Lean statement

Order-six Hadamard Fourier vanishing and a five-basis upper bound

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.MUB6.fourier_and_family_bound

Lean statement file: MUBSix.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.

theorem fourier_and_family_bound :    (∀ H : CMatrix, IsHadamard H → ¬Equivalent H tao →      ∀ π : Equiv.Perm Coord, g H (permuteCharge π alpha) = 0) ∧    (∀ n : ℕ, Attainable n → n ≤ 5) := 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.MUB6.Coord line 10
```
abbrev Coord := Fin 6
```

abbreviation OAI.MUB6.CMatrix line 11

abbrev CMatrix := Matrix Coord Coord ℂ

abbreviation OAI.MUB6.Charge line 12
```
abbrev Charge := Coord → ℤ
```

definition OAI.MUB6.IsHadamard line 14

def IsHadamard (H : CMatrix) : Prop :=  (∀ i j, Complex.normSq (H i j) = 1) ∧  H.conjTranspose * H = (6 : ℂ) • (1 : CMatrix)

definition OAI.MUB6.character line 18

def character (x : Coord → ℂ) (a : Charge) : ℂ := ∏ i, x i ^ a i

definition OAI.MUB6.g line 20

def g (H : CMatrix) (a : Charge) : ℂ :=  (6 : ℂ)⁻¹ * ∑ k, character (fun i => H i k) a

definition OAI.MUB6.alpha line 23

def alpha : Charge := ![1, 1, 1, -1, -1, -1]

definition OAI.MUB6.permuteCharge line 25

def permuteCharge (π : Equiv.Perm Coord) (a : Charge) : Charge := a ∘ π

definition OAI.MUB6.Equivalent line 28

def Equivalent (H K : CMatrix) : Prop :=  ∃ (r c : Equiv.Perm Coord) (u v : Coord → ℂ),    (∀ i, Complex.normSq (u i) = 1) ∧ (∀ j, Complex.normSq (v j) = 1) ∧    ∀ i j, K i j = u i * H (r i) (c j) * v j

definition OAI.MUB6.omega line 33

def omega : ℂ := Complex.exp (2 * Real.pi * Complex.I / 3)

definition OAI.MUB6.taoExponent line 37

def taoExponent : Matrix Coord Coord ℕ :=  !![0, 0, 0, 0, 0, 0;     0, 0, 1, 2, 2, 1;     0, 1, 0, 1, 2, 2;     0, 2, 1, 0, 1, 2;     0, 2, 2, 1, 0, 1;     0, 1, 2, 2, 1, 0]

definition OAI.MUB6.tao line 45

def tao : CMatrix := fun i j => omega ^ taoExponent i j

abbreviation OAI.MUB6.Space line 47

abbrev Space := EuclideanSpace ℂ Coord

abbreviation OAI.MUB6.Basis line 48

abbrev Basis := OrthonormalBasis Coord ℂ Space

definition OAI.MUB6.IsMUBFamily line 50

def IsMUBFamily {n : ℕ} (B : Fin n → Basis) : Prop :=  ∀ r s, r ≠ s → ∀ i j,    Complex.normSq (inner ℂ (B r i) (B s j)) = (1 : ℝ) / 6

definition OAI.MUB6.Attainable line 54

def Attainable (n : ℕ) : Prop := ∃ B : Fin n → Basis, IsMUBFamily B

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: equivalent, fourier, cubic, permutation, pi, complex, alpha, matrix and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: vanishing, coordinate, 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.

Theorem 1.1 of 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
\[
 g_H(\pi\alpha)=0\qquad\text{for every coordinate permutation }\pi.
\]

Other possible paper matches, closest wording first

Proposition 4.1 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: entry, unit, cubic, permuting, complex, matrix and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: matrices, order and six.
Theorem 1.2 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: 1/6, family and orthonormal. Found in the paper’s title, which its main theorem states: fourier, complex and hadamard. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bases.
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: permutation, pi, alpha, matrix and hadamard.
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.
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: alpha, matrix and hadamard.
Proposition 5.2 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: bounds, pi and alpha. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: upper.

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: 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…

No verdict reached Their term: “REFUSED”.
Re-run through Comparator did not reach a verdict (the job stopped before Comparator ran (see the run log)).
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.

Built; Comparator not run Their term: “closure-pass”.
The solution was built and its dependency closure checked; Comparator was not run.
Reported undated, 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 16981 s; peak memory 26.1 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

“The paper claims that at most three mutually unbiased orthonormal bases exist in $\mathbb C^6$. The linked formalization proves a weaker family bound: every family in its mutually unbiased bases model has at most five members. It also proves a Fourier character-sum vanishing statement for order-six complex Hadamard matrices not equivalent to the Tao matrix, uniformly over coordinate permutations.”
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

66 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: 37 to its own declarations, 18 to Mathlib’s documentation. 17 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 fourier_and_family_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 16 other declarations of the file; with the theorem itself they span 35 lines. 9 of them are named in the statement itself.

Names in the namespace OAI.MUB6 are listed without it.

abbrev Coordline 10in the statementlibrary
abbrev CMatrixline 11in the statementlibrary
abbrev Chargeline 12library
def IsHadamardline 14in the statementlibrary
def characterline 18library
def gline 20in the statementlibrary
def alphaline 23in the statementlibrary
def permuteChargeline 25in the statementlibrary
def Equivalentline 28in the statementlibrary
def omegaline 33library
def taoExponentline 37library
def taoline 45in the statementlibrary
abbrev Spaceline 47library
abbrev Basisline 48library
def IsMUBFamilyline 50library
def Attainableline 54in the statementlibrary
theorem fourier_and_family_boundline 56selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperators ComplexConjugatenamespace MUB6abbrev Coord := Fin 6abbrev CMatrix := Matrix Coord Coord ℂabbrev Charge := Coord → ℤdef IsHadamard (H : CMatrix) : Prop :=  (∀ i j, Complex.normSq (H i j) = 1) ∧  H.conjTranspose * H = (6 : ℂ) • (1 : CMatrix)def character (x : Coord → ℂ) (a : Charge) : ℂ := ∏ i, x i ^ a idef g (H : CMatrix) (a : Charge) : ℂ :=  (6 : ℂ)⁻¹ * ∑ k, character (fun i => H i k) adef alpha : Charge := ![1, 1, 1, -1, -1, -1]def permuteCharge (π : Equiv.Perm Coord) (a : Charge) : Charge := a ∘ π/-- Full Hadamard equivalence, with arbitrary unit complex phases. -/def Equivalent (H K : CMatrix) : Prop :=  ∃ (r c : Equiv.Perm Coord) (u v : Coord → ℂ),    (∀ i, Complex.normSq (u i) = 1) ∧ (∀ j, Complex.normSq (v j) = 1) ∧    ∀ i j, K i j = u i * H (r i) (c j) * v jdef omega : ℂ := Complex.exp (2 * Real.pi * Complex.I / 3)/-- The five-cycle exponent matrix in equation (cubic-matrix).The entries 2 represent the exponents -1 modulo three. -/def taoExponent : Matrix Coord Coord ℕ :=  !![0, 0, 0, 0, 0, 0;     0, 0, 1, 2, 2, 1;     0, 1, 0, 1, 2, 2;     0, 2, 1, 0, 1, 2;     0, 2, 2, 1, 0, 1;     0, 1, 2, 2, 1, 0]def tao : CMatrix := fun i j => omega ^ taoExponent i jabbrev Space := EuclideanSpace ℂ Coordabbrev Basis := OrthonormalBasis Coord ℂ Spacedef IsMUBFamily {n : ℕ} (B : Fin n → Basis) : Prop :=  ∀ r s, r ≠ s → ∀ i j,    Complex.normSq (inner ℂ (B r i) (B s j)) = (1 : ℝ) / 6def Attainable (n : ℕ) : Prop := ∃ B : Fin n → Basis, IsMUBFamily Btheorem fourier_and_family_bound :    (∀ H : CMatrix, IsHadamard H → ¬Equivalent H tao →      ∀ π : Equiv.Perm Coord, g H (permuteCharge π alpha) = 0) ∧    (∀ n : ℕ, Attainable n → n ≤ 5) := by  sorryend MUB6endend 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

Show the TeX

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

Show the TeX

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/MUBSix.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.Analysis.MutuallyUnbiased.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: 16 local vocabulary declarations
Lines of Lean, without comments or blank lines: 35
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