Lean statement

Ordinary Elliott cancellation under uniform nonpretentiousness

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.OrdinaryCorrelations.binary_corrected_elliott

Lean statement file: OrdinaryElliott.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 007 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.

theorem binary_corrected_elliott    (f₁ f₂ : ℕ → ℂ) (hf₁ : OneBounded f₁) (hf₂ : OneBounded f₂)    (hm₁ : Multiplicative f₁) (hm₂ : Multiplicative f₂)    (hNP : UniformlyNonpretentious f₁ ∨ UniformlyNonpretentious f₂)    (h₁ h₂ : ℕ) (hne : h₁ ≠ h₂) :    Tendsto (shiftAverage f₁ f₂ h₁ h₂) atTop (nhds 0) := 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.OrdinaryCorrelations.OneBounded line 11

def OneBounded (f : ℕ → ℂ) : Prop := ∀ n, ‖f n‖ ≤ 1

definition OAI.OrdinaryCorrelations.Multiplicative line 13

def Multiplicative (f : ℕ → ℂ) : Prop :=  ∀ m n : ℕ, 0 < m → 0 < n → m.Coprime n → f (m * n) = f m * f n

definition OAI.OrdinaryCorrelations.distanceSq line 16

def distanceSq (f : ℕ → ℂ) {q : ℕ} (χ : DirichletCharacter ℂ q)    (t X : ℝ) : ℝ :=  ∑ p ∈ (Finset.Icc 2 ⌊X⌋₊).filter Nat.Prime,    (1 - (f p * star (χ (p : ZMod q) *      Complex.exp ((t * Real.log (p : ℝ) : ℝ) * Complex.I))).re) / (p : ℝ)

definition OAI.OrdinaryCorrelations.distance line 22

def distance (f : ℕ → ℂ) {q : ℕ} (χ : DirichletCharacter ℂ q)    (t X : ℝ) : ℝ := Real.sqrt (distanceSq f χ t X)

definition OAI.OrdinaryCorrelations.UniformlyNonpretentious line 25

def UniformlyNonpretentious (f : ℕ → ℂ) : Prop :=  ∀ (q : ℕ), 0 < q → ∀ χ : DirichletCharacter ℂ q,    Tendsto (fun N : ℕ => sInf ((fun t : ℝ => distance f χ t (N : ℝ)) ''      Set.Icc (-(N : ℝ)) (N : ℝ))) atTop atTop

definition OAI.OrdinaryCorrelations.shiftAverage line 30

def shiftAverage (f₁ f₂ : ℕ → ℂ) (h₁ h₂ N : ℕ) : ℂ :=  (N : ℂ)⁻¹ * ∑ n ∈ Finset.Icc 1 N, f₁ (n + h₁) * f₂ (n + h₂)

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: elliott, binary, corrected, multiplicative, integers and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least.

Theorem 1.2 of Ordinary two-point correlations of multiplicative functions

Theorem (Binary corrected Elliott)

Let f_1,f_2:\N\to\mathbb D be multiplicative, and suppose that at least one satisfies (eq:unp). For every fixed pair of distinct nonnegative integers h_1,h_2,

\frac1N\sum_{n=1}^N f_1(n+h_1)f_2(n+h_2)\longrightarrow0 \qquad(N\longrightarrow\infty,\ N\in\N).

Section 1, Introduction Read it in the paper TeX source, line 64

Show the TeX

Let \(f_1,f_2:\N\to\mathbb D\) be multiplicative, and suppose that
at least one satisfies \eqref{eq:unp}. For every fixed pair of distinct
nonnegative integers \(h_1,h_2\),
\[
 \frac1N\sum_{n=1}^N f_1(n+h_1)f_2(n+h_2)\longrightarrow0
 \qquad(N\longrightarrow\infty,\ N\in\N).
\]

Other possible paper matches, closest wording first

Proposition 18.4 of Ordinary two-point correlations of multiplicative functions. Words of the Lean statement, its names split into words, that also appear in the theorem: uniformly, nonpretentious, real, multiplicative, integers and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: correlations and least.
Theorem 12.2 of Ordinary two-point correlations of multiplicative functions. Words of the Lean statement, its names split into words, that also appear in the theorem: bounded, averaged, multiplicative, log and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: function.
Corollary 1.3 of Ordinary two-point correlations of multiplicative functions. Words of the Lean statement, its names split into words, that also appear in the theorem: multiplicative, integers and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: affine, correlations and least.
Proposition 11.4 of Ordinary two-point correlations of multiplicative functions. Words of the Lean statement, its names split into words, that also appear in the theorem: uniformly, nonpretentious, multiplicative and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: function and least.
Theorem 1.1 of Ordinary two-point correlations of multiplicative functions. Words of the Lean statement, its names split into words, that also appear in the theorem: real, integers and log. Found in the paper’s title, which its main theorem states: multiplicative. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: affine and correlations.
Theorem 5.4 of Ordinary two-point correlations of multiplicative functions. Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt, averaged, exp and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least.

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 007’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 found; affine forms take natural-number coefficients (a₁,a₂>0, b₁,b₂≥0), which loses nothing up to re-indexing; custom non-pretentiousness definition checked against the standard one.
Definitions the group asks readers to compare with the paper: def UniformlyNonpretentious (f : ℕ → ℂ) : Prop := ∀ (q : ℕ), 0 < q → ∀ (χ : DirichletCharacter ℂ q) (K : ℝ), ∀ᶠ N : ℕ in atTop, ∀ t : ℝ, |t| ≤ (N : ℝ) → K ≤ squaredDistance f (characterTwist χ t) N (custom but matches the standard definition; two files use different but equivalent encodings)
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

“For two one-bounded multiplicative functions, if at least one is uniformly nonpretentious against all Dirichlet-character twists with frequency $|t|\le N$, then their shifted and nonproportional affine correlation sums divided by $N$ tend to zero.”
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

44 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: 9 to its own declarations, 20 to Mathlib’s documentation. 7 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 binary_corrected_elliott 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 6 other declarations of the file; with the theorem itself they span 23 lines. 4 of them are named in the statement itself.

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

def OneBoundedline 11in the statementlibrary
def Multiplicativeline 13in the statementlibrary
def distanceSqline 16library
def distanceline 22library
def UniformlyNonpretentiousline 25in the statementlibrary
def shiftAverageline 30in the statementlibrary
theorem binary_corrected_elliottline 33selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperatorsopen Filternamespace OrdinaryCorrelationsdef OneBounded (f : ℕ → ℂ) : Prop := ∀ n, ‖f n‖ ≤ 1def Multiplicative (f : ℕ → ℂ) : Prop :=  ∀ m n : ℕ, 0 < m → 0 < n → m.Coprime n → f (m * n) = f m * f ndef distanceSq (f : ℕ → ℂ) {q : ℕ} (χ : DirichletCharacter ℂ q)    (t X : ℝ) : ℝ :=  ∑ p ∈ (Finset.Icc 2 ⌊X⌋₊).filter Nat.Prime,    (1 - (f p * star (χ (p : ZMod q) *      Complex.exp ((t * Real.log (p : ℝ) : ℝ) * Complex.I))).re) / (p : ℝ)def distance (f : ℕ → ℂ) {q : ℕ} (χ : DirichletCharacter ℂ q)    (t X : ℝ) : ℝ := Real.sqrt (distanceSq f χ t X)def UniformlyNonpretentious (f : ℕ → ℂ) : Prop :=  ∀ (q : ℕ), 0 < q → ∀ χ : DirichletCharacter ℂ q,    Tendsto (fun N : ℕ => sInf ((fun t : ℝ => distance f χ t (N : ℝ)) ''      Set.Icc (-(N : ℝ)) (N : ℝ))) atTop atTopdef shiftAverage (f₁ f₂ : ℕ → ℂ) (h₁ h₂ N : ℕ) : ℂ :=  (N : ℂ)⁻¹ * ∑ n ∈ Finset.Icc 1 N, f₁ (n + h₁) * f₂ (n + h₂)theorem binary_corrected_elliott    (f₁ f₂ : ℕ → ℂ) (hf₁ : OneBounded f₁) (hf₂ : OneBounded f₂)    (hm₁ : Multiplicative f₁) (hm₂ : Multiplicative f₂)    (hNP : UniformlyNonpretentious f₁ ∨ UniformlyNonpretentious f₂)    (h₁ h₂ : ℕ) (hne : h₁ ≠ h₂) :    Tendsto (shiftAverage f₁ f₂ h₁ h₂) atTop (nhds 0) := by  sorryend OrdinaryCorrelationsendend 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.

Ordinary two-point correlations of multiplicative functions

Theorem (A logarithmic saving for affine Liouville correlations)

There is an absolute constant c>0 such that, for every fixed choice of integers a_1,a_2\ge1 and b_1,b_2\ge0 with a_1b_2-a_2b_1\ne0, there is a constant C_{a_1,a_2,b_1,b_2} for which

\left|\sum_{1\le n\le X} \lambda(a_1n+b_1)\lambda(a_2n+b_2)\right| \le C_{a_1,a_2,b_1,b_2}\frac{X}{(\log X)^c} \qquad(X\ge3).

The cutoff X may be any real number. No coprimality condition is imposed on the coefficients.

Section 1, Introduction Read it in the paper TeX source, line 20

Show the TeX

There is an absolute constant \(c>0\) such that, for every fixed choice
of integers \(a_1,a_2\ge1\) and \(b_1,b_2\ge0\) with
\(a_1b_2-a_2b_1\ne0\), there is a constant
\(C_{a_1,a_2,b_1,b_2}\) for which
\[
 \left|\sum_{1\le n\le X}
       \lambda(a_1n+b_1)\lambda(a_2n+b_2)\right|
 \le C_{a_1,a_2,b_1,b_2}\frac{X}{(\log X)^c}
 \qquad(X\ge3).
\]
The cutoff \(X\) may be any real number. No coprimality condition is
imposed on the coefficients.

Theorem (Binary corrected Elliott)

Let f_1,f_2:\N\to\mathbb D be multiplicative, and suppose that at least one satisfies (eq:unp). For every fixed pair of distinct nonnegative integers h_1,h_2,

\frac1N\sum_{n=1}^N f_1(n+h_1)f_2(n+h_2)\longrightarrow0 \qquad(N\longrightarrow\infty,\ N\in\N).

Section 1, Introduction Read it in the paper TeX source, line 64

Show the TeX

Let \(f_1,f_2:\N\to\mathbb D\) be multiplicative, and suppose that
at least one satisfies \eqref{eq:unp}. For every fixed pair of distinct
nonnegative integers \(h_1,h_2\),
\[
 \frac1N\sum_{n=1}^N f_1(n+h_1)f_2(n+h_2)\longrightarrow0
 \qquad(N\longrightarrow\infty,\ N\in\N).
\]

Corollary (Affine correlations)

Let f_1,f_2:\N\to\mathbb D be multiplicative, with at least one satisfying (eq:unp). For all fixed integers a_1,a_2\ge1 and b_1,b_2\ge0 with a_1b_2-a_2b_1\ne0,

\frac1N\sum_{n=1}^N f_1(a_1n+b_1)f_2(a_2n+b_2) \longrightarrow0.

Section 1, Introduction Read it in the paper TeX source, line 81

Show the TeX

Let \(f_1,f_2:\N\to\mathbb D\) be multiplicative, with at least one
satisfying \eqref{eq:unp}. For all fixed integers \(a_1,a_2\ge1\) and
\(b_1,b_2\ge0\) with \(a_1b_2-a_2b_1\ne0\),
\[
 \frac1N\sum_{n=1}^N f_1(a_1n+b_1)f_2(a_2n+b_2)
 \longrightarrow0.
\]

Configuration and file facts

File: lean/ComparatorChallenges/OrdinaryElliott.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 007 link it.
Solution module: OAI.NumberTheory.OrdinaryCorrelations.Elliott.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: 6 local vocabulary declarations
Lines of Lean, without comments or blank lines: 23
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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