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Brennan’s conjecture and the integral-means spectrum

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves Brennan’s conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, \(|\phi'|^s\) is area-integrable for \(4/3\lt s\lt 4\). The sharp universal integral-means identity is \(B_{\mathcal S}(t)=|t|-1\) for t ≤ −2. A strict bound \(B_b(-1)\lt 1/4\) for bounded univalent functions disproves Kraetzer’s prediction at that parameter. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

The two independent assessments disagree on how much of the result described above is expressed by the Lean statements. Read the assessments

The paper states

Theorem 1.1 (Brennan’s conjecture and inverse-square means)

In Brennan’s conjecture and sharp inverse-square integral means, 24 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

The other possible matches are Theorem 1.1 (Uniform inverse-first-power bound) in A strict inverse-first-power bound for univalent functions () and Corollary 1.2 (Sharpness of the open range) in the same paper ().

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).
Reported proof verification
Three Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
Two current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Brennan’s conjecture and sharp inverse-square integral means

24 September 2026, current edition.

Read it herePDF

We prove Brennan’s conjecture: if a simply connected plane domain admits a conformal bijection φ onto the unit disk, then \(|\varphi'|^s\) is area-integrable for every \(4/3\lt s\lt 4\). We also prove the sharp inverse-square integral-means exponent \(B_{\mathcal S}(-2)=1\) for the normalized schlicht class \(\mathcal S\).

A strict inverse-first-power bound for univalent functions

24 September 2026, current edition.

Read it herePDF

We prove a uniform upper bound for inverse-first-power integral means of normalized univalent disk maps with exponent strictly below 1/4. Consequently, the bounded universal integral-means spectrum satisfies \(B_b(-1)\lt 1/4\), disproving Kraetzer’s conjectured spectrum at p = −1.

Three Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

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

Brennan

The paper states

It is paired with Theorem 1.1 and Corollary 1.2, each shown below.

Theorem 1.1 (Brennan’s conjecture and inverse-square means)

In Brennan’s conjecture and sharp inverse-square integral means, 24 September 2026

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).

The words it shares with Theorem 1.1: 2/3, connected, boundary, points, univalent, simply, means, 4/3, 1/2, integral, square, brennans, map, inverse; with Corollary 1.2: integrability, 2/3, 4/3, inverse, area.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Brennan and inverse-square integral-means bounds

Brennan.lean, the release lists it as a main result.

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

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

BrennanSharp

The paper states

Theorem 1.1 (Brennan’s conjecture and inverse-square means)

In Brennan’s conjecture and sharp inverse-square integral means, 24 September 2026

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).

The words they share: boundary, brennan, conjecture, connected, domain, inverse, means, simply, square, univalent.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Sharp endpoint divergence

BrennanSharp.lean, the release lists it as a main result.

def SharpEndpointStatement : Prop :=  Schlicht koebeMap ∧  areaMoment koebeMap disk (-2) = ⊤ ∧  areaMoment koebeMap disk (2 / 3) = ⊤ ∧  ¬ IntegrableOn (fun z => ‖deriv koebeInverse z‖ ^ (4 : ℝ)) koebeDomain volume ∧  ¬ IntegrableOn (fun z => ‖deriv koebeInverse z‖ ^ (4 / 3 : ℝ)) koebeDomain volumetheorem sharp_endpoints : SharpEndpointStatement := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

StrictMeans

The paper states

The words they share: 1/4, kraetzers, bound, 1/2, univalent, power, first, constants, varepsilon, inverse.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Strict inverse-first-power bound and consequences

StrictMeans.lean, the release lists it as a main result.

theorem main :    (∃ ε C : ℝ, 0 < ε ∧ ε < 1 / 4 ∧      ∀ (f : ℂ → ℂ), NormalizedUnivalent f →        ∀ r : ℝ, 1 / 2 ≤ r → r < 1 →          integralMean (-1) f r ≤ C * (1 - r) ^ (-(1 / 4 : ℝ) + ε)) ∧    boundedSpectrum (-1) < (1 / 4 : EReal) ∧    (¬ ∀ p : ℝ, boundedSpectrum p = kraetzerPrediction p) := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

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

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: 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.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).

The gap they name: The headline claims B_S(t) = |t| − 1 for every t ≤ −2. Lean states it only at t = −2. The manuscript states the full range as a corollary, 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.

“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

Brennan’s conjecture asserts that $|\phi'|^s$ is area-integrable for $4/3<s<4$ when $\phi$ conformally maps a simply connected plane domain with nontrivial spherical boundary onto the disk. The formalization establishes this interval, along with area integrability of $|f'|^t$ for univalent disk maps when $-2<t<2/3$.

It also gives the uniform inverse-square radial-mean bound with exponent $-1-\varepsilon$ for every $\varepsilon>0$ and the spectrum value $B_{\mathcal S}(-2)=1$. The Koebe map and its inverse give divergence at all four boundary exponents. Kraetzer’s proposed integral-means spectrum predicts $B_b(-1)=1/4$ for bounded univalent maps. The formalized result proves $B_b(-1)<1/4$, contradicting that prediction. The underlying estimate gives constants $0<\varepsilon<1/4$ and $C<\infty$ such that the normalized circular mean of $|f'|^{-1}$ is at most $C(1-r)^{-1/4+\varepsilon}$ for every normalized univalent disk map and $1/2\le r<1$. The bound requires neither bounded image nor boundary regularity, and no numerical value of $\varepsilon$ is specified.

AI-drafted comparisons, with no recorded human review

  • Brennan and inverse-square integral-means bounds Brennan

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: Brennan’s conjecture for 4/3 < s < 4, the equivalent range −2 < t < 2/3 for univalent disk maps, the inverse-square integral-means bound and the value B_S(−2) = 1.

  • Sharp endpoint divergence BrennanSharp

    Unreviewed AI draft: states a supporting result. States the paper’s sharpness corollary: the Koebe function and its inverse make the integrals diverge at the four endpoint exponents. This complements the main theorem rather than following from it.

  • Strict inverse-first-power bound and consequences StrictMeans

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem of the strict-bound paper: constants 0 < ε < 1/4 and C with the circular mean of |f′|^{-1} at most C(1 − r)^{-1/4+ε}, with the consequence B_b(−1) < 1/4 against Kraetzer’s prediction.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Brennan and inverse-square integral-means bounds Brennan

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 517 s; peak memory 6.5 GiB).11 October 2026Original report

Sharp endpoint divergence BrennanSharp

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 229 s; peak memory 6.3 GiB).11 October 2026Original report

Strict inverse-first-power bound and consequences StrictMeans

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

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Offer to write an explanationCompare one Lean statement with its paper: Brennan, BrennanSharp, StrictMeansSuggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

  • Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source

    A Formal Conjectures maintainer, after source inspection, links the integral means result to Brennan’s conjecture at the endpoint value.

Questions awaiting review

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Identify the theorem the catalogue describes

  • Family 072 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Address a question raised in an independent assessment

Compare the proposed paper and Lean matches

  • Family 072 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form