Lean statement

Brennan and inverse-square integral-means bounds

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.Brennan.main_theorem

Lean statement file: Brennan.leanShow on the mapFamily 072. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorems 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 MainStatement : Prop :=  (∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ f : ℂ → ℂ, Schlicht f →    ∀ r : ℝ, 1 / 2 ≤ r → r < 1 →      integralMean f (-2) r ≤ C * (1 - r) ^ (-1 - ε)) ∧  spectrum (-2) = 1 ∧  (∀ W : Set ℂ, IsOpen W → IsConnected W → SimplyConnectedSpace W →    SphericalBoundaryNontrivial W → ∀ φ : ℂ → ℂ,      DifferentiableOn ℂ φ W → Set.BijOn φ W disk →      ∀ s : ℝ, 4 / 3 < s → s < 4 →        IntegrableOn (fun z => ‖deriv φ z‖ ^ s) W volume) ∧  (∀ f : ℂ → ℂ, UnivalentOn f disk → ∀ t : ℝ,    -2 < t → t < 2 / 3 →      IntegrableOn (fun z => ‖deriv f z‖ ^ t) disk volume)⋮theorem main_theorem : MainStatement := 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.

definition OAI.Brennan.disk line 10
```
def disk : Set ℂ := Metric.ball 0 1
```

definition OAI.Brennan.UnivalentOn line 12

def UnivalentOn (f : ℂ → ℂ) (U : Set ℂ) : Prop :=  DifferentiableOn ℂ f U ∧ Set.InjOn f U

definition OAI.Brennan.Schlicht line 15

def Schlicht (f : ℂ → ℂ) : Prop :=  UnivalentOn f disk ∧ f 0 = 0 ∧ deriv f 0 = 1

definition OAI.Brennan.integralMean line 18

def integralMean (f : ℂ → ℂ) (t r : ℝ) : ℝ :=  (2 * Real.pi)⁻¹ * ∫ θ in -Real.pi..Real.pi,    ‖deriv f ((r : ℂ) * Complex.exp ((θ : ℂ) * Complex.I))‖ ^ t

definition OAI.Brennan.beta line 22

def beta (f : ℂ → ℂ) (t : ℝ) : EReal :=  Filter.limsup (fun r : ℝ =>    ((Real.log (integralMean f t r) / Real.log (1 / (1 - r)) : ℝ) : EReal))    (𝓝[<] (1 : ℝ))

definition OAI.Brennan.spectrum line 27

def spectrum (t : ℝ) : EReal :=  sSup {b | ∃ f : ℂ → ℂ, Schlicht f ∧ beta f t = b}

definition OAI.Brennan.SphericalBoundaryNontrivial line 30

def SphericalBoundaryNontrivial (W : Set ℂ) : Prop :=  Set.Nontrivial (frontier (((↑) : ℂ → OnePoint ℂ) '' W))

Possible paper matches for the parts of the Lean statement

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

Statements in Brennan’s conjecture and sharp inverse-square integral means come first because the release’s name for this Lean statement shares words with that paper’s title.

The Lean statement joins several assertions with “and”. Atlas selected possible matches to two paper results from their shared wording, listed below. This does not establish that their mathematical meanings agree.

Theorem 1.1 of Brennan’s conjecture and sharp inverse-square integral means

Words of the Lean statement, its names split into words, that also appear in the theorem: 2/3, connected, boundary, points, univalent, simply, means, 4/3 and one more. Found in the paper’s title, which its main theorem states: integral. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: square, brennans, map and inverse.

Theorem (Brennan’s conjecture and inverse-square means)

For every \varepsilon>0 there is C_\varepsilon<\infty such that

M_{-2}[f'](r)\le C_\varepsilon(1-r)^{-1-\varepsilon} \qquad\left(f\in\SSS,\ \frac12\le r<1\right).

Moreover, B_{\SSS}(-2)=1.

Let W\subset\CC be any simply connected domain whose boundary in the Riemann sphere contains at least two points. Every conformal bijection \varphi:W\to\DD satisfies

\int_W|\varphi'(z)|^s\dd A(z)<\infty \qquad\left(\frac43<s<4\right).

Equivalently, every univalent map f:\DD\to\CC satisfies

\int_{\DD}|f'(w)|^t\dd A(w)<\infty \qquad\left(-2<t<\frac23\right).

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

Show the TeX

For every $\varepsilon>0$ there is $C_\varepsilon<\infty$ such that
\begin{equation}
 M_{-2}[f'](r)\le C_\varepsilon(1-r)^{-1-\varepsilon}
 \qquad\left(f\in\SSS,\ \frac12\le r<1\right).
\end{equation}
Moreover, $B_{\SSS}(-2)=1$.

Let $W\subset\CC$ be any simply connected domain whose boundary in
the Riemann sphere contains at least two points. Every conformal
bijection $\varphi:W\to\DD$ satisfies
\[
 \int_W|\varphi'(z)|^s\dd A(z)<\infty
 \qquad\left(\frac43<s<4\right).
\]
Equivalently, every univalent map $f:\DD\to\CC$ satisfies
\[
 \int_{\DD}|f'(w)|^t\dd A(w)<\infty
 \qquad\left(-2<t<\frac23\right).
\]

Corollary 1.2 of Brennan’s conjecture and sharp inverse-square integral means

Words of the Lean statement, its names split into words, that also appear in the theorem: integrability, 2/3 and 4/3. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: inverse and area.

Corollary (Sharpness of the open range)

The Koebe function f(z)=z/(1-z)^2 belongs to \SSS and satisfies

\int_{\DD}|f'|^{-2}\dd A=\infty, \qquad \int_{\DD}|f'|^{2/3}\dd A=\infty.

Its inverse therefore fails area integrability at both s=4 and s=4/3.

Section 1, Introduction Read it in the paper TeX source, line 82

Show the TeX

The Koebe function $f(z)=z/(1-z)^2$ belongs to $\SSS$ and satisfies
\[
 \int_{\DD}|f'|^{-2}\dd A=\infty,
 \qquad
 \int_{\DD}|f'|^{2/3}\dd A=\infty.
\]
Its inverse therefore fails area integrability at both $s=4$ and $s=4/3$.

Other possible paper matches, closest wording first

Proposition 2.3 of A strict inverse-first-power bound for univalent functions. Words of the Lean statement, its names split into words, that also appear in the theorem: means, disk, beta and 1/2.
Corollary 7.4 of Brennan’s conjecture and sharp inverse-square integral means. Words of the Lean statement, its names split into words, that also appear in the theorem: connected, boundary, points, simply and 4/3. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: map and bounds.
Proposition 2.4 of Brennan’s conjecture and sharp inverse-square integral means. Words of the Lean statement, its names split into words, that also appear in the theorem: disk and 1/2.
Theorem 1.1 of A strict inverse-first-power bound for univalent functions. Words of the Lean statement, its names split into words, that also appear in the theorem: 1/2. Found in the paper’s title, which its main theorem states: univalent. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bound and inverse.
Proposition 3.1 of A strict inverse-first-power bound for univalent functions. Words of the Lean statement, its names split into words, that also appear in the theorem: log and beta.
Corollary 7.1 of Brennan’s conjecture and sharp inverse-square integral means. Words of the Lean statement, its names split into words, that also appear in the theorem: real and 1/2.

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 517 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 072’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 disagree 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: Brennan's conjecture and the strict Kraetzer bound are stated; the sharp spectrum identity B_S(t) = |t|-1 is stated only at t = -2, not for all t <= -2.
Definitions the group asks readers to compare with the paper: Custom integral-means spectrum: beta f t := limsup_{r→1⁻} log(integralMean f t r) / log(1/(1-r)) and spectrum t := sSup {b | ∃ f, Schlicht f ∧ beta f t = b} in EReal (standard definition, checked); kraetzerPrediction p := if |p| ≤ 2 then p^2/4 else |p| - 1
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: The headline claims B_S(t) = |t| − 1 for every t ≤ −2. Lean states it only at t = −2 (Brennan.lean:37). The manuscript states the full range as a corollary (sections/06-integrability.tex:56), so the headline matches the paper and the Lean is narrower by that corollary. Mathematically the gap is a short classical step (B_S(t)+t is nondecreasing by the distortion theorem, and the Koebe function gives the lower bound), but it is not stated.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization establishes this interval, along with area integrability of $|f'|^t$ for univalent disk maps when $-2<t<2/3$.”
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

53 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: 14 to its own declarations, 34 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 main_theorem 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 31 lines. 1 of them is named in the statement itself.

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

def diskline 10library
def UnivalentOnline 12library
def Schlichtline 15library
def integralMeanline 18library
def betaline 22library
def spectrumline 27library
def SphericalBoundaryNontrivialline 30library
def MainStatementline 33in the statementlibrary
theorem main_theoremline 47selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen Set MeasureTheory Filteropen scoped Topologynamespace Brennandef disk : Set ℂ := Metric.ball 0 1def UnivalentOn (f : ℂ → ℂ) (U : Set ℂ) : Prop :=  DifferentiableOn ℂ f U ∧ Set.InjOn f Udef Schlicht (f : ℂ → ℂ) : Prop :=  UnivalentOn f disk ∧ f 0 = 0 ∧ deriv f 0 = 1def integralMean (f : ℂ → ℂ) (t r : ℝ) : ℝ :=  (2 * Real.pi)⁻¹ * ∫ θ in -Real.pi..Real.pi,    ‖deriv f ((r : ℂ) * Complex.exp ((θ : ℂ) * Complex.I))‖ ^ tdef beta (f : ℂ → ℂ) (t : ℝ) : EReal :=  Filter.limsup (fun r : ℝ =>    ((Real.log (integralMean f t r) / Real.log (1 / (1 - r)) : ℝ) : EReal))    (𝓝[<] (1 : ℝ))def spectrum (t : ℝ) : EReal :=  sSup {b | ∃ f : ℂ → ℂ, Schlicht f ∧ beta f t = b}def SphericalBoundaryNontrivial (W : Set ℂ) : Prop :=  Set.Nontrivial (frontier (((↑) : ℂ → OnePoint ℂ) '' W))def MainStatement : Prop :=  (∀ ε : ℝ, 0 < ε → ∃ C : ℝ, ∀ f : ℂ → ℂ, Schlicht f →    ∀ r : ℝ, 1 / 2 ≤ r → r < 1 →      integralMean f (-2) r ≤ C * (1 - r) ^ (-1 - ε)) ∧  spectrum (-2) = 1 ∧  (∀ W : Set ℂ, IsOpen W → IsConnected W → SimplyConnectedSpace W →    SphericalBoundaryNontrivial W → ∀ φ : ℂ → ℂ,      DifferentiableOn ℂ φ W → Set.BijOn φ W disk →      ∀ s : ℝ, 4 / 3 < s → s < 4 →        IntegrableOn (fun z => ‖deriv φ z‖ ^ s) W volume) ∧  (∀ f : ℂ → ℂ, UnivalentOn f disk → ∀ t : ℝ,    -2 < t → t < 2 / 3 →      IntegrableOn (fun z => ‖deriv f z‖ ^ t) disk volume)theorem main_theorem : MainStatement := by  sorryend Brennanendend 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.

Brennan’s conjecture and sharp inverse-square integral means

Theorem (Brennan’s conjecture and inverse-square means)

For every \varepsilon>0 there is C_\varepsilon<\infty such that

M_{-2}[f'](r)\le C_\varepsilon(1-r)^{-1-\varepsilon} \qquad\left(f\in\SSS,\ \frac12\le r<1\right).

Moreover, B_{\SSS}(-2)=1.

Let W\subset\CC be any simply connected domain whose boundary in the Riemann sphere contains at least two points. Every conformal bijection \varphi:W\to\DD satisfies

\int_W|\varphi'(z)|^s\dd A(z)<\infty \qquad\left(\frac43<s<4\right).

Equivalently, every univalent map f:\DD\to\CC satisfies

\int_{\DD}|f'(w)|^t\dd A(w)<\infty \qquad\left(-2<t<\frac23\right).

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

Show the TeX

For every $\varepsilon>0$ there is $C_\varepsilon<\infty$ such that
\begin{equation}
 M_{-2}[f'](r)\le C_\varepsilon(1-r)^{-1-\varepsilon}
 \qquad\left(f\in\SSS,\ \frac12\le r<1\right).
\end{equation}
Moreover, $B_{\SSS}(-2)=1$.

Let $W\subset\CC$ be any simply connected domain whose boundary in
the Riemann sphere contains at least two points. Every conformal
bijection $\varphi:W\to\DD$ satisfies
\[
 \int_W|\varphi'(z)|^s\dd A(z)<\infty
 \qquad\left(\frac43<s<4\right).
\]
Equivalently, every univalent map $f:\DD\to\CC$ satisfies
\[
 \int_{\DD}|f'(w)|^t\dd A(w)<\infty
 \qquad\left(-2<t<\frac23\right).
\]

Corollary (Sharpness of the open range)

The Koebe function f(z)=z/(1-z)^2 belongs to \SSS and satisfies

\int_{\DD}|f'|^{-2}\dd A=\infty, \qquad \int_{\DD}|f'|^{2/3}\dd A=\infty.

Its inverse therefore fails area integrability at both s=4 and s=4/3.

Section 1, Introduction Read it in the paper TeX source, line 82

Show the TeX

The Koebe function $f(z)=z/(1-z)^2$ belongs to $\SSS$ and satisfies
\[
 \int_{\DD}|f'|^{-2}\dd A=\infty,
 \qquad
 \int_{\DD}|f'|^{2/3}\dd A=\infty.
\]
Its inverse therefore fails area integrability at both $s=4$ and $s=4/3$.

A strict inverse-first-power bound for univalent functions

Theorem (Uniform inverse-first-power bound)

There exist constants 0<\varepsilon<1/4 and C<\infty such that

M_{-1}[f'](r)\le C(1-r)^{-1/4+\varepsilon} \qquad\text{for every }f\in\SSS\text{ and every }1/2\le r<1.

Consequently B_b(-1)<1/4, and Kraetzer’s conjectured identity (eq:kraetzer) is false.

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

Show the TeX

There exist constants \(0<\varepsilon<1/4\) and \(C<\infty\) such that
\begin{equation}
 M_{-1}[f'](r)\le C(1-r)^{-1/4+\varepsilon}
 \qquad\text{for every }f\in\SSS\text{ and every }1/2\le r<1.
\end{equation}
Consequently \(B_b(-1)<1/4\), and Kraetzer's conjectured identity
\eqref{eq:kraetzer} is false.

Configuration and file facts

File: lean/ComparatorChallenges/Brennan.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/IntegralMeans/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: 8 local vocabulary declarations
Lines of Lean, without comments or blank lines: 31
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