Lean statement

Failure of polynomial removal for ordered binary matrices

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.Problem348.no_polynomial_removal_bound

Lean statement file: MatrixRemoval.leanShow on the mapFamily 190. 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 no_polynomial_removal_bound :    ∀ c C : ℝ, 0 < c → 0 < C →  ∃ n : ℕ, 1 ≤ n ∧    ∃ ε : ℝ, 0 < ε ∧ ε < 1 ∧      ∃ A : BinaryMatrix n,        fixedDistance A ≥ ε ∧        (copyCount fixedH A : ℝ) <          c * Real.rpow ε C * ((n : ℝ) ^ 132) := 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.

abbreviation OAI.Problem348.BinaryMatrix line 9

abbrev BinaryMatrix (n : ℕ) := Fin n → Fin n → Bool

abbreviation OAI.Problem348.IncreasingMap line 11

abbrev IncreasingMap (k n : ℕ) := {f : Fin k → Fin n // StrictMono f}

definition OAI.Problem348.orderedCopies line 13

def orderedCopies {k n : ℕ} (H : BinaryMatrix k) (A : BinaryMatrix n) :    Finset (IncreasingMap k n × IncreasingMap k n) := by  classical  exact Finset.univ.filter (fun rc =>    ∀ i j, A ((rc.1).val i) ((rc.2).val j) = H i j)

definition OAI.Problem348.copyCount line 19

def copyCount {k n : ℕ} (H : BinaryMatrix k) (A : BinaryMatrix n) : ℕ :=  (orderedCopies H A).card

definition OAI.Problem348.HFree line 22

def HFree {k n : ℕ} (H : BinaryMatrix k) (A : BinaryMatrix n) : Prop :=  copyCount H A = 0

definition OAI.Problem348.hammingDistance line 25

def hammingDistance {n : ℕ} (A B : BinaryMatrix n) : ℕ := by  classical  exact (Finset.univ.filter (fun rc : Fin n × Fin n =>    A rc.1 rc.2 ≠ B rc.1 rc.2)).card

definition OAI.Problem348.etaBit line 30

def etaBit (a b : ℕ) : Bool :=  decide (((a / (2 ^ (5 - b))) % 2) = 1)

definition OAI.Problem348.anchor64 line 33

def anchor64 (u v : Fin 64) : Bool :=  if 32 ≤ u.val ∧ 59 ≤ v.val then    etaBit (u.val - 32) (v.val - 58)  else    decide (u.val ≠ v.val)

definition OAI.Problem348.fixedH line 39

def fixedH : BinaryMatrix 66 := fun u v =>

Show the remaining 12 lines

  if hu : u.val < 64 then    if hv : v.val < 64 then      anchor64 ⟨u.val, hu⟩ ⟨v.val, hv⟩    else      decide ((v.val = 64 ∧ u.val = 0) ∨        (v.val = 65 ∧ u.val = 1))  else    if v.val < 64 then      decide ((u.val = 64 ∧ v.val = 0) ∨        (u.val = 65 ∧ v.val = 1))    else      decide (u.val = 65 ∨ v.val = 64)

definition OAI.Problem348.fixedMinEdits line 53

def fixedMinEdits {n : ℕ} (A : BinaryMatrix n) : ℕ := by  classical  exact    (Finset.univ.image (fun B : BinaryMatrix n =>      if HFree fixedH B then hammingDistance A B else n * n + 1)).min'      (by simp)

definition OAI.Problem348.fixedDistance line 60

def fixedDistance {n : ℕ} (A : BinaryMatrix n) : ℝ :=  (fixedMinEdits A : ℝ) / ((n : ℝ) ^ 2)

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: bound, polynomial, removal, binary, matrix and integer. Found in the paper’s title, which its main theorem states: ordered. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: times.

Theorem 1.1 of Polynomial removal fails for ordered binary matrices

Theorem

There is a fixed binary 66\times66 matrix H with the following property. For every integer h\ge1, put

n_h=(386h+2)2^h,\qquad \epsilon_h=(386h+2)^{-2}.

There is a binary n_h\times n_h matrix A_h such that

\operatorname{dist}_H(A_h)\ge\epsilon_h, \qquad \frac{N_H(A_h)}{n_h^{132}}\le\epsilon_h2^{-h}.

Consequently, no positive constants c,C give a polynomial removal bound for this fixed H.

Section 1, Introduction Read it in the paper TeX source, line 33

Show the TeX

There is a fixed binary $66\times66$ matrix $H$ with the following
property. For every integer $h\ge1$, put
\[
 n_h=(386h+2)2^h,\qquad \epsilon_h=(386h+2)^{-2}.
\]
There is a binary $n_h\times n_h$ matrix $A_h$ such that
\[
 \operatorname{dist}_H(A_h)\ge\epsilon_h,
 \qquad
 \frac{N_H(A_h)}{n_h^{132}}\le\epsilon_h2^{-h}.
\]
Consequently, no positive constants $c,C$ give a polynomial removal
bound for this fixed $H$.

Other possible paper matches, closest wording first

Proposition 3.1 of Polynomial removal fails for ordered binary matrices. Words of the Lean statement, its names split into words, that also appear in the theorem: free, binary, matrix and order. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least and times.
Proposition 4.6 of Polynomial removal fails for ordered binary matrices. Words of the Lean statement, its names split into words, that also appear in the theorem: maps, copy, integer and ordered. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: pattern and times.
Corollary 4.5 of Polynomial removal fails for ordered binary matrices. Words of the Lean statement, its names split into words, that also appear in the theorem: anchor, copy and ordered. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.
Corollary 5.1 of Polynomial removal fails for ordered binary matrices. Words of the Lean statement, its names split into words, that also appear in the theorem: copy, integer and ordered. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: induced and 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 190’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: no_polynomial_removal_bound : ∀ c C > 0, ∃ n ≥ 1, ∃ ε ∈ (0,1), ∃ A : BinaryMatrix n, fixedDistance A ≥ ε ∧ copyCount fixedH A < c * ε^C * n^132 for the explicit 66x66 fixedH; copyCount counts pairs of strictly increasing row/column maps with A (r i) (c j) = H i j (ordered, zeros and ones matched) and fixedMinEdits is a true min Hamming distance to pattern-free matrices (junk branch n*n+1 exceeds any distance). Matches the summary and docs.
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 disproves a polynomial removal bound for one explicit $66\times66$ binary pattern. For every $c,C>0$, there is an $n\times n$ binary matrix at normalized edit distance at least $\varepsilon>0$ from being pattern-free but with fewer than $c\varepsilon^C n^{132}$ induced ordered copies.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a weaker form of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

77 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: 25 to its own declarations, 21 to Mathlib’s documentation. 12 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 no_polynomial_removal_bound 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 11 other declarations of the file; with the theorem itself they span 52 lines. 4 of them are named in the statement itself.

Names in the namespace OAI.Problem348 are listed without it.

abbrev BinaryMatrixline 9in the statementlibrary
abbrev IncreasingMapline 11library
def orderedCopiesline 13library
def copyCountline 19in the statementlibrary
def HFreeline 22library
def hammingDistanceline 25library
def etaBitline 30library
def anchor64line 33library
def fixedHline 39in the statementlibrary
def fixedMinEditsline 53library
def fixedDistanceline 60in the statementlibrary
theorem no_polynomial_removal_boundline 63selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionnamespace Problem348abbrev BinaryMatrix (n : ℕ) := Fin n → Fin n → Boolabbrev IncreasingMap (k n : ℕ) := {f : Fin k → Fin n // StrictMono f}def orderedCopies {k n : ℕ} (H : BinaryMatrix k) (A : BinaryMatrix n) :    Finset (IncreasingMap k n × IncreasingMap k n) := by  classical  exact Finset.univ.filter (fun rc =>    ∀ i j, A ((rc.1).val i) ((rc.2).val j) = H i j)def copyCount {k n : ℕ} (H : BinaryMatrix k) (A : BinaryMatrix n) : ℕ :=  (orderedCopies H A).carddef HFree {k n : ℕ} (H : BinaryMatrix k) (A : BinaryMatrix n) : Prop :=  copyCount H A = 0def hammingDistance {n : ℕ} (A B : BinaryMatrix n) : ℕ := by  classical  exact (Finset.univ.filter (fun rc : Fin n × Fin n =>    A rc.1 rc.2 ≠ B rc.1 rc.2)).carddef etaBit (a b : ℕ) : Bool :=  decide (((a / (2 ^ (5 - b))) % 2) = 1)def anchor64 (u v : Fin 64) : Bool :=  if 32 ≤ u.val ∧ 59 ≤ v.val then    etaBit (u.val - 32) (v.val - 58)  else    decide (u.val ≠ v.val)def fixedH : BinaryMatrix 66 := fun u v =>  if hu : u.val < 64 then    if hv : v.val < 64 then      anchor64 ⟨u.val, hu⟩ ⟨v.val, hv⟩    else      decide ((v.val = 64 ∧ u.val = 0) ∨        (v.val = 65 ∧ u.val = 1))  else    if v.val < 64 then      decide ((u.val = 64 ∧ v.val = 0) ∨        (u.val = 65 ∧ v.val = 1))    else      decide (u.val = 65 ∨ v.val = 64)def fixedMinEdits {n : ℕ} (A : BinaryMatrix n) : ℕ := by  classical  exact    (Finset.univ.image (fun B : BinaryMatrix n =>      if HFree fixedH B then hammingDistance A B else n * n + 1)).min'      (by simp)def fixedDistance {n : ℕ} (A : BinaryMatrix n) : ℝ :=  (fixedMinEdits A : ℝ) / ((n : ℝ) ^ 2)theorem no_polynomial_removal_bound :    ∀ c C : ℝ, 0 < c → 0 < C →  ∃ n : ℕ, 1 ≤ n ∧    ∃ ε : ℝ, 0 < ε ∧ ε < 1 ∧      ∃ A : BinaryMatrix n,        fixedDistance A ≥ ε ∧        (copyCount fixedH A : ℝ) <          c * Real.rpow ε C * ((n : ℝ) ^ 132) := by  sorryend Problem348endend 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.

Polynomial removal fails for ordered binary matrices