Lean statement

Cube-fiber cancellation for order-six Hadamard row ratios

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.HadamardSix.rowRatio_cubeFiber_sum_zero

Lean statement file: HadamardCubeFiber.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 266 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.

lemma rowRatio_cubeFiber_sum_zero {H : Matrix6} (hH : IsComplexHadamard H)    (hs : IsComplexHadamard (entrywiseSquare H)) {i j : Index} (hij : i ≠ j)    (a b : ℂ) (hne : a ≠ b)    (hab : ∀ k, rowRatio H i j k ^ 3 = a ∨ rowRatio H i j k ^ 3 = b) :    ∑ k ∈ cubeFiber (rowRatio H i j) a, H i k * star (H j k) = 0 := 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.HadamardSix.Index line 11
```
abbrev Index := Fin 6
```

abbreviation OAI.HadamardSix.Matrix6 line 12

abbrev Matrix6 := Matrix Index Index ℂ

definition OAI.HadamardSix.Unimodular line 14

def Unimodular (H : Matrix6) : Prop := ∀ i j, ‖H i j‖ = 1

definition OAI.HadamardSix.IsComplexHadamard line 16

def IsComplexHadamard (H : Matrix6) : Prop :=  Unimodular H ∧ H.conjTranspose * H = (6 : ℂ) • (1 : Matrix6)

definition OAI.HadamardSix.entrywiseSquare line 19

def entrywiseSquare (H : Matrix6) : Matrix6 := fun i j => H i j ^ 2

definition OAI.HadamardSix.cubeFiber line 21

noncomputable def cubeFiber (v : Index → ℂ) (z : ℂ) : Finset Index := by  classical  exact Finset.univ.filter (fun i => v i ^ 3 = z)

definition OAI.HadamardSix.rowRatio line 25

noncomputable def rowRatio (H : Matrix6) (i j : Index) : Index → ℂ :=  fun k => H i k * star (H j k)

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: cube, rows, entrywise, square, complex, matrix and hadamard. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: order and six.

Statements in Exact Fourier certificates for complex Hadamard matrices of order six come first because the release’s name for this Lean statement shares words with that paper’s title.

Proposition 4.1 of Exact Fourier certificates for complex Hadamard matrices of order six

Proposition (Classification of the cubic alternative)

Suppose that H and its entrywise square H^{\circ2} are Hadamard matrices of order six. By multiplying rows and columns by unit complex numbers and permuting rows and columns, one can transform H into the matrix T in (eq:cubic-matrix). In particular, every entry of the transformed matrix is a cube root of unity.

Section 4, The cubic alternative Read it in the paper TeX source, line 12

Show the TeX

Suppose that $H$ and its entrywise square $H^{\circ2}$ are Hadamard
matrices of order six. By multiplying rows and columns by unit complex
numbers and permuting rows and columns, one can transform $H$ into the
matrix $T$ in \eqref{eq:cubic-matrix}.
In particular, every entry of the transformed matrix is a cube root of
unity.

Other possible paper matches, closest wording first

Corollary 3.1 of Exact Fourier certificates for complex Hadamard matrices of order six. Words of the Lean statement, its names split into words, that also appear in the theorem: entrywise, row, square, matrix and hadamard.
Theorem 1.1 of Exact Fourier certificates for complex Hadamard matrices of order six. Words of the Lean statement, its names split into words, that also appear in the theorem: complex, matrix and hadamard. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: vanishing, fourier, order and six.
Proposition 5.2 of Exact Fourier certificates for complex Hadamard matrices of order six. Words of the Lean statement, its names split into words, that also appear in the theorem: matrix and hadamard. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: mutually, unbiased and bases.
Theorem 1.2 of Exact Fourier certificates for complex Hadamard matrices of order six. Found in the paper’s title, which its main theorem states: complex and hadamard. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bases.
Proposition 2.10 of The maximum number of mutually unbiased bases in dimension six. Words of the Lean statement, its names split into words, that also appear in the theorem: complex, matrix and hadamard. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: selected and ordering.
Proposition 3.4 of Exact Fourier certificates for complex Hadamard matrices of order six. Words of the Lean statement, its names split into words, that also appear in the theorem: matrix and hadamard.

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

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. 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

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 54 s; peak memory 6.1 GiB).
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 266’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
Their note: MUB6.fourier_and_family_bound states (∀ H, IsHadamard H → ¬ Equivalent H tao → ∀ π, g H (permuteCharge π alpha) = 0) ∧ (∀ n, Attainable n → n ≤ 5): the Fourier-vanishing statement (the Matolcsi-Ruzsa-Weiner companion) and only the weak family bound n ≤ 5 for mutually unbiased orthonormal bases of ℂ⁶ (IsMUBFamily B := ∀ r ≠ s, ∀ i j, |⟨B r i, B s j⟩|² = 1/6). Not stated: N(6) = 3, i.e. neither Attainable 3 nor the exclusion of four bases (docs: 'does not establish the paper's upper bound of three or its computer-assisted exclusion of four'). n ≤ 5 is the non-existence of 6 MUBs (with Weiner's gap theorem, not in Lean, this would give N(6) <= 4); the binary64 certificate for excluding four is outside the Lean statement. Cone is 2.83M lines of OAI, the largest in this range.
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: N(6) ≤ 3 (no four MUBs in ℂ⁶) and the attainability of three; Lean proves only ≤ 5.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“This is a supporting Fourier cancellation statement. The paper's full character-sum vanishing theorem and its mutually unbiased bases bound are outside this selected statement.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a supporting result”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

37 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: 22 to its own declarations, 6 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 rowRatio_cubeFiber_sum_zero 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 17 lines. 6 of them are named in the statement itself.

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

abbrev Indexline 11in the statementlibrary
abbrev Matrix6line 12in the statementlibrary
def Unimodularline 14library
def IsComplexHadamardline 16in the statementlibrary
def entrywiseSquareline 19in the statementlibrary
def cubeFiberline 21in the statementlibrary
def rowRatioline 25in the statementlibrary
lemma rowRatio_cubeFiber_sum_zeroline 28selected theoremlibrary

import Mathlibnamespace OAI/-! Cancellation of row ratios on a prescribed cubic fibre. -/open scoped BigOperators Matrixnamespace HadamardSixabbrev Index := Fin 6abbrev Matrix6 := Matrix Index Index ℂdef Unimodular (H : Matrix6) : Prop := ∀ i j, ‖H i j‖ = 1def IsComplexHadamard (H : Matrix6) : Prop :=  Unimodular H ∧ H.conjTranspose * H = (6 : ℂ) • (1 : Matrix6)def entrywiseSquare (H : Matrix6) : Matrix6 := fun i j => H i j ^ 2noncomputable def cubeFiber (v : Index → ℂ) (z : ℂ) : Finset Index := by  classical  exact Finset.univ.filter (fun i => v i ^ 3 = z)noncomputable def rowRatio (H : Matrix6) (i j : Index) : Index → ℂ :=  fun k => H i k * star (H j k)lemma rowRatio_cubeFiber_sum_zero {H : Matrix6} (hH : IsComplexHadamard H)    (hs : IsComplexHadamard (entrywiseSquare H)) {i j : Index} (hij : i ≠ j)    (a b : ℂ) (hne : a ≠ b)    (hab : ∀ k, rowRatio H i j k ^ 3 = a ∨ rowRatio H i j k ^ 3 = b) :    ∑ k ∈ cubeFiber (rowRatio H i j) a, H i k * star (H j k) = 0 := by  sorryend HadamardSixend 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 maximum number of mutually unbiased bases in dimension six

Theorem (Computer-assisted)

Under the arithmetic and compiler conditions in Appendix sec:rounding, the fresh complete execution documented in Section execute:observed certifies that there are no four pairwise mutually unbiased orthonormal bases in \C^6. Consequently, N(6)=3.

Section 1, Introduction Read it in the paper TeX source, line 13

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Under the arithmetic and compiler conditions in
Appendix~\ref{sec:rounding}, the fresh complete execution documented in
Section~\ref{execute:observed} certifies that there are no four pairwise mutually unbiased orthonormal bases in
$\C^6$. Consequently, $N(6)=3$.

Corollary (Adjoint-closed Cartan subalgebras)

Under the arithmetic and compiler conditions of Appendix sec:rounding, using the complete execution documented in Section execute:observed as in Theorem thm:main, the maximum number of pairwise Killing-orthogonal Cartan subalgebras of \mathfrak{sl}_6(\C) that are all closed under the same standard adjoint X\mapsto X^*=\overline X^{\,T} is three.

Section 1, Introduction Read it in the paper TeX source, line 230

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Under the arithmetic and compiler conditions of
Appendix~\ref{sec:rounding}, using the complete execution documented in
Section~\ref{execute:observed} as in Theorem~\ref{thm:main},
the maximum number of pairwise Killing-orthogonal Cartan subalgebras of
$\mathfrak{sl}_6(\C)$ that are all closed under the same standard adjoint
$X\mapsto X^*=\overline X^{\,T}$ is three.

Exact Fourier certificates for complex Hadamard matrices of order six

Theorem (Fourier vanishing)

Let H be a complex Hadamard matrix of order six. If H is not equivalent to the cubic matrix T in (eq:cubic-matrix), then

g_H(\pi\alpha)=0\qquad\text{for every coordinate permutation }\pi.

Section 1, Introduction Read it in the paper TeX source, line 65

Show the TeX

Let $H$ be a complex Hadamard matrix of order six. If $H$ is not equivalent
to the cubic matrix $T$ in \eqref{eq:cubic-matrix}, then
\[
 g_H(\pi\alpha)=0\qquad\text{for every coordinate permutation }\pi.
\]

Theorem (No complete family)

There do not exist seven orthonormal bases B_r=(b_{r,0},\ldots,b_{r,5}) of \C^6, 1\le r\le7, such that

\bigl|\langle b_{r,i},b_{s,j}\rangle\bigr|^2=\frac16 \qquad(r\ne s,\ 0\le i,j\le5).

Section 1, Introduction Read it in the paper TeX source, line 142

Show the TeX

There do not exist seven orthonormal bases
$B_r=(b_{r,0},\ldots,b_{r,5})$ of $\C^6$, $1\le r\le7$, such that
\[
 \bigl|\langle b_{r,i},b_{s,j}\rangle\bigr|^2=\frac16
 \qquad(r\ne s,\ 0\le i,j\le5).
\]

Configuration and file facts

File: lean/ComparatorChallenges/HadamardCubeFiber.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 266 link it.
Solution module: OAI.LinearAlgebra.Hadamard.Main
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: 17
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.

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