D_E=\infty, all integer generations are available. When D_E<\infty, there is a largest scale \ell_0 with D_E\le\ell_0<2D_E, and finitely many cells at that scale.

All constants depend only on d,n,C_{\rm AD}.

Section 1, The quantitative theorem and normalized pairings Read it in the paper TeX source, line 270

Show the TeX

If $D_E>0$, there are a common $\mu$-null set $N\subset E$ and
families $\mathcal D_k$ of Borel cells in $E\setminus N$ such that:
\begin{enumerate}[label=(\roman*)]
\item each $\mathcal D_k$ partitions $E\setminus N$, and cells of
different generations are disjoint or nested;
\item writing $\ell(Q)=2^{-k}$ for $Q\in\mathcal D_k$, there are
centers $z_Q\in Q$ and constants $0<c_0\le1/2$, $C_0\ge2$ for which
\[
 (E\setminus N)\cap B(z_Q,c_0\ell(Q))\subset Q
 \subset E\cap B(z_Q,C_0\ell(Q)),\qquad
 c_1\ell(Q)^n\le\mu(Q)\le C_1\ell(Q)^n;
\]
\item every non-top cell has one parent, and every cell has finitely
many children. When
$D_E=\infty$, all integer generations are available. When
$D_E<\infty$, there is a largest scale $\ell_0$ with
$D_E\le\ell_0<2D_E$, and finitely many cells at that scale.
\end{enumerate}
All constants depend only on $d,n,C_{\rm AD}$.

Configuration and file facts

File: lean/ComparatorChallenges/RieszQuantitative.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/RieszRectifiability/Uniformity/Rectifiability.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: 8 local vocabulary declarations
Lines of Lean, without comments or blank lines: 34
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