Lean statement

Ordinary Liouville and multiplicative two-point correlations

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.

Theorems selected for verification: OAI.OrdinaryTwoPointCorrelations.liouville_log_saving OAI.OrdinaryTwoPointCorrelations.binary_corrected_elliott OAI.OrdinaryTwoPointCorrelations.affine_corrected_elliott

Lean statement file: OrdinaryTwoPointCorrelations.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 liouville_log_saving :    ∃ c : ℝ, 0 < c ∧ ∀ a₁ a₂ b₁ b₂ : ℕ,      0 < a₁ → 0 < a₂ → a₁ * b₂ ≠ a₂ * b₁ →      ∃ C : ℝ, 0 < C ∧ ∀ X : ℝ, 3 ≤ X →        ‖affineSum liouville liouville a₁ a₂ b₁ b₂ ⌊X⌋₊‖ ≤          C * X / Real.rpow (Real.log X) c := by  sorry⋮theorem binary_corrected_elliott :    ∀ f₁ f₂ : ℕ → ℂ, Multiplicative f₁ → Multiplicative f₂ →      OneBounded f₁ → OneBounded f₂ →      (UniformlyNonpretentious f₁ ∨ UniformlyNonpretentious f₂) →      ∀ h₁ h₂ : ℕ, h₁ ≠ h₂ →        Tendsto (fun N : ℕ => correlationSum f₁ f₂ h₁ h₂ N / (N : ℂ)) atTop (nhds 0) := by  sorry⋮theorem affine_corrected_elliott :    ∀ f₁ f₂ : ℕ → ℂ, Multiplicative f₁ → Multiplicative f₂ →      OneBounded f₁ → OneBounded f₂ →      (UniformlyNonpretentious f₁ ∨ UniformlyNonpretentious f₂) →      ∀ a₁ a₂ b₁ b₂ : ℕ, 0 < a₁ → 0 < a₂ → a₁ * b₂ ≠ a₂ * b₁ →        Tendsto (fun N : ℕ => affineSum f₁ f₂ a₁ a₂ b₁ b₂ N / (N : ℂ)) 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.TwoPointCorrelations.liouville line 11

def liouville (n : ℕ) : ℂ := ArithmeticFunction.liouville n

definition OAI.TwoPointCorrelations.correlationSum line 13

def correlationSum (f g : ℕ → ℂ) (h₁ h₂ N : ℕ) : ℂ :=  ∑ n ∈ Finset.Icc 1 N, f (n + h₁) * g (n + h₂)

definition OAI.TwoPointCorrelations.affineSum line 16

def affineSum (f g : ℕ → ℂ) (a₁ a₂ b₁ b₂ N : ℕ) : ℂ :=  ∑ n ∈ Finset.Icc 1 N, f (a₁ * n + b₁) * g (a₂ * n + b₂)

definition OAI.TwoPointCorrelations.Multiplicative line 19

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.TwoPointCorrelations.OneBounded line 22

def OneBounded (f : ℕ → ℂ) : Prop := ∀ n : ℕ, 0 < n → ‖f n‖ ≤ 1

definition OAI.TwoPointCorrelations.primesUpTo line 24

def primesUpTo (N : ℕ) : Finset ℕ := (Finset.range (N + 1)).filter Nat.Prime

definition OAI.TwoPointCorrelations.squaredDistance line 26

def squaredDistance (f g : ℕ → ℂ) (N : ℕ) : ℝ :=  ∑ p ∈ primesUpTo N, (1 - (f p * conj (g p)).re) / (p : ℝ)

definition OAI.TwoPointCorrelations.characterTwist line 29

def characterTwist {q : ℕ} (χ : DirichletCharacter ℂ q)    (t : ℝ) (n : ℕ) : ℂ :=  χ (n : ZMod q) * Complex.exp (((t * Real.log (n : ℝ) : ℝ) : ℂ) * Complex.I)

definition OAI.TwoPointCorrelations.UniformlyNonpretentious line 33

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

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: absolute, affine, constant, correlations, fixed, liouville, logarithmic and saving.

Theorem 1.1 of 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.

Other possible paper matches, closest wording first

Theorem 1.2 of Ordinary two-point correlations of multiplicative functions. Words of the Lean statement, its names split into words, that also appear in the theorem: elliott, binary, corrected, multiplicative, integers and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: pair.
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, correlations, multiplicative, integers and one more.
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: affine, correlations, multiplicative, integers and one.
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, function, multiplicative, log and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: implied, absolute and constant.
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, function, multiplicative and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: depend, implied and constant.
Proposition 2.1 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, correlations, log and one. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: absolute.

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 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

“The formalization proves ordinary two-point cancellation for multiplicative functions. For the Liouville function on every fixed pair of nonproportional affine forms, the correlation sum up to $X$ is $O(X/(\log X)^c)$ for an absolute $c>0$, with the implied constant depending on the forms.”
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

71 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: 16 to its own declarations, 21 to Mathlib’s documentation. 12 declared names also link to the same declaration in OpenAI’s library. 5 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. 4 of them could name more than one declaration. 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 liouville_log_saving 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 8 of the file’s 9 other declarations; with the theorem itself they span 37 lines. 5 of them are named in the statement itself.

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

def liouvilleline 11in the statementlibrary
def correlationSumline 13in the statementlibrary
def affineSumline 16in the statementlibrary
def OneBoundedline 22in the statementlibrary
def primesUpToline 24library
def squaredDistanceline 26library
def characterTwistline 29library
def UniformlyNonpretentiousline 33in the statementlibrary
theorem OAI.OrdinaryTwoPointCorrelations.liouville_log_savingline 43selected theoremlibrary
theorem OAI.OrdinaryTwoPointCorrelations.binary_corrected_elliottline 51selected theoremlibrary
theorem OAI.OrdinaryTwoPointCorrelations.affine_corrected_elliottline 59selected theoremlibrary

1 declaration the statement does not reach by name

These definitions and lemmas were not found by following names in the statement. Lean may still use them through rules such as instances or macros.

def Multiplicative line 19

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperators ComplexConjugateopen Filternamespace TwoPointCorrelationsdef liouville (n : ℕ) : ℂ := ArithmeticFunction.liouville ndef correlationSum (f g : ℕ → ℂ) (h₁ h₂ N : ℕ) : ℂ :=  ∑ n ∈ Finset.Icc 1 N, f (n + h₁) * g (n + h₂)def affineSum (f g : ℕ → ℂ) (a₁ a₂ b₁ b₂ N : ℕ) : ℂ :=  ∑ n ∈ Finset.Icc 1 N, f (a₁ * n + b₁) * g (a₂ * n + b₂)def Multiplicative (f : ℕ → ℂ) : Prop :=  ∀ m n : ℕ, 0 < m → 0 < n → m.Coprime n → f (m * n) = f m * f ndef OneBounded (f : ℕ → ℂ) : Prop := ∀ n : ℕ, 0 < n → ‖f n‖ ≤ 1def primesUpTo (N : ℕ) : Finset ℕ := (Finset.range (N + 1)).filter Nat.Primedef squaredDistance (f g : ℕ → ℂ) (N : ℕ) : ℝ :=  ∑ p ∈ primesUpTo N, (1 - (f p * conj (g p)).re) / (p : ℝ)def characterTwist {q : ℕ} (χ : DirichletCharacter ℂ q)    (t : ℝ) (n : ℕ) : ℂ :=  χ (n : ZMod q) * Complex.exp (((t * Real.log (n : ℝ) : ℝ) : ℂ) * Complex.I)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) Nend TwoPointCorrelationsnamespace OrdinaryTwoPointCorrelationsopen TwoPointCorrelationstheorem liouville_log_saving :    ∃ c : ℝ, 0 < c ∧ ∀ a₁ a₂ b₁ b₂ : ℕ,      0 < a₁ → 0 < a₂ → a₁ * b₂ ≠ a₂ * b₁ →      ∃ C : ℝ, 0 < C ∧ ∀ X : ℝ, 3 ≤ X →        ‖affineSum liouville liouville a₁ a₂ b₁ b₂ ⌊X⌋₊‖ ≤          C * X / Real.rpow (Real.log X) c := by  sorrytheorem binary_corrected_elliott :    ∀ f₁ f₂ : ℕ → ℂ, Multiplicative f₁ → Multiplicative f₂ →      OneBounded f₁ → OneBounded f₂ →      (UniformlyNonpretentious f₁ ∨ UniformlyNonpretentious f₂) →      ∀ h₁ h₂ : ℕ, h₁ ≠ h₂ →        Tendsto (fun N : ℕ => correlationSum f₁ f₂ h₁ h₂ N / (N : ℂ)) atTop (nhds 0) := by  sorrytheorem affine_corrected_elliott :    ∀ f₁ f₂ : ℕ → ℂ, Multiplicative f₁ → Multiplicative f₂ →      OneBounded f₁ → OneBounded f₂ →      (UniformlyNonpretentious f₁ ∨ UniformlyNonpretentious f₂) →      ∀ a₁ a₂ b₁ b₂ : ℕ, 0 < a₁ → 0 < a₂ → a₁ * b₂ ≠ a₂ * b₁ →        Tendsto (fun N : ℕ => affineSum f₁ f₂ a₁ a₂ b₁ b₂ N / (N : ℂ)) atTop (nhds 0) := by  sorryend OrdinaryTwoPointCorrelationsendend 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/OrdinaryTwoPointCorrelations.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.TwoPointCorrelations.FinalMain
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: 39
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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