Lean statement

Joint Dickman law and equal ordering densities for consecutive integers

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.JointDickmanPaper.joint_law OAI.JointDickmanPaper.increasing_order OAI.JointDickmanPaper.decreasing_order

Lean statement file: JointDickman.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 012 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 joint_law (a b : ℝ) (ha : 0 < a) (ha1 : a < 1) (hb : 0 < b) (hb1 : b < 1) :    Tendsto (realDensity (fun n =>      (n.maxPrimeFac : ℝ) ≤ (n : ℝ) ^ a ∧ ((n + 1).maxPrimeFac : ℝ) ≤ (n : ℝ) ^ b))      atTop (𝓝 (Erdos970.NumberTheoryLean.Dickman.rho (1 / a) *        Erdos970.NumberTheoryLean.Dickman.rho (1 / b))) := by  sorry⋮theorem increasing_order :    Tendsto (realDensity (fun n => n.maxPrimeFac < (n + 1).maxPrimeFac))      atTop (𝓝 (1 / 2)) := by  sorry⋮theorem decreasing_order :    Tendsto (realDensity (fun n => (n + 1).maxPrimeFac < n.maxPrimeFac))      atTop (𝓝 (1 / 2)) := 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.Erdos970.NumberTheoryLean.DelayConstruction.kernel line 9

noncomputable def kernel (f : ℝ → ℝ) (t : ℝ) : ℝ := f (t - 1) / max 1 t

definition OAI.Erdos970.NumberTheoryLean.DelayConstruction.stepApprox line 11

noncomputable def stepApprox (σ : ℝ) : ℕ → ℝ → ℝ  | 0 => fun _ => 1  | n + 1 => fun u => 1 + σ * ∫ t in (1 : ℝ)..max 1 u, kernel (stepApprox σ n) t

definition OAI.Erdos970.NumberTheoryLean.DelayConstruction.delay line 15

noncomputable def delay (σ u : ℝ) : ℝ := stepApprox σ ⌈u⌉₊ u

definition OAI.Erdos970.NumberTheoryLean.Dickman.rho line 22

noncomputable def rho : ℝ → ℝ := DelayConstruction.delay (-1)

definition OAI.JointDickman.empiricalCount line 28

noncomputable def empiricalCount (P : ℕ → Prop) (N : ℕ) : ℕ := by  classical  exact ((Finset.Icc 2 N).filter P).card

definition OAI.JointDickman.realDensity line 33

noncomputable def realDensity (P : ℕ → Prop) (X : ℝ) : ℝ :=  (empiricalCount P ⌊X⌋₊ : ℝ) / X

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: dickman, joint, ordinary and real.

Theorem 1.1 of The joint Dickman law for consecutive integers

Theorem (Joint Dickman law)

For every fixed a,b\in(0,1),

\lim_{X\to\infty}\frac1X \#\{2\leq n\leq X:\Pplus(n)\leq n^a, \ \Pplus(n+1)\leq n^b\} =\rho(1/a)\rho(1/b).

Here the limit is taken over all real X, with ordinary, unweighted counting.

Section 1, Introduction Read it in the paper TeX source, line 19

Show the TeX

For every fixed $a,b\in(0,1)$,
\[
 \lim_{X\to\infty}\frac1X
 \#\{2\leq n\leq X:\Pplus(n)\leq n^a,
                         \ \Pplus(n+1)\leq n^b\}
 =\rho(1/a)\rho(1/b).
\]
Here the limit is taken over all real $X$, with ordinary, unweighted
counting.

Other possible paper matches, closest wording first

Corollary 1.2 of The joint Dickman law for consecutive integers. Words of the Lean statement, its names split into words, that also appear in the theorem: natural, density, reverse, 1/2, ordering and one.
Proposition 10.3 of The joint Dickman law for consecutive integers. Found in the paper’s title, which its main theorem states: dickman, joint, law and integers.
Proposition 3.3 of The joint Dickman law for consecutive integers. Words of the Lean statement, its names split into words, that also appear in the theorem: primes, 1/2, order, law, integer and real.
Theorem 2.4 of The joint Dickman law for consecutive integers. Words of the Lean statement, its names split into words, that also appear in the theorem: function and real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: tending.
Proposition 7.1 of The joint Dickman law for consecutive integers. Words of the Lean statement, its names split into words, that also appear in the theorem: threshold and one.
Proposition 6.1 of The joint Dickman law for consecutive integers. Words of the Lean statement, its names split into words, that also appear in the theorem: normalization and kernel.

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 012’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; headline and the 1/2 corollary are stated. Dickman rho is a custom delay-equation construction (checked faithful).
Definitions the group asks readers to compare with the paper: noncomputable def rho : ℝ → ℝ := DelayConstruction.delay (-1) with stepApprox σ (n+1) u = 1 + σ * ∫ t in 1..max 1 u, kernel (stepApprox σ n) t (custom Dickman function; correct by inspection, no bridge lemma to a Mathlib object)
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 the joint Dickman law in ordinary natural density: for every $0<a,b<1$, the density of integers satisfying $P^+(n)\le n^a$ and $P^+(n+1)\le n^b$ tends to $\rho(1/a)\rho(1/b)$, where $\rho$ is the Dickman function.”
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

65 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: 10 to its own declarations, 8 to Mathlib’s documentation. 8 declared names also link to the same declaration in OpenAI’s library. 3 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. 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 joint_law 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 25 lines. 2 of them are named in the statement itself.

def OAI.Erdos970.NumberTheoryLean.DelayConstruction.kernelline 9library
def OAI.Erdos970.NumberTheoryLean.DelayConstruction.stepApproxline 11library
def OAI.Erdos970.NumberTheoryLean.DelayConstruction.delayline 15library
def OAI.Erdos970.NumberTheoryLean.Dickman.rholine 22in the statement
def OAI.JointDickman.empiricalCountline 28library
def OAI.JointDickman.realDensityline 33in the statementlibrary
theorem OAI.JointDickmanPaper.joint_lawline 44selected theoremlibrary
theorem OAI.JointDickmanPaper.increasing_orderline 52selected theoremlibrary
theorem OAI.JointDickmanPaper.decreasing_orderline 58selected theoremlibrary

import Mathlibnamespace OAI/-! # The joint Dickman law for consecutive integers -/namespace Erdos970.NumberTheoryLean.DelayConstructionnoncomputable def kernel (f : ℝ → ℝ) (t : ℝ) : ℝ := f (t - 1) / max 1 tnoncomputable def stepApprox (σ : ℝ) : ℕ → ℝ → ℝ  | 0 => fun _ => 1  | n + 1 => fun u => 1 + σ * ∫ t in (1 : ℝ)..max 1 u, kernel (stepApprox σ n) tnoncomputable def delay (σ u : ℝ) : ℝ := stepApprox σ ⌈u⌉₊ uend Erdos970.NumberTheoryLean.DelayConstructionnamespace Erdos970.NumberTheoryLean.Dickman/-- The continuous delay-equation construction of the Dickman function. -/noncomputable def rho : ℝ → ℝ := DelayConstruction.delay (-1)end Erdos970.NumberTheoryLean.Dickmannamespace JointDickmannoncomputable def empiricalCount (P : ℕ → Prop) (N : ℕ) : ℕ := by  classical  exact ((Finset.Icc 2 N).filter P).card/-- Ordinary natural density over `2 ≤ n ≤ X`, normalized by the real endpoint. -/noncomputable def realDensity (P : ℕ → Prop) (X : ℝ) : ℝ :=  (empiricalCount P ⌊X⌋₊ : ℝ) / Xend JointDickmannamespace JointDickmanPaperopen Filter JointDickmanopen scoped Topology/-- Theorem 1: ordinary natural density, with both thresholds based at `n`. -/theorem joint_law (a b : ℝ) (ha : 0 < a) (ha1 : a < 1) (hb : 0 < b) (hb1 : b < 1) :    Tendsto (realDensity (fun n =>      (n.maxPrimeFac : ℝ) ≤ (n : ℝ) ^ a ∧ ((n + 1).maxPrimeFac : ℝ) ≤ (n : ℝ) ^ b))      atTop (𝓝 (Erdos970.NumberTheoryLean.Dickman.rho (1 / a) *        Erdos970.NumberTheoryLean.Dickman.rho (1 / b))) := by  sorry/-- Corollary 2: the increasing strict ordering has natural density one half. -/theorem increasing_order :    Tendsto (realDensity (fun n => n.maxPrimeFac < (n + 1).maxPrimeFac))      atTop (𝓝 (1 / 2)) := by  sorry/-- The reverse strict ordering also has natural density one half. -/theorem decreasing_order :    Tendsto (realDensity (fun n => (n + 1).maxPrimeFac < n.maxPrimeFac))      atTop (𝓝 (1 / 2)) := by  sorryend JointDickmanPaperend 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 joint Dickman law for consecutive integers

Theorem (Joint Dickman law)

For every fixed a,b\in(0,1),

\lim_{X\to\infty}\frac1X \#\{2\leq n\leq X:\Pplus(n)\leq n^a, \ \Pplus(n+1)\leq n^b\} =\rho(1/a)\rho(1/b).

Here the limit is taken over all real X, with ordinary, unweighted counting.

Section 1, Introduction Read it in the paper TeX source, line 19

Show the TeX

For every fixed $a,b\in(0,1)$,
\[
 \lim_{X\to\infty}\frac1X
 \#\{2\leq n\leq X:\Pplus(n)\leq n^a,
                         \ \Pplus(n+1)\leq n^b\}
 =\rho(1/a)\rho(1/b).
\]
Here the limit is taken over all real $X$, with ordinary, unweighted
counting.

Corollary

One has

\lim_{X\to\infty}\frac1X \#\{2\leq n\leq X:\Pplus(n)<\Pplus(n+1)\}=\frac12.

The reverse ordering has the same natural density.

Section 1, Introduction Read it in the paper TeX source, line 43

Show the TeX

One has
\[
 \lim_{X\to\infty}\frac1X
 \#\{2\leq n\leq X:\Pplus(n)<\Pplus(n+1)\}=\frac12.
\]
The reverse ordering has the same natural density.

Configuration and file facts

File: lean/ComparatorChallenges/JointDickman.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 012 link it.
Solution module: OAI.NumberTheory.JointDickman.PaperMain
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: 25
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