Lean statement

Complete sharp Crouzeix theorem on arbitrary Hilbert spaces

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.CrouzeixHilbert.hilbert

Lean statement file: CrouzeixHilbert.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 325 link it.

See the closest candidates 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 hilbert (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]    [CompleteSpace H] (A : Operator H) :    (Nontrivial H → NonzeroConclusion A) ∧    (Subsingleton H → ZeroConclusion A) ∧ SharpConstant := 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.CrouzeixHilbert.Operator line 16

abbrev Operator (H : Type u) [NormedAddCommGroup H] [NormedSpace ℂ H] := H →L[ℂ] H

abbreviation OAI.CrouzeixHilbert.Coeff line 17

abbrev Coeff (m : ℕ) := Matrix (Fin m) (Fin m) ℂ

definition OAI.CrouzeixHilbert.numericalRange line 21

def numericalRange (A : Operator H) : Set ℂ :=  {z | ∃ x : H, ‖x‖ = 1 ∧ ⟪x, A x⟫_ℂ = z}

definition OAI.CrouzeixHilbert.numericalClosure line 24

def numericalClosure (A : Operator H) : Set ℂ := closure (numericalRange A)

definition OAI.CrouzeixHilbert.supNorm line 26

def supNorm {E : Type*} [NormedAddCommGroup E] (S : Set ℂ) (F : ℂ → E) : ℝ :=  sSup (insert 0 ((fun z => ‖F z‖) '' S))

abbreviation OAI.CrouzeixHilbert.Amplification line 29

abbrev Amplification (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]    (m : ℕ) := UniformSpace.Completion (H ⊗[ℂ] EuclideanSpace ℂ (Fin m))

definition OAI.CrouzeixHilbert.tensorOperator line 32

def tensorOperator (A : Operator H) {m : ℕ} (B : Coeff m) :    Operator (Amplification H m) :=  (TensorProduct.mapL A (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin m) B)).completion

definition OAI.CrouzeixHilbert.matrixPolynomial line 36

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

definition OAI.CrouzeixHilbert.polynomialEval line 39

def polynomialEval (A : Operator H) {m d : ℕ} (B : Fin (d + 1) → Coeff m) :    Operator (Amplification H m) :=  ∑ k : Fin (d + 1), tensorOperator (A ^ (k : ℕ)) (B k)

structure OAI.CrouzeixHilbert.SmoothContour line 43

structure SmoothContour where  path : ℝ → ℂ  smooth : ContDiff ℝ 1 path  closed : path 0 = path 1

definition OAI.CrouzeixHilbert.SmoothContour.index line 48

def SmoothContour.index (Γ : SmoothContour) (z : ℂ) : ℂ :=  (2 * (Real.pi : ℂ) * Complex.I)⁻¹ *    ∫ t in (0 : ℝ)..1, deriv Γ.path t / (Γ.path t - z)

structure OAI.CrouzeixHilbert.CalculusContour line 52

structure CalculusContour (K U : Set ℂ) extends SmoothContour where  avoids : ∀ t ∈ Icc (0 : ℝ) 1, path t ∈ U \ K  index_inside : ∀ z ∈ K, toSmoothContour.index z = 1  index_outside : ∀ z ∉ U, toSmoothContour.index z = 0

definition OAI.CrouzeixHilbert.contourEval line 57

def contourEval (A : Operator H) (Γ : SmoothContour) (f : ℂ → ℂ) : Operator H :=  (2 * (Real.pi : ℂ) * Complex.I)⁻¹ •    ∫ t in (0 : ℝ)..1, (deriv Γ.path t * f (Γ.path t)) •      Ring.inverse (algebraMap ℂ (Operator H) (Γ.path t) - A)

definition OAI.CrouzeixHilbert.holomorphicEval line 62

def holomorphicEval (A : Operator H) (U : Set ℂ) (f : ℂ → ℂ) : Operator H := by  classical  exact if h : Nonempty (CalculusContour (numericalClosure A) U) then    contourEval A (Classical.choice h).toSmoothContour f  else 0

definition OAI.CrouzeixHilbert.matrixHolomorphicEval line 68

def matrixHolomorphicEval (A : Operator H) {m : ℕ} (U : Set ℂ)    (F : ℂ → Coeff m) : Operator (Amplification H m) :=  ∑ i, ∑ j, tensorOperator (holomorphicEval A U (fun z => F z i j))    (Matrix.single i j 1)

definition OAI.CrouzeixHilbert.EntrywiseHolomorphic line 73

def EntrywiseHolomorphic {m : ℕ} (U : Set ℂ) (F : ℂ → Coeff m) : Prop :=  ∀ i j, DifferentiableOn ℂ (fun z => F z i j) U

abbreviation OAI.CrouzeixHilbert.RationalMatrix line 76

abbrev RationalMatrix (m : ℕ) := Matrix (Fin m) (Fin m) (RatFunc ℂ)

definition OAI.CrouzeixHilbert.PolesOutside line 78

def PolesOutside {m : ℕ} (K : Set ℂ) (R : RationalMatrix m) : Prop :=  ∀ i j z, z ∈ K → (R i j).denom.eval z ≠ 0

definition OAI.CrouzeixHilbert.matrixRationalFunction line 81

def matrixRationalFunction {m : ℕ} (R : RationalMatrix m) (z : ℂ) : Coeff m :=  fun i j => RatFunc.eval (RingHom.id ℂ) z (R i j)

definition OAI.CrouzeixHilbert.rationalEval line 84

def rationalEval (A : Operator H) (r : RatFunc ℂ) : Operator H :=  (Polynomial.aeval A r.num) * Ring.inverse (Polynomial.aeval A r.denom)

definition OAI.CrouzeixHilbert.matrixRationalEval line 87

def matrixRationalEval (A : Operator H) {m : ℕ} (R : RationalMatrix m) :    Operator (Amplification H m) :=  ∑ i, ∑ j, tensorOperator (rationalEval A (R i j)) (Matrix.single i j 1)

definition OAI.CrouzeixHilbert.NonzeroConclusion line 91

def NonzeroConclusion (A : Operator H) : Prop :=

Show the remaining 28 lines

  IsCompact (numericalClosure A) ∧  Convex ℝ (numericalClosure A) ∧  spectrum ℂ A ⊆ numericalClosure A ∧  (∀ (m : ℕ), 0 < m → ∀ (d : ℕ) (B : Fin (d + 1) → Coeff m),    ‖polynomialEval A B‖ ≤ 2 * supNorm (numericalRange A) (matrixPolynomial B) ∧    supNorm (numericalRange A) (matrixPolynomial B) =      supNorm (numericalClosure A) (matrixPolynomial B) ∧    ∃ z ∈ numericalClosure A,      supNorm (numericalClosure A) (matrixPolynomial B) = ‖matrixPolynomial B z‖) ∧  (∀ (m : ℕ), 0 < m → ∀ (U : Set ℂ) (F : ℂ → Coeff m),    IsOpen U → numericalClosure A ⊆ U → EntrywiseHolomorphic U F →    Nonempty (CalculusContour (numericalClosure A) U) ∧    (∀ Γ : CalculusContour (numericalClosure A) U, ∀ i j,      holomorphicEval A U (fun z => F z i j) =        contourEval A Γ.toSmoothContour (fun z => F z i j)) ∧    ‖matrixHolomorphicEval A U F‖ ≤ 2 * supNorm (numericalClosure A) F ∧    ∃ z ∈ numericalClosure A, supNorm (numericalClosure A) F = ‖F z‖) ∧  (∀ (m : ℕ), 0 < m → ∀ (R : RationalMatrix m),    PolesOutside (numericalClosure A) R →    (∀ i j, IsUnit (Polynomial.aeval A (R i j).denom)) ∧    ‖matrixRationalEval A R‖ ≤      2 * supNorm (numericalClosure A) (matrixRationalFunction R) ∧    (∃ z ∈ numericalClosure A,      supNorm (numericalClosure A) (matrixRationalFunction R) =        ‖matrixRationalFunction R z‖) ∧    ∃ U : Set ℂ, IsOpen U ∧ numericalClosure A ⊆ U ∧      EntrywiseHolomorphic U (matrixRationalFunction R) ∧      matrixRationalEval A R = matrixHolomorphicEval A U (matrixRationalFunction R))

definition OAI.CrouzeixHilbert.ZeroConclusion line 121

def ZeroConclusion (A : Operator H) : Prop :=

Show the remaining 14 lines

  numericalRange A = ∅ ∧  (∀ (m d : ℕ) (B : Fin (d + 1) → Coeff m),    polynomialEval A B = 0 ∧    supNorm (numericalRange A) (matrixPolynomial B) = 0 ∧    ‖polynomialEval A B‖ ≤ 2 * supNorm (numericalRange A) (matrixPolynomial B)) ∧  (∀ (m : ℕ) (U : Set ℂ) (F : ℂ → Coeff m),    matrixHolomorphicEval A U F = 0 ∧    supNorm (numericalRange A) F = 0 ∧    ‖matrixHolomorphicEval A U F‖ ≤ 2 * supNorm (numericalRange A) F) ∧  (∀ (m : ℕ) (R : RationalMatrix m),    matrixRationalEval A R = 0 ∧    supNorm (numericalRange A) (matrixRationalFunction R) = 0 ∧    ‖matrixRationalEval A R‖ ≤      2 * supNorm (numericalRange A) (matrixRationalFunction R))

definition OAI.CrouzeixHilbert.SharpConstant line 137

def SharpConstant : Prop :=  ∃ (A : Operator (EuclideanSpace ℂ (Fin 2))) (B : Fin 2 → Coeff 1),    supNorm (numericalRange A) (matrixPolynomial (d := 1) B) = 1 ∧    ‖polynomialEval A (d := 1) B‖ = 2

No confident match with a paper statement

No paper theorem is paired with this Lean statement.

Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.

Statements in A direct proof of the complete Crouzeix inequality come first because the AI draft reading of OpenAI’s note shares words with that paper’s title.

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: nonzero, outside, tensor, poles, range, calculus and 21 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: separability, assumption, supremum, evaluation, two, inequalities and three more.
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: extends, map, hilbert, operators, space, matrix and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: near, finite, bounded and valued.
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: product, real, extends, compact, convex, function and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounded.
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: choices, sharp, complete, matrix, constant, integers and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: inequality.
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: constant, integers and polynomial. Found in the paper’s title, which its main theorem states: complete. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two.

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.

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

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 formalization proves the complete Crouzeix inequality with sharp constant two for every bounded operator on an arbitrary complex Hilbert space. For every matrix-valued polynomial, its operator evaluation has norm at most twice its supremum norm on the numerical range. The same bound holds for finite matrix-valued functions holomorphic near the closure of the numerical range and for rational functions with poles outside that closure. No separability assumption is required.”
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

158 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: 133 to its own declarations, 77 to Mathlib’s documentation. 25 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 hilbert 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 24 other declarations of the file; with the theorem itself they span 106 lines. 4 of them are named in the statement itself.

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

abbrev Operatorline 16in the statementlibrary
abbrev Coeffline 17library
def numericalRangeline 21library
def numericalClosureline 24library
def supNormline 26library
abbrev Amplificationline 29library
def tensorOperatorline 32library
def matrixPolynomialline 36library
def polynomialEvalline 39library
structure SmoothContourline 43library
def SmoothContour.indexline 48library
structure CalculusContourline 52library
def contourEvalline 57library
def holomorphicEvalline 62library
def matrixHolomorphicEvalline 68library
def EntrywiseHolomorphicline 73library
abbrev RationalMatrixline 76library
def PolesOutsideline 78library
def matrixRationalFunctionline 81library
def rationalEvalline 84library
def matrixRationalEvalline 87library
def NonzeroConclusionline 91in the statementlibrary
def ZeroConclusionline 121in the statementlibrary
def SharpConstantline 137in the statementlibrary
theorem hilbertline 148selected theoremlibrary

import Mathlibnamespace OAI/-! The sharp complete inequality for bounded operators on complex Hilbert spaces. -/noncomputable sectionopen scoped TensorProduct Matrix.Norms.L2Operator InnerProductSpaceopen Setnamespace CrouzeixHilbertuniverse uabbrev Operator (H : Type u) [NormedAddCommGroup H] [NormedSpace ℂ H] := H →L[ℂ] Habbrev Coeff (m : ℕ) := Matrix (Fin m) (Fin m) ℂvariable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]def numericalRange (A : Operator H) : Set ℂ :=  {z | ∃ x : H, ‖x‖ = 1 ∧ ⟪x, A x⟫_ℂ = z}def numericalClosure (A : Operator H) : Set ℂ := closure (numericalRange A)def supNorm {E : Type*} [NormedAddCommGroup E] (S : Set ℂ) (F : ℂ → E) : ℝ :=  sSup (insert 0 ((fun z => ‖F z‖) '' S))abbrev Amplification (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]    (m : ℕ) := UniformSpace.Completion (H ⊗[ℂ] EuclideanSpace ℂ (Fin m))def tensorOperator (A : Operator H) {m : ℕ} (B : Coeff m) :    Operator (Amplification H m) :=  (TensorProduct.mapL A (Matrix.toEuclideanCLM (𝕜 := ℂ) (n := Fin m) B)).completiondef matrixPolynomial {m d : ℕ} (B : Fin (d + 1) → Coeff m) (z : ℂ) : Coeff m :=  ∑ k : Fin (d + 1), z ^ (k : ℕ) • B kdef polynomialEval (A : Operator H) {m d : ℕ} (B : Fin (d + 1) → Coeff m) :    Operator (Amplification H m) :=  ∑ k : Fin (d + 1), tensorOperator (A ^ (k : ℕ)) (B k)structure SmoothContour where  path : ℝ → ℂ  smooth : ContDiff ℝ 1 path  closed : path 0 = path 1def SmoothContour.index (Γ : SmoothContour) (z : ℂ) : ℂ :=  (2 * (Real.pi : ℂ) * Complex.I)⁻¹ *    ∫ t in (0 : ℝ)..1, deriv Γ.path t / (Γ.path t - z)structure CalculusContour (K U : Set ℂ) extends SmoothContour where  avoids : ∀ t ∈ Icc (0 : ℝ) 1, path t ∈ U \ K  index_inside : ∀ z ∈ K, toSmoothContour.index z = 1  index_outside : ∀ z ∉ U, toSmoothContour.index z = 0def contourEval (A : Operator H) (Γ : SmoothContour) (f : ℂ → ℂ) : Operator H :=  (2 * (Real.pi : ℂ) * Complex.I)⁻¹ •    ∫ t in (0 : ℝ)..1, (deriv Γ.path t * f (Γ.path t)) •      Ring.inverse (algebraMap ℂ (Operator H) (Γ.path t) - A)def holomorphicEval (A : Operator H) (U : Set ℂ) (f : ℂ → ℂ) : Operator H := by  classical  exact if h : Nonempty (CalculusContour (numericalClosure A) U) then    contourEval A (Classical.choice h).toSmoothContour f  else 0def matrixHolomorphicEval (A : Operator H) {m : ℕ} (U : Set ℂ)    (F : ℂ → Coeff m) : Operator (Amplification H m) :=  ∑ i, ∑ j, tensorOperator (holomorphicEval A U (fun z => F z i j))    (Matrix.single i j 1)def EntrywiseHolomorphic {m : ℕ} (U : Set ℂ) (F : ℂ → Coeff m) : Prop :=  ∀ i j, DifferentiableOn ℂ (fun z => F z i j) Uabbrev RationalMatrix (m : ℕ) := Matrix (Fin m) (Fin m) (RatFunc ℂ)def PolesOutside {m : ℕ} (K : Set ℂ) (R : RationalMatrix m) : Prop :=  ∀ i j z, z ∈ K → (R i j).denom.eval z ≠ 0def matrixRationalFunction {m : ℕ} (R : RationalMatrix m) (z : ℂ) : Coeff m :=  fun i j => RatFunc.eval (RingHom.id ℂ) z (R i j)def rationalEval (A : Operator H) (r : RatFunc ℂ) : Operator H :=  (Polynomial.aeval A r.num) * Ring.inverse (Polynomial.aeval A r.denom)def matrixRationalEval (A : Operator H) {m : ℕ} (R : RationalMatrix m) :    Operator (Amplification H m) :=  ∑ i, ∑ j, tensorOperator (rationalEval A (R i j)) (Matrix.single i j 1)def NonzeroConclusion (A : Operator H) : Prop :=  IsCompact (numericalClosure A) ∧  Convex ℝ (numericalClosure A) ∧  spectrum ℂ A ⊆ numericalClosure A ∧  (∀ (m : ℕ), 0 < m → ∀ (d : ℕ) (B : Fin (d + 1) → Coeff m),    ‖polynomialEval A B‖ ≤ 2 * supNorm (numericalRange A) (matrixPolynomial B) ∧    supNorm (numericalRange A) (matrixPolynomial B) =      supNorm (numericalClosure A) (matrixPolynomial B) ∧    ∃ z ∈ numericalClosure A,      supNorm (numericalClosure A) (matrixPolynomial B) = ‖matrixPolynomial B z‖) ∧  (∀ (m : ℕ), 0 < m → ∀ (U : Set ℂ) (F : ℂ → Coeff m),    IsOpen U → numericalClosure A ⊆ U → EntrywiseHolomorphic U F →    Nonempty (CalculusContour (numericalClosure A) U) ∧    (∀ Γ : CalculusContour (numericalClosure A) U, ∀ i j,      holomorphicEval A U (fun z => F z i j) =        contourEval A Γ.toSmoothContour (fun z => F z i j)) ∧    ‖matrixHolomorphicEval A U F‖ ≤ 2 * supNorm (numericalClosure A) F ∧    ∃ z ∈ numericalClosure A, supNorm (numericalClosure A) F = ‖F z‖) ∧  (∀ (m : ℕ), 0 < m → ∀ (R : RationalMatrix m),    PolesOutside (numericalClosure A) R →    (∀ i j, IsUnit (Polynomial.aeval A (R i j).denom)) ∧    ‖matrixRationalEval A R‖ ≤      2 * supNorm (numericalClosure A) (matrixRationalFunction R) ∧    (∃ z ∈ numericalClosure A,      supNorm (numericalClosure A) (matrixRationalFunction R) =        ‖matrixRationalFunction R z‖) ∧    ∃ U : Set ℂ, IsOpen U ∧ numericalClosure A ⊆ U ∧      EntrywiseHolomorphic U (matrixRationalFunction R) ∧      matrixRationalEval A R = matrixHolomorphicEval A U (matrixRationalFunction R))def ZeroConclusion (A : Operator H) : Prop :=  numericalRange A = ∅ ∧  (∀ (m d : ℕ) (B : Fin (d + 1) → Coeff m),    polynomialEval A B = 0 ∧    supNorm (numericalRange A) (matrixPolynomial B) = 0 ∧    ‖polynomialEval A B‖ ≤ 2 * supNorm (numericalRange A) (matrixPolynomial B)) ∧  (∀ (m : ℕ) (U : Set ℂ) (F : ℂ → Coeff m),    matrixHolomorphicEval A U F = 0 ∧    supNorm (numericalRange A) F = 0 ∧    ‖matrixHolomorphicEval A U F‖ ≤ 2 * supNorm (numericalRange A) F) ∧  (∀ (m : ℕ) (R : RationalMatrix m),    matrixRationalEval A R = 0 ∧    supNorm (numericalRange A) (matrixRationalFunction R) = 0 ∧    ‖matrixRationalEval A R‖ ≤      2 * supNorm (numericalRange A) (matrixRationalFunction R))def SharpConstant : Prop :=  ∃ (A : Operator (EuclideanSpace ℂ (Fin 2))) (B : Fin 2 → Coeff 1),    supNorm (numericalRange A) (matrixPolynomial (d := 1) B) = 1 ∧    ‖polynomialEval A (d := 1) B‖ = 2end CrouzeixHilbertnamespace CrouzeixHilbertuniverse utheorem hilbert (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]    [CompleteSpace H] (A : Operator H) :    (Nontrivial H → NonzeroConclusion A) ∧    (Subsingleton H → ZeroConclusion A) ∧ SharpConstant := by  sorryend CrouzeixHilbertendend 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

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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

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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

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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/CrouzeixHilbert.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 325 link it.
Solution module: OAI.Analysis.Crouzeix.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: 24 local vocabulary declarations
Lines of Lean, without comments or blank lines: 106
Second proof-verification program (nanoda): Off in OpenAI's configuration.

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