Polynomial removal fails for ordered binary matrices
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Disproves polynomial ordered binary matrix removal with one fixed \(66\times66\) zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance. Copies preserve row and column orders and match both zeros and ones. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
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
We construct a fixed \(66\times66\) binary matrix for which ordered matrix removal has no polynomial bound. This disproves the polynomial ordered binary matrix-removal conjecture. Ordered copies preserve the separate row and column orders and match both zeros and ones; removal permits changing entries in either direction.
One Lean statement in this family
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MatrixRemoval.lean, the release lists it as a main result.
theoremno_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) := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Other mathematical statements in OpenAI’s Lean library
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Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
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.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: 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.
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. Row and column indices are independently increasing, and every zero and one entry must match. The construction gives an explicit sequence of such matrices, allowing edits in both directions.
AI-drafted comparisons, with no recorded human review
Failure of polynomial removal for ordered binary matricesMatrixRemoval
Unreviewed AI draft: states a weaker form of a paper’s theorem. States the qualitative conclusion of the main theorem for the explicit 66 × 66 pattern: no constants c, C give a polynomial removal bound. The theorem’s explicit sizes n_h, distances ε_h and copy bound ε_h 2^{-h} are not asserted.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Failure of polynomial removal for ordered binary matricesMatrixRemoval
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
These reports examine claims in the papers using exact calculations for finite claims, independently of the release’s Lean proofs. Read each report for the exact claim it covers.
Theorem 1.1 of Polynomial removal fails for ordered binary matrices, 25 September 2026
Reported by
cert-machine
Method
Exact computation
Their verdict
“CERTIFIED” Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: Theorem thm:main at h = 1 and h = 2 (A_1 776 x 776, A_2 3,096 x 3,096), both inequalities decided exactly (N_H(A_h) = 2 m^130 by an exhaustive ordered-copy search; dist_H(A_h) >= eps_h by the sampling argument with every finite step checked, for h <= 6), and the finite lemmas of Section 3; the theorem for every h (the headline) rests on the paper’s uniform proofs and is not decided here”
What these papers cite in the release, and what cites them
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
Follow-up work by others
wzn1118, A 4x4 obstruction to polynomial ordered-matrix removal, with complete Lean 4 source (issue 1), 11 October 2026. Source
Claimed 4x4 obstruction to polynomial removal for ordered binary matrices with 550 Lean modules, improving the release’s 66x66 pattern.
References outside the release
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release
Explain a result, compare statements or propose a correction
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Field-specific journal”. The page says that nothing on it is verified.
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Read the Lean statement against the paper
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Identify the theorem the catalogue describes
Family 190The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Compare the proposed paper and Lean matches
Family 190Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing.Open the form
Why this passage is shown
Theorem 1.1
In Polynomial removal fails for ordered binary matrices, 25 September 2026
How it was found
Its words were compared with the words of OpenAI’s catalogue description.
Words they share
binary, bound, fixed, matrix, polynomial, removal
Review
Atlas records no review confirming that this is the theorem OpenAI’s description refers to.
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
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.
The challenge statements state the headline as OpenAI wrote it.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Theorem 1.1 of Polynomial removal fails for ordered binary matrices, 25 September 2026
Exact calculations for finite claims, cert-machine
Their verdict
“CERTIFIED”
Claim tested or proved
Exact code written independently from the paper confirms that the published object has the claimed property.
What it covers
“a finite component: Theorem thm:main at h = 1 and h = 2 (A_1 776 x 776, A_2 3,096 x 3,096), both inequalities decided exactly (N_H(A_h) = 2 m^130 by an exhaustive ordered-copy search; dist_H(A_h) >= eps_h by the sampling argument with every finite step checked, for h <= 6), and the finite lemmas of Section 3; the theorem for every h (the headline) rests on the paper’s uniform proofs and is not decided here”