Lean statement

Maximal triangular Hilbert transform bound

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.TriangularHilbert.main_estimate

Lean statement file: TriangularHilbert.leanShow on the mapFamily 082. 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 MainEstimate : Prop :=  ∃ C : ℝ≥0, ∀ F G : Plane → ℂ,    MemLp F 3 volume → MemLp G 3 volume →    (∀ᵐ z ∂volume, GoodPoint F G z) ∧    AEMeasurable (maximal F G) volume ∧    maximalNorm F G ≤ (C : ℝ≥0∞) * eLpNorm F 3 volume * eLpNorm G 3 volume⋮theorem main_estimate : MainEstimate := 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.TriangularHilbert.Plane line 12
```
abbrev Plane := ℝ × ℝ
```

structure OAI.TriangularHilbert.Endpoints line 14

structure Endpoints where  lower : ℝ  upper : ℝ  lower_pos : 0 < lower  lower_lt_upper : lower < upper

definition OAI.TriangularHilbert.annulus line 20

def annulus (ε R : ℝ) : Set ℝ := {t | ε < |t| ∧ |t| < R}

definition OAI.TriangularHilbert.integrand line 22

def integrand (F G : Plane → ℂ) (z : Plane) (t : ℝ) : ℂ :=  F (z.1 + t, z.2) * G (z.1, z.2 + t) / (t : ℂ)

definition OAI.TriangularHilbert.truncation line 25

def truncation (F G : Plane → ℂ) (E : Endpoints) (z : Plane) : ℂ :=  ∫ t in annulus E.lower E.upper, integrand F G z t

definition OAI.TriangularHilbert.GoodPoint line 28

def GoodPoint (F G : Plane → ℂ) (z : Plane) : Prop :=  ∀ n : ℕ, IntegrableOn (integrand F G z)    (annulus (1 / ((n : ℝ) + 2)) ((n : ℝ) + 2))

definition OAI.TriangularHilbert.maximal line 32

def maximal (F G : Plane → ℂ) (z : Plane) : ℝ≥0∞ := by  classical  exact if GoodPoint F G z then    ⨆ E : Endpoints, ENNReal.ofReal ‖truncation F G E z‖ else 0

definition OAI.TriangularHilbert.maximalNorm line 37

def maximalNorm (F G : Plane → ℂ) : ℝ≥0∞ :=  (∫⁻ z : Plane, maximal F G z ^ ((3 : ℝ) / 2)) ^ ((2 : ℝ) / 3)

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: 3/2, endpoints, truncation and complex. Found in the paper’s title, which its main theorem states: point and maximal. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: hard, supremum, absolute, finite and constant.

Statements in The maximal triangular Hilbert transform at the symmetric point come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of The maximal triangular Hilbert transform at the symmetric point

Theorem

There is an absolute constant C such that, for every pair of complex functions F,G\in L^3(\R^2),

\|B_*(F,G)\|_{L^{3/2}(\R^2)} \le C\|F\|_{L^3(\R^2)}\|G\|_{L^3(\R^2)}.

The supremum is over both finite hard truncation endpoints.

Section 1, Introduction Read it in the paper TeX source, line 16

Show the TeX

There is an absolute constant $C$ such that, for every pair of complex
functions $F,G\in L^3(\R^2)$,
\[
 \|B_*(F,G)\|_{L^{3/2}(\R^2)}
 \le C\|F\|_{L^3(\R^2)}\|G\|_{L^3(\R^2)}.
\]
The supremum is over both finite hard truncation endpoints.

Other possible paper matches, closest wording first

Proposition 6.5 of The maximal triangular Hilbert transform at the symmetric point. Words of the Lean statement, its names split into words, that also appear in the theorem: norm, 3/2, endpoints, truncation and complex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: hard, supremum, absolute, finite and constant.
Theorem 1.2 of Annular variation of the triangular Hilbert transform at the symmetric point. Words of the Lean statement, its names split into words, that also appear in the theorem: annulus, integer, 3/2, estimate and complex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: absolute and constant.
Proposition 6.1 of The maximal triangular Hilbert transform at the symmetric point. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, endpoints, truncations and complex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: arbitrary and constant.
Corollary 9.1 of Annular variation of the triangular Hilbert transform at the symmetric point. Words of the Lean statement, its names split into words, that also appear in the theorem: maximal, measurable, endpoints, truncation and complex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: hard, supremum, absolute, finite and constant.
Proposition 7.4 of The maximal triangular Hilbert transform at the symmetric point. Words of the Lean statement, its names split into words, that also appear in the theorem: integrable, estimates, truncation and complex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounded, finite and constants.
Proposition 6.2 of The maximal triangular Hilbert transform at the symmetric point. Words of the Lean statement, its names split into words, that also appear in the theorem: measurable, 3/2 and complex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: supremum, absolute 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

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 082’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: Maximal L^3 x L^3 -> L^{3/2} bound only; annular r-variation (r > 2), a.e./norm convergence and the trilinear symmetric-point estimate are not stated.
Definitions the group asks readers to compare with the paper: maximal F G z := if GoodPoint F G z then ⨆ E : Endpoints, ENNReal.ofReal ‖truncation F G E z‖ else 0 (junk value 0 off the good set; the statement itself asserts ∀ᵐ z, GoodPoint F G z)
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: Not stated: the annular r-variation estimate (every r > 2), the a.e. and norm convergence of the truncations, and the trilinear L^3 x L^3 x L^3 estimate. Also not stated: the dyadic triangular Hilbert form bound of the family's second manuscript, which the family title names but the headline text does not.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result proves the symmetric maximal estimate for the triangular Hilbert transform: for arbitrary complex $F,G\in L^3(\mathbb R^2)$, the $L^{3/2}$ norm of the supremum over all finite hard-truncation intervals is at most $C\|F\|_3\|G\|_3$ 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

57 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: 24 to its own declarations, 14 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_estimate 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 28 lines. 1 of them is named in the statement itself.

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

abbrev Planeline 12library
structure Endpointsline 14library
def annulusline 20library
def integrandline 22library
def truncationline 25library
def GoodPointline 28library
def maximalline 32library
def maximalNormline 37library
def MainEstimateline 40in the statementlibrary
theorem main_estimateline 50selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheoryopen scoped ENNReal NNRealnamespace TriangularHilbertabbrev Plane := ℝ × ℝstructure Endpoints where  lower : ℝ  upper : ℝ  lower_pos : 0 < lower  lower_lt_upper : lower < upperdef annulus (ε R : ℝ) : Set ℝ := {t | ε < |t| ∧ |t| < R}def integrand (F G : Plane → ℂ) (z : Plane) (t : ℝ) : ℂ :=  F (z.1 + t, z.2) * G (z.1, z.2 + t) / (t : ℂ)def truncation (F G : Plane → ℂ) (E : Endpoints) (z : Plane) : ℂ :=  ∫ t in annulus E.lower E.upper, integrand F G z tdef GoodPoint (F G : Plane → ℂ) (z : Plane) : Prop :=  ∀ n : ℕ, IntegrableOn (integrand F G z)    (annulus (1 / ((n : ℝ) + 2)) ((n : ℝ) + 2))def maximal (F G : Plane → ℂ) (z : Plane) : ℝ≥0∞ := by  classical  exact if GoodPoint F G z then    ⨆ E : Endpoints, ENNReal.ofReal ‖truncation F G E z‖ else 0def maximalNorm (F G : Plane → ℂ) : ℝ≥0∞ :=  (∫⁻ z : Plane, maximal F G z ^ ((3 : ℝ) / 2)) ^ ((2 : ℝ) / 3)def MainEstimate : Prop :=  ∃ C : ℝ≥0, ∀ F G : Plane → ℂ,    MemLp F 3 volume → MemLp G 3 volume →    (∀ᵐ z ∂volume, GoodPoint F G z) ∧    AEMeasurable (maximal F G) volume ∧    maximalNorm F G ≤ (C : ℝ≥0∞) * eLpNorm F 3 volume * eLpNorm G 3 volumeopen MeasureTheory Filteropen scoped ENNRealtheorem main_estimate : MainEstimate := by  sorryend TriangularHilbertendend 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.

The maximal triangular Hilbert transform at the symmetric point

Theorem

There is an absolute constant C such that, for every pair of complex functions F,G\in L^3(\R^2),

\|B_*(F,G)\|_{L^{3/2}(\R^2)} \le C\|F\|_{L^3(\R^2)}\|G\|_{L^3(\R^2)}.

The supremum is over both finite hard truncation endpoints.

Section 1, Introduction Read it in the paper TeX source, line 16

Show the TeX

There is an absolute constant $C$ such that, for every pair of complex
functions $F,G\in L^3(\R^2)$,
\[
 \|B_*(F,G)\|_{L^{3/2}(\R^2)}
 \le C\|F\|_{L^3(\R^2)}\|G\|_{L^3(\R^2)}.
\]
The supremum is over both finite hard truncation endpoints.

Configuration and file facts

File: lean/ComparatorChallenges/TriangularHilbert.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/TriangularHilbert/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: 9 local vocabulary declarations
Lines of Lean, without comments or blank lines: 28
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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