Lean statement

Positive lower density of prime-ratio increases

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.Problem344.large_gaps_and_ratio_density

Lean statement file: PrimeGaps.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 026 link it.

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.

theorem large_gaps_and_ratio_density :    (∀ C : ℝ, 0 < C →      ∃ c : ℝ, 0 < c ∧ ∃ N₀ : ℕ, ∀ N : ℕ, N₀ ≤ N →        c * (N : ℝ) ≤ ((LargePrimeGaps.largeGapIndices C N).card : ℝ)) ∧    0 < lowerAsymptoticDensity ratioIncreaseIndices := 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.LargePrimeGaps.prime line 7

noncomputable def prime (n : ℕ) : ℕ := Nat.nth Nat.Prime (n - 1)

definition OAI.LargePrimeGaps.largeGapIndices line 9

noncomputable def largeGapIndices (C : ℝ) (N : ℕ) : Finset ℕ := by  classical  exact (Finset.Icc 1 N).filter fun n =>    C * Real.log (prime n : ℝ) < (prime (n + 1) : ℝ) - (prime n : ℝ)

definition OAI.Problem344.primeAt line 20

def primeAt (n : ℕ) : ℕ :=  Nat.nth Nat.Prime (n - 1)

definition OAI.Problem344.initialCount line 23

def initialCount (A : Set ℕ) (N : ℕ) : ℕ := by  classical  exact ((Finset.Icc 1 N).filter (fun n => n ∈ A)).card

definition OAI.Problem344.lowerAsymptoticDensity line 27

def lowerAsymptoticDensity (A : Set ℕ) : ℝ :=  sSup {d : ℝ | ∃ N0 : ℕ, ∀ N : ℕ, N0 ≤ N →    d ≤ (initialCount A N : ℝ) / (N : ℝ)}

definition OAI.Problem344.ratioIncreaseIndices line 31

def ratioIncreaseIndices : Set ℕ :=  {n | 1 ≤ n ∧    (primeAt n : ℝ) / (n : ℝ) <      (primeAt (n + 1) : ℝ) / ((n + 1 : ℕ) : ℝ)}

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.

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 Positive lower density of large prime gaps

Words of the Lean statement, its names split into words, that also appear in the theorem: log, real and integer. Found in the paper’s title, which its main theorem states: large, gaps, prime, density and lower.

Theorem

For every fixed real C>0, there are constants c(C)>0 and N_0(C) such that, for every integer N\ge N_0(C),

\#\{1\le n\le N:p_{n+1}-p_n>C\log p_n\}\ge c(C)N.

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

For every fixed real $C>0$, there are constants $c(C)>0$ and $N_0(C)$
such that, for every integer $N\ge N_0(C)$,
\[
 \#\{1\le n\le N:p_{n+1}-p_n>C\log p_n\}\ge c(C)N.
\]

Corollary 1.2 of Positive lower density of large prime gaps

Words of the Lean statement, its names split into words, that also appear in the theorem: asymptotic, density and lower. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.

Corollary

The set

\left\{n\ge1:\frac{p_n}{n}<\frac{p_{n+1}}{n+1}\right\}

has positive lower asymptotic density.

Section 1, Introduction Read it in the paper TeX source, line 38

Show the TeX

The set
\[
 \left\{n\ge1:\frac{p_n}{n}<\frac{p_{n+1}}{n+1}\right\}
\]
has positive lower asymptotic density.

Other possible paper matches, closest wording first

Proposition 4.2 of Positive lower density of large prime gaps. Words of the Lean statement, its names split into words, that also appear in the theorem: increasing.
Proposition 5.1 of Positive lower density of large prime gaps. Words of the Lean statement, its names split into words, that also appear in the theorem: prime, log, real and integers. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.
Proposition 2.1 of Positive lower density of large prime gaps. Words of the Lean statement, its names split into words, that also appear in the theorem: integers.
Proposition 3.1 of Positive lower density of large prime gaps. Words of the Lean statement, its names split into words, that also appear in the theorem: real.

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 026’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; challenge states the large-gap theorem and the ratio corollary. Lower density is a custom sSup-defined liminf (checked).
Definitions the group asks readers to compare with the paper: def lowerAsymptoticDensity (A : Set ℕ) : ℝ := sSup {d : ℝ | ∃ N0 : ℕ, ∀ N : ℕ, N0 ≤ N → d ≤ (initialCount A N : ℝ) / (N : ℝ)} (custom liminf-style density; correct since the set is nonempty and bounded by 1)
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 formalized supplement proves that the indices $n$ for which $p_n/n<p_{n+1}/(n+1)$ have positive lower asymptotic density, where $p_n$ is the $n$th prime. Thus the normalized prime sequence has a positive-density set of increases. This is the prime-ratio corollary associated with the paper's theorem on a positive lower density of large prime gaps.”
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

47 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, 12 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 large_gaps_and_ratio_density 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. 3 of them are named in the statement itself.

Names in the namespace OAI.Problem344 are listed without it.

def OAI.LargePrimeGaps.primeline 7library
def OAI.LargePrimeGaps.largeGapIndicesline 9in the statementlibrary
def primeAtline 20library
def initialCountline 23library
def lowerAsymptoticDensityline 27in the statementlibrary
def ratioIncreaseIndicesline 31in the statementlibrary
theorem large_gaps_and_ratio_densityline 36selected theoremlibrary

import Mathlibnamespace OAInamespace LargePrimeGapsnoncomputable def prime (n : ℕ) : ℕ := Nat.nth Nat.Prime (n - 1)noncomputable def largeGapIndices (C : ℝ) (N : ℕ) : Finset ℕ := by  classical  exact (Finset.Icc 1 N).filter fun n =>    C * Real.log (prime n : ℝ) < (prime (n + 1) : ℝ) - (prime n : ℝ)end LargePrimeGapsnoncomputable sectionnamespace Problem344def primeAt (n : ℕ) : ℕ :=  Nat.nth Nat.Prime (n - 1)def initialCount (A : Set ℕ) (N : ℕ) : ℕ := by  classical  exact ((Finset.Icc 1 N).filter (fun n => n ∈ A)).carddef lowerAsymptoticDensity (A : Set ℕ) : ℝ :=  sSup {d : ℝ | ∃ N0 : ℕ, ∀ N : ℕ, N0 ≤ N →    d ≤ (initialCount A N : ℝ) / (N : ℝ)}def ratioIncreaseIndices : Set ℕ :=  {n | 1 ≤ n ∧    (primeAt n : ℝ) / (n : ℝ) <      (primeAt (n + 1) : ℝ) / ((n + 1 : ℕ) : ℝ)}theorem large_gaps_and_ratio_density :    (∀ C : ℝ, 0 < C →      ∃ c : ℝ, 0 < c ∧ ∃ N₀ : ℕ, ∀ N : ℕ, N₀ ≤ N →        c * (N : ℝ) ≤ ((LargePrimeGaps.largeGapIndices C N).card : ℝ)) ∧    0 < lowerAsymptoticDensity ratioIncreaseIndices := by  sorryend Problem344endend 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.

Positive lower density of large prime gaps

Theorem

For every fixed real C>0, there are constants c(C)>0 and N_0(C) such that, for every integer N\ge N_0(C),

\#\{1\le n\le N:p_{n+1}-p_n>C\log p_n\}\ge c(C)N.

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

For every fixed real $C>0$, there are constants $c(C)>0$ and $N_0(C)$
such that, for every integer $N\ge N_0(C)$,
\[
 \#\{1\le n\le N:p_{n+1}-p_n>C\log p_n\}\ge c(C)N.
\]

Corollary

The set

\left\{n\ge1:\frac{p_n}{n}<\frac{p_{n+1}}{n+1}\right\}

has positive lower asymptotic density.

Section 1, Introduction Read it in the paper TeX source, line 38

Show the TeX

The set
\[
 \left\{n\ge1:\frac{p_n}{n}<\frac{p_{n+1}}{n+1}\right\}
\]
has positive lower asymptotic density.

Configuration and file facts

File: lean/ComparatorChallenges/PrimeGaps.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 026 link it.
Solution module: OAI.NumberTheory.PrimeGaps.RatioCorollary
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.

Plectis Atlas