Lean statement

Positive density of power-free polynomial values

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.QuarticPowerFree.allDegrees

Lean statement file: PowerFreeValues.leanShow on the mapFamily 020. 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.

theorem allDegrees (f : Polynomial ℤ)    (hirr : Irreducible (f.map (Int.castRingHom ℚ)))    (hdeglo : 4 ≤ f.natDegree)    (hlocal : LocallyAdmissible f (f.natDegree - 2)) :    DensityStatement f (f.natDegree - 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.QuarticPowerFree.PowerFree line 12

def PowerFree (k : ℕ) (a : ℤ) : Prop :=  ∀ p : ℕ, p.Prime → ¬ (p : ℤ) ^ k ∣ a

definition OAI.QuarticPowerFree.rho line 15

def rho (f : Polynomial ℤ) (q : ℕ) : ℕ :=  ((Finset.range q).filter fun a : ℕ => (q : ℤ) ∣ f.eval (a : ℤ)).card

definition OAI.QuarticPowerFree.count line 18

noncomputable def count (f : Polynomial ℤ) (k : ℕ) (X : ℝ) : ℕ := by  classical  exact ((Finset.Icc 1 ⌊X⌋₊).filter fun n : ℕ => PowerFree k (f.eval (n : ℤ))).card

definition OAI.QuarticPowerFree.LocallyAdmissible line 22

def LocallyAdmissible (f : Polynomial ℤ) (k : ℕ) : Prop :=  ∀ p : ℕ, p.Prime → rho f (p ^ k) < p ^ k

definition OAI.QuarticPowerFree.localFactor line 25

noncomputable def localFactor (f : Polynomial ℤ) (k : ℕ) (p : Nat.Primes) : ℝ :=  1 - (rho f (p.val ^ k) : ℝ) / (p.val : ℝ) ^ k

definition OAI.QuarticPowerFree.densityConstant line 28

noncomputable def densityConstant (f : Polynomial ℤ) (k : ℕ) : ℝ :=  ∏' p : Nat.Primes, localFactor f k p

definition OAI.QuarticPowerFree.DensityStatement line 31

def DensityStatement (f : Polynomial ℤ) (k : ℕ) : Prop :=  Multipliable (localFactor f k) ∧  0 < densityConstant f k ∧  (fun X : ℝ => (count f k X : ℝ) - densityConstant f k * X) =o[atTop]    (fun X : ℝ => X)

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: count, factor, free, power, rho, polynomials, integers, prime and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: products, values and put.

Corollary 1.3 of Squarefree values of quartics and power-free values of polynomials

Corollary (Separable products and simultaneous values)

Fix an integer k\ge2. Let F\in\Z[x]\setminus\{0\} be separable over \Q, with every nonconstant irreducible factor of degree at most k+2. If \rho_F(p^k)<p^k for every prime p, then

S_{F,k}(N)=c_{F,k}N+o_{F,k}(N),\qquad c_{F,k}=\prod_p\left(1-\frac{\rho_F(p^k)}{p^k}\right)>0.

More generally, let \mathbf F=(F_1,\ldots,F_r), with r\ge1, be a fixed family of nonzero integer polynomials, each separable over \Q with all nonconstant irreducible factors of degree at most k+2. Factors may be shared by different F_i. Put

\Omega_p=\{a\in\Z/p^k\Z:p^k\mid F_i(a)\text{ for at least one }i\}.

Let S_{\mathbf F,k}(N) count the integers 1\le n\le N for which every F_i(n) is k-power-free. If \#\Omega_p<p^k for every prime p, then

S_{\mathbf F,k}(N)=c_{\mathbf F,k}N+o_{\mathbf F,k}(N),\qquad c_{\mathbf F,k}=\prod_p\left(1-\frac{\#\Omega_p}{p^k}\right)>0.

Section 1, Introduction Read it in the paper TeX source, line 136

Show the TeX

Fix an integer $k\ge2$. Let $F\in\Z[x]\setminus\{0\}$ be separable over
$\Q$, with every nonconstant irreducible factor of degree at most $k+2$.
If $\rho_F(p^k)<p^k$ for every prime $p$, then
\[
 S_{F,k}(N)=c_{F,k}N+o_{F,k}(N),\qquad
 c_{F,k}=\prod_p\left(1-\frac{\rho_F(p^k)}{p^k}\right)>0.
\]
More generally, let $\mathbf F=(F_1,\ldots,F_r)$, with $r\ge1$, be a fixed
family of nonzero integer polynomials, each separable over $\Q$ with all
nonconstant irreducible factors of degree at most $k+2$. Factors may be
shared by different $F_i$. Put
\[
 \Omega_p=\{a\in\Z/p^k\Z:p^k\mid F_i(a)\text{ for at least one }i\}.
\]
Let $S_{\mathbf F,k}(N)$ count the integers $1\le n\le N$ for which every
$F_i(n)$ is $k$-power-free. If $\#\Omega_p<p^k$ for every prime $p$, then
\[
 S_{\mathbf F,k}(N)=c_{\mathbf F,k}N+o_{\mathbf F,k}(N),\qquad
 c_{\mathbf F,k}=\prod_p\left(1-\frac{\#\Omega_p}{p^k}\right)>0.
\]

Other possible paper matches, closest wording first

Theorem 1.1 of Squarefree values of quartics and power-free values of polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: local, rho, irreducible and degree. Found in the paper’s title, which its main theorem states: free, polynomials and power. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: put.
Corollary 1.2 of Squarefree values of quartics and power-free values of polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: rho, power, prime, irreducible and degree. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: th and put.
Theorem 3.1 of Squarefree values of quartics and power-free values of polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: constants, integers, irreducible and degree. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: assume, numbers, positive and put.
Proposition 2.1 of Squarefree values of quartics and power-free values of polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: rho, prime, irreducible and degree. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: assume and positive.
Proposition 8.2 of Squarefree values of quartics and power-free values of polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: polynomials, integer and degree. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: value and put.
Proposition 10.1 of Squarefree values of quartics and power-free values of polynomials. Words of the Lean statement, its names split into words, that also appear in the theorem: rational and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: numbers and positive.

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 020’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; statement is for every d >= 4 (stronger than the new range 4..8 alone).
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 result proves positive-density power-free values for every integer polynomial $f$ irreducible over $\mathbb Q$ of degree $d\ge4$. Put $k=d-2$ and assume no prime $k$th power divides every value of $f$. Then the number of positive integers $n\le X$ for which $f(n)$ is $k$-free is $c_fX+o_f(X)$, where $c_f>0$ is the convergent product of local factors.”
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

46 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, 15 to Mathlib’s documentation. 8 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 allDegrees 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 7 other declarations of the file; with the theorem itself they span 24 lines. 2 of them are named in the statement itself.

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

def PowerFreeline 12library
def rholine 15library
def countline 18library
def LocallyAdmissibleline 22in the statementlibrary
def localFactorline 25library
def densityConstantline 28library
def DensityStatementline 31in the statementlibrary
theorem allDegreesline 37selected theoremlibrary

import Mathlibnamespace OAIopen Filter Asymptoticsopen scoped Topologynamespace QuarticPowerFreedef PowerFree (k : ℕ) (a : ℤ) : Prop :=  ∀ p : ℕ, p.Prime → ¬ (p : ℤ) ^ k ∣ adef rho (f : Polynomial ℤ) (q : ℕ) : ℕ :=  ((Finset.range q).filter fun a : ℕ => (q : ℤ) ∣ f.eval (a : ℤ)).cardnoncomputable def count (f : Polynomial ℤ) (k : ℕ) (X : ℝ) : ℕ := by  classical  exact ((Finset.Icc 1 ⌊X⌋₊).filter fun n : ℕ => PowerFree k (f.eval (n : ℤ))).carddef LocallyAdmissible (f : Polynomial ℤ) (k : ℕ) : Prop :=  ∀ p : ℕ, p.Prime → rho f (p ^ k) < p ^ knoncomputable def localFactor (f : Polynomial ℤ) (k : ℕ) (p : Nat.Primes) : ℝ :=  1 - (rho f (p.val ^ k) : ℝ) / (p.val : ℝ) ^ knoncomputable def densityConstant (f : Polynomial ℤ) (k : ℕ) : ℝ :=  ∏' p : Nat.Primes, localFactor f k pdef DensityStatement (f : Polynomial ℤ) (k : ℕ) : Prop :=  Multipliable (localFactor f k) ∧  0 < densityConstant f k ∧  (fun X : ℝ => (count f k X : ℝ) - densityConstant f k * X) =o[atTop]    (fun X : ℝ => X)theorem allDegrees (f : Polynomial ℤ)    (hirr : Irreducible (f.map (Int.castRingHom ℚ)))    (hdeglo : 4 ≤ f.natDegree)    (hlocal : LocallyAdmissible f (f.natDegree - 2)) :    DensityStatement f (f.natDegree - 2) := by  sorryend QuarticPowerFreeend 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.

Squarefree values of quartics and power-free values of polynomials

Theorem

Let f\in\Z[x] be irreducible over \Q, of degree d with 4\le d\le8, and put k=d-2. If (eq:local-admissibility) holds, then

S_{f,k}(X)=c_{f,k}X+o_f(X),\qquad c_{f,k}=\prod_p\left(1-\frac{\rho_f(p^k)}{p^k}\right)>0.

No primitivity or sign condition on the leading coefficient is required.

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

Show the TeX

Let $f\in\Z[x]$ be irreducible over $\Q$, of degree $d$ with
$4\le d\le8$, and put $k=d-2$. If \eqref{eq:local-admissibility} holds,
then
\[
 S_{f,k}(X)=c_{f,k}X+o_f(X),\qquad
 c_{f,k}=\prod_p\left(1-\frac{\rho_f(p^k)}{p^k}\right)>0.
\]
No primitivity or sign condition on the leading coefficient is required.

Corollary

Let f\in\Z[x] be irreducible over \Q of degree d\ge4, and put k=d-2. If f has no fixed prime kth-power divisor, then

S_{f,k}(X)=c_{f,k}X+o_f(X),\qquad c_{f,k}=\prod_p\left(1-\frac{\rho_f(p^k)}{p^k}\right)>0.

Section 1, Introduction Read it in the paper TeX source, line 113

Show the TeX

Let $f\in\Z[x]$ be irreducible over $\Q$ of degree $d\ge4$, and
put $k=d-2$. If $f$ has no fixed prime $k$th-power divisor, then
\[
 S_{f,k}(X)=c_{f,k}X+o_f(X),\qquad
 c_{f,k}=\prod_p\left(1-\frac{\rho_f(p^k)}{p^k}\right)>0.
\]

Corollary (Separable products and simultaneous values)

Fix an integer k\ge2. Let F\in\Z[x]\setminus\{0\} be separable over \Q, with every nonconstant irreducible factor of degree at most k+2. If \rho_F(p^k)<p^k for every prime p, then

S_{F,k}(N)=c_{F,k}N+o_{F,k}(N),\qquad c_{F,k}=\prod_p\left(1-\frac{\rho_F(p^k)}{p^k}\right)>0.

\Omega_p=\{a\in\Z/p^k\Z:p^k\mid F_i(a)\text{ for at least one }i\}.

Let S_{\mathbf F,k}(N) count the integers 1\le n\le N for which every F_i(n) is k-power-free. If \#\Omega_p<p^k for every prime p, then

S_{\mathbf F,k}(N)=c_{\mathbf F,k}N+o_{\mathbf F,k}(N),\qquad c_{\mathbf F,k}=\prod_p\left(1-\frac{\#\Omega_p}{p^k}\right)>0.

Section 1, Introduction Read it in the paper TeX source, line 136

Show the TeX

Fix an integer $k\ge2$. Let $F\in\Z[x]\setminus\{0\}$ be separable over
$\Q$, with every nonconstant irreducible factor of degree at most $k+2$.
If $\rho_F(p^k)<p^k$ for every prime $p$, then
\[
 S_{F,k}(N)=c_{F,k}N+o_{F,k}(N),\qquad
 c_{F,k}=\prod_p\left(1-\frac{\rho_F(p^k)}{p^k}\right)>0.
\]
More generally, let $\mathbf F=(F_1,\ldots,F_r)$, with $r\ge1$, be a fixed
family of nonzero integer polynomials, each separable over $\Q$ with all
nonconstant irreducible factors of degree at most $k+2$. Factors may be
shared by different $F_i$. Put
\[
 \Omega_p=\{a\in\Z/p^k\Z:p^k\mid F_i(a)\text{ for at least one }i\}.
\]
Let $S_{\mathbf F,k}(N)$ count the integers $1\le n\le N$ for which every
$F_i(n)$ is $k$-power-free. If $\#\Omega_p<p^k$ for every prime $p$, then
\[
 S_{\mathbf F,k}(N)=c_{\mathbf F,k}N+o_{\mathbf F,k}(N),\qquad
 c_{\mathbf F,k}=\prod_p\left(1-\frac{\#\Omega_p}{p^k}\right)>0.
\]

Proposition

Let f\in\Z[x] be primitive and irreducible, with positive leading coefficient and degree 4\le d\le8. Put k=d-2. There exists \delta=\delta_f>0 such that

\#\{n\in\Z:X<n\le2X,\ p^k\mid f(n) \text{ for some prime }p>X\} \ll_f X^{1-\delta}.

Section 1, Introduction Read it in the paper TeX source, line 197

Show the TeX

Let $f\in\Z[x]$ be primitive and irreducible, with positive leading
coefficient and degree $4\le d\le8$. Put $k=d-2$. There exists
$\delta=\delta_f>0$ such that
\[
 \#\{n\in\Z:X<n\le2X,\ p^k\mid f(n)
          \text{ for some prime }p>X\}
 \ll_f X^{1-\delta}.
\]

Configuration and file facts

File: lean/ComparatorChallenges/PowerFreeValues.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/NumberTheory/PowerFree/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: 7 local vocabulary declarations
Lines of Lean, without comments or blank lines: 24
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