Lean statement

Complete Crouzeix inequality and sharpness

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.

Theorems selected for verification: OAI.DirectCrouzeix.complete_crouzeix OAI.DirectCrouzeix.uniform_sharpness

Lean statement file: DirectCrouzeix.leanShow on the mapFamily 325. 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 complete_crouzeix : UniversalBound 2 := by  sorry⋮theorem uniform_sharpness (c : ℝ) (hc : UniversalBound c) : 2 ≤ c := 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.DirectCrouzeix.numericalRange line 11

def numericalRange {n : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) : Set ℂ :=  {z | ∃ u : EuclideanSpace ℂ (Fin n), ‖u‖ = 1 ∧    inner ℂ u (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℂ) A u) = z}

definition OAI.DirectCrouzeix.polynomialValue line 15

def polynomialValue {m d : ℕ} (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ)    (z : ℂ) : Matrix (Fin m) (Fin m) ℂ :=  ∑ k : Fin (d + 1), z ^ (k : ℕ) • B k

definition OAI.DirectCrouzeix.tensorEvaluation line 19

def tensorEvaluation {n m d : ℕ} (A : Matrix (Fin n) (Fin n) ℂ)    (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) :    Matrix (Fin n × Fin m) (Fin n × Fin m) ℂ :=  ∑ k : Fin (d + 1), (A ^ (k : ℕ)) ⊗ₖ (B k)

definition OAI.DirectCrouzeix.rangeMaximum line 24

def rangeMaximum {n m d : ℕ} (A : Matrix (Fin n) (Fin n) ℂ)    (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) : ℝ :=  sSup ((fun z => ‖polynomialValue B z‖) '' numericalRange A)

definition OAI.DirectCrouzeix.UniversalBound line 28

def UniversalBound (c : ℝ) : Prop :=  ∀ (n m d : ℕ), 0 < n → 0 < m →    ∀ (A : Matrix (Fin n) (Fin n) ℂ)      (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ),      ‖tensorEvaluation A B‖ ≤ c * rangeMaximum A 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: real, bound, valued, integers and polynomial. Found in the paper’s title, which its main theorem states: crouzeix and complete. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.

Statements in A direct proof of the complete Crouzeix inequality come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 3.2 of A direct proof of the complete Crouzeix inequality

Theorem (The bound on an analytic convex domain)

For positive integers n,m, let \Omega\subset\C be bounded, open, and convex with regular real-analytic Jordan boundary, and let A\in M_n(\C) satisfy W(A)\subset\Omega. For every M_m(\C)-valued polynomial F with \|F(z)\|\le1 on \partial\Omega, one has \|F[A]\|\le2.

Section 3, Singular vectors and the amplified estimate Read it in the paper TeX source, line 323

Show the TeX

For positive integers $n,m$, let $\Omega\subset\C$ be bounded, open, and convex with regular real-analytic Jordan boundary, and let $A\in M_n(\C)$ satisfy $W(A)\subset\Omega$. For every $M_m(\C)$-valued polynomial $F$ with $\|F(z)\|\le1$ on $\partial\Omega$, one has $\|F[A]\|\le2$.

Other possible paper matches, closest wording first

Corollary 6.3 of The complete Crouzeix theorem: optimal similarity and a common positive boundary representation. Words of the Lean statement, its names split into words, that also appear in the theorem: tensor, range, numerical, complex, evaluation, space and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: norm, size, inequalities, coefficient, constant and positive.
Theorem 1.1 of A direct proof of the complete Crouzeix inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: integers and polynomial. Found in the paper’s title, which its main theorem states: crouzeix and complete. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant and positive.
Theorem 2.1 of A direct proof of the complete Crouzeix inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: real, bounded and polynomial. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: degree, coefficients, constant and positive.
Proposition 2.2 of A direct proof of the complete Crouzeix inequality. Words of the Lean statement, its names split into words, that also appear in the theorem: valued, integer and polynomial. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: norm, coefficient and positive.
Corollary 6.1 of The complete Crouzeix theorem: optimal similarity and a common positive boundary representation. Words of the Lean statement, its names split into words, that also appear in the theorem: complete, matrix, integers and polynomial. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: smaller, degree, inequality, constant and positive.
Proposition 3.3 of The complete Crouzeix theorem: optimal similarity and a common positive boundary representation. Words of the Lean statement, its names split into words, that also appear in the theorem: space, matrix, bounded, valued and polynomial. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: size and coefficient.

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 213 s; peak memory 6.5 GiB).
Reported 11 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 325’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: CrouzeixHilbert.hilbert states, for every complete complex inner-product space H (any dimension, no separability) and bounded A, ‖Σ_k A^k ⊗ B_k‖ ≤ 2·sup_{W(A)} ‖Σ z^k B_k‖ for all m > 0, degrees d and complex m×m coefficients (numerical range {⟨x,Ax⟩ : ‖x‖ = 1}, amplification = completed H ⊗ ℂ^m), plus an explicit SharpConstant witness (A on ℂ², scalar degree-1 polynomial, sup = 1 and ‖P[A]‖ = 2); CompleteCrouzeix/DirectCrouzeix give the matrix case with UniversalBound 2 ∧ ∀ C, UniversalBound C → 2 ≤ C. HilbertCrouzeix (compactness/convexity of the closed numerical range, finite compressions, trivial space) is supporting-only and StructuralCrouzeix (optimal similarity and boundary density for regular analytic convex Jordan domains) is an extra finite-dimensional result.
Definitions the group asks readers to compare with the paper: HilbertCrouzeix: def chosenContour (K U : Set ℂ) : ℝ → ℂ := Classical.epsilon (IsCauchyContour K U) feeds holomorphicCalculus, but the only theorem using it (zeroConclusion) is for a subsingleton H, where every operator is 0, so the choice cannot matter ; CrouzeixHilbert: holomorphicEval A U f := if h : Nonempty (CalculusContour (numericalClosure A) U) then contourEval A (Classical.choice h).toSmoothContour f else 0; harmless because NonzeroConclusion itself asserts the contour exists and that the value equals contourEval A Γ for every contour Γ ; supNorm S F := sSup (insert 0 ((fun z => ‖F z‖) '' S)) (the insert 0 only handles the empty numerical range of the zero space)
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 complete Crouzeix inequality bounds a matrix-valued polynomial evaluated at a matrix by its maximum norm on the numerical range. The formalized result proves the bound with constant $2$ for every positive matrix and coefficient size, every polynomial degree, and arbitrary complex coefficients. It also proves that no smaller universal constant works. Normality of the matrix and nonempty interior of its numerical range are not assumed.”
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

44 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: 6 to its own declarations, 45 to Mathlib’s documentation. 7 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 complete_crouzeix 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 5 other declarations of the file; with the theorem itself they span 22 lines. 1 of them is named in the statement itself.

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

def numericalRangeline 11library
def polynomialValueline 15library
def tensorEvaluationline 19library
def rangeMaximumline 24library
def UniversalBoundline 28in the statementlibrary
theorem complete_crouzeixline 34selected theoremlibrary
theorem uniform_sharpnessline 37selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped Matrix Matrix.Norms.L2Operator Kroneckernamespace DirectCrouzeixdef numericalRange {n : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) : Set ℂ :=  {z | ∃ u : EuclideanSpace ℂ (Fin n), ‖u‖ = 1 ∧    inner ℂ u (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℂ) A u) = z}def polynomialValue {m d : ℕ} (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ)    (z : ℂ) : Matrix (Fin m) (Fin m) ℂ :=  ∑ k : Fin (d + 1), z ^ (k : ℕ) • B kdef tensorEvaluation {n m d : ℕ} (A : Matrix (Fin n) (Fin n) ℂ)    (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) :    Matrix (Fin n × Fin m) (Fin n × Fin m) ℂ :=  ∑ k : Fin (d + 1), (A ^ (k : ℕ)) ⊗ₖ (B k)def rangeMaximum {n m d : ℕ} (A : Matrix (Fin n) (Fin n) ℂ)    (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ) : ℝ :=  sSup ((fun z => ‖polynomialValue B z‖) '' numericalRange A)def UniversalBound (c : ℝ) : Prop :=  ∀ (n m d : ℕ), 0 < n → 0 < m →    ∀ (A : Matrix (Fin n) (Fin n) ℂ)      (B : Fin (d + 1) → Matrix (Fin m) (Fin m) ℂ),      ‖tensorEvaluation A B‖ ≤ c * rangeMaximum A Btheorem complete_crouzeix : UniversalBound 2 := by  sorrytheorem uniform_sharpness (c : ℝ) (hc : UniversalBound c) : 2 ≤ c := by  sorryend DirectCrouzeixendend 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.

A direct proof of the complete Crouzeix inequality

Theorem

For every pair of positive integers n,m, every A\in M_n(\C), every nonnegative integer d, and every polynomial P(z)=\sum_{k=0}^d B_kz^k with B_k\in M_m(\C),

\|P[A]\|\le2\max_{z\in W(A)}\|P(z)\|.

The constant two cannot be decreased uniformly over n,m, and d.

Section 1, Introduction Read it in the paper TeX source, line 58

Show the TeX

For every pair of positive integers $n,m$, every $A\in M_n(\C)$,
every nonnegative integer $d$, and every polynomial
$P(z)=\sum_{k=0}^d B_kz^k$ with $B_k\in M_m(\C)$,
\begin{equation}
 \|P[A]\|\le2\max_{z\in W(A)}\|P(z)\|.
\end{equation}
The constant two cannot be decreased uniformly over $n,m$, and $d$.

The complete Crouzeix theorem: optimal similarity and a common positive boundary representation

Theorem (The optimal metric)

For every admissible \Omega, every finite n\ge1, every A\in M_n(\C) with W(A)\subset\Omega, and the interior conformal coordinate f above, the problem (eq:minimum) is strictly feasible and has an attained minimum 1\le\kappa^2\le4. If H is a minimizing metric, then

S=H^{1/2},\qquad A'=SAS^{-1},\qquad D=Sf(A)S^{-1}=f(A')

satisfy

D^*D\preceq I,\qquad \rho(D)<1, \qquad \|S\|\|S^{-1}\|=\kappa\le2.

Section 1, The structural question Read it in the paper TeX source, line 116

Show the TeX

For every admissible $\Omega$, every finite $n\ge1$, every
$A\in M_n(\C)$ with $W(A)\subset\Omega$, and the interior conformal
coordinate $f$ above, the problem \eqref{eq:minimum} is strictly feasible
and has an attained minimum $1\le\kappa^2\le4$. If $H$ is a minimizing
metric, then
\[
 S=H^{1/2},\qquad A'=SAS^{-1},\qquad D=Sf(A)S^{-1}=f(A')
\]
satisfy
\[
 D^*D\preceq I,\qquad \rho(D)<1,
 \qquad \|S\|\|S^{-1}\|=\kappa\le2.
\]

Theorem (A common positive representation)

With the same \Omega,A,f,G and any chosen minimizing metric in Theorem thm:structural, there is one continuous function \Lambda:\T\to M_n(\C) satisfying

\Lambda(t)\succeq0,\qquad \int_\T\Lambda(t)\,d\sigma(t)=I_n,

such that, for every positive integer m and every M_m(\C)-valued function v holomorphic on a neighborhood of \overline\Omega,

v[A']=\int_\T\Lambda(t)\otimes v(G(t))\,d\sigma(t), \qquad \|v[A]\|\le\kappa\max_{\overline\Omega}\|v\|.

The same S, \Lambda, and \kappa work for all m and v.

Section 1, The structural question Read it in the paper TeX source, line 132

Show the TeX

With the same $\Omega,A,f,G$ and any chosen minimizing metric in
Theorem~\ref{thm:structural}, there is one continuous function
$\Lambda:\T\to M_n(\C)$ satisfying
\[
 \Lambda(t)\succeq0,\qquad \int_\T\Lambda(t)\,d\sigma(t)=I_n,
\]
such that, for every positive integer $m$ and every $M_m(\C)$-valued
function $v$ holomorphic on a neighborhood of $\overline\Omega$,
\begin{equation}
 v[A']=\int_\T\Lambda(t)\otimes v(G(t))\,d\sigma(t),
 \qquad
 \|v[A]\|\le\kappa\max_{\overline\Omega}\|v\|.
\end{equation}
The same $S$, $\Lambda$, and $\kappa$ work for all $m$ and $v$.

Configuration and file facts

File: lean/ComparatorChallenges/DirectCrouzeix.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Analysis/DirectCrouzeix/CompleteBound.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: 5 local vocabulary declarations
Lines of Lean, without comments or blank lines: 22
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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