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.StructuralCrouzeixReference.challenge
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.
noncomputabledefmatrixAnalyticEval (A : Matrix n n ℂ) (f : ℂ → ℂ) : Matrix n n ℂ := Matrix.toLinAlgEquiv'.symm (primaryEval (Matrix.toLinAlgEquiv' A) f)
defcompleteAnalyticEval (D : Matrix n n ℂ) (F : ℂ → Matrix m m ℂ) : Matrix (n×m) (n×m) ℂ :=fun i j => matrixAnalyticEval D (fun z => F z i.2 j.2) i.1 j.1
structureIsAdmissible (U : Set ℂ) : Propwhere isOpen : IsOpen U bounded : Bornology.IsBounded U convex : Convex ℝ U nonempty : U.Nonempty jordan : Nonempty (Circle ≃ₜ ↥(frontier U)) regular : ∀ p ∈ frontier U, RegularAnalyticBoundaryAt U p
∀ (U : Set ℂ), IsAdmissible U → (∀ a ∈ U, Nonempty (DiskCoordinate U a)) ∧ Nonempty (ExteriorCoordinate U) ∧ ∀ (a : ℂ), a ∈ U → ∀ (f : DiskCoordinate U a) (G : ExteriorCoordinate U), ∀ (n : ℕ), 0 < n → ∀ (A : Matrix (Fin n) (Fin n) ℂ), numericalRange A ⊆ U →let T := matrixAnalyticEval A f.toDisk StrictlyFeasible T ∧ ∃ κ : ℝ, 1 ≤ κ ∧ κ ≤ 2 ∧ (∃ H : Matrix (Fin n) (Fin n) ℂ, IsMinimizing T (κ ^ 2) H) ∧ ∀ H : Matrix (Fin n) (Fin n) ℂ, IsMinimizing T (κ ^ 2) H →let S := CFC.sqrt Hlet A' := S * A * S⁻¹let D := S * T * S⁻¹ H.PosDef ∧ IsUnit S ∧ D = matrixAnalyticEval A' f.toDisk ∧ Dᴴ * D ≤ 1 ∧ spectralRadius ℂ D < 1 ∧ ‖S‖ * ‖S⁻¹‖ = κ ∧ (∀ R : Matrix (Fin n) (Fin n) ℂ, IsUnit R → (R * T * R⁻¹)ᴴ * (R * T * R⁻¹) ≤ 1 → κ ≤ ‖R‖ * ‖R⁻¹‖) ∧ ∃ Λ : C(UnitAddCircle, Matrix (Fin n) (Fin n) ℂ), (∀ t, (Λ t).PosSemidef) ∧ (∫ t, Λ t ∂AddCircle.haarAddCircle) = 1 ∧ ∀ (m : ℕ), 0 < m → ∀ v : ℂ → Matrix (Fin m) (Fin m) ℂ, AnalyticOnNhd ℂ v (closure U) → completeAnalyticEval A' v = (∫ t, Λ t ⊗ₖ v (boundaryPoint G t) ∂AddCircle.haarAddCircle) ∧ ‖completeAnalyticEval A v‖ ≤ κ * analyticSupNorm U v
Possible paper matches for the parts of the Lean statement
Atlas selected possible matches by comparing words in each part of the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Statements in The complete Crouzeix theorem: optimal similarity and a common positive boundary representation come first because the release’s name for this Lean statement shares words with that paper’s title.
The Lean statement joins several assertions with “and”. Atlas selected possible matches to two paper results from their shared wording, listed below. This does not establish that their mathematical meanings agree.
Theorem 1.2 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: strictly, coordinate, feasible, interior, admissible, eq, minimizing and metric. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: attained, optimal and finite.
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
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 1.3 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: structural, representation, minimizing, metric, max and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: continuous, valued, one and positive.
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
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 $v$.
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: outside, range, zero, spectral, numerical, units and eleven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two, optimal, finite, valued and positive.
Proposition 3.1 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: circle, admissible, scalar, map, leading, exterior and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: continuous, finite and one.
Theorem 3.2 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: real, jordan, regular, analytic, domain, convex and three more. Found in the paper’s title, which its main theorem states: crouzeix, direct and complete. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: valued, one and positive.
Proposition 2.1 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: representation, eq, minimizing, metric and bound. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: density and one.
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, regular, analytic, scalars, domain, leading and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: including, continuous, one 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: map, space, boundary, matrix and bounded. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite, valued and one.
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.
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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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.
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.
Finds the whole advertised result stated or impliedTheir 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)
OpenAI’s own note on what this Lean statement covers
“For finite matrices in a bounded convex domain with regular real-analytic Jordan boundary, the structural result gives an attained optimal similarity with condition number at most two and one continuous positive boundary density of mass the identity representing every matrix-valued analytic test. The linked supporting statements include numerical-range geometry and finite-compression identities.”
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
216 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: 33 to its own declarations, 140 to Mathlib’s documentation. 19 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
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.
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 $d$.
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
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
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 $v$.
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
18 local vocabulary declarations
Lines of Lean, without comments or blank lines
119
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Optimal similarity and boundary representation
Family 325: The complete Crouzeix conjecture
Candidate pairing
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
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.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.