Lean statement

Numerical-range geometry and finite-compression support

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.HilbertCrouzeix.numericalClosure_geometry OAI.HilbertCrouzeix.exists_finite_compression OAI.HilbertCrouzeix.rational_denominator_isUnit OAI.HilbertCrouzeix.zeroConclusion

Lean statement file: HilbertCrouzeix.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 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 numericalClosure_geometry (A : H →L[ℂ] H) :    IsCompact (numericalClosure A) ∧ Convex ℝ (numericalClosure A) ∧      spectrum ℂ A ⊆ numericalClosure A := by  sorry⋮theorem exists_finite_compression {m d : ℕ} (A : H →L[ℂ] H)    (B : Fin (d + 1) → Coeff m) (ξ : H ⊗[ℂ] CoeffSpace m) :    ∃ (E : Submodule ℂ H) (hE : FiniteDimensional ℂ E),      letI : FiniteDimensional ℂ E := hE⋮theorem rational_denominator_isUnit [Nontrivial H] (A : H →L[ℂ] H)    {m : ℕ} (R : RationalMatrix m) (hR : numericalClosure A ⊆ poleFreeSet R)    (i j : Fin m) : IsUnit (Polynomial.aeval A (R i j).denom) := by  sorry⋮theorem zeroConclusion [Subsingleton H] (A : H →L[ℂ] H) : ZeroConclusion A := 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.HilbertCrouzeix.Coeff line 14

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

abbreviation OAI.HilbertCrouzeix.CoeffSpace line 15

abbrev CoeffSpace (m : ℕ) := EuclideanSpace ℂ (Fin m)

definition OAI.HilbertCrouzeix.numericalRange line 19

def numericalRange (A : H →L[ℂ] H) : Set ℂ :=  {z | ∃ x : H, ‖x‖ = 1 ∧ inner ℂ x (A x) = z}

definition OAI.HilbertCrouzeix.numericalClosure line 22

def numericalClosure (A : H →L[ℂ] H) : Set ℂ := closure (numericalRange A)

abbreviation OAI.HilbertCrouzeix.Amplification line 24

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

definition OAI.HilbertCrouzeix.tensorOp line 27

def tensorOp {m : ℕ} (A : H →L[ℂ] H) (B : Coeff m) :    Amplification H m →L[ℂ] Amplification H m :=  (TensorProduct.mapL A (Matrix.toEuclideanCLM (n := Fin m) (𝕜 := ℂ) B)).completion

definition OAI.HilbertCrouzeix.matrixPolynomial line 31

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

definition OAI.HilbertCrouzeix.polynomialEval line 34

def polynomialEval {m d : ℕ} (A : H →L[ℂ] H) (B : Fin (d + 1) → Coeff m) :    Amplification H m →L[ℂ] Amplification H m :=  ∑ k : Fin (d + 1), tensorOp (A ^ k.val) (B k)

definition OAI.HilbertCrouzeix.normSup line 38

def normSup {E : Type*} [Norm E] (S : Set ℂ) (f : ℂ → E) : ℝ :=  if S.Nonempty then sSup ((fun z => ‖f z‖) '' S) else 0

definition OAI.HilbertCrouzeix.contourIntegral line 40

def contourIntegral {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]    (γ : ℝ → ℂ) (f : ℂ → E) : E :=  ∫ t : ℝ in (0 : ℝ)..1, deriv γ t • f (γ t)

definition OAI.HilbertCrouzeix.contourIndex line 44

def contourIndex (γ : ℝ → ℂ) (z : ℂ) : ℂ :=  (2 * (Real.pi : ℂ) * Complex.I)⁻¹ * contourIntegral γ (fun w => (w - z)⁻¹)

definition OAI.HilbertCrouzeix.IsCauchyContour line 47

def IsCauchyContour (K U : Set ℂ) (γ : ℝ → ℂ) : Prop :=  ContDiff ℝ 1 γ ∧ γ 0 = γ 1 ∧  (∀ t ∈ Set.Icc (0 : ℝ) 1, γ t ∈ U \ K) ∧  (∀ z ∈ K, contourIndex γ z = 1) ∧  (∀ z ∉ U, contourIndex γ z = 0)

definition OAI.HilbertCrouzeix.chosenContour line 53

def chosenContour (K U : Set ℂ) : ℝ → ℂ :=  Classical.epsilon (IsCauchyContour K U)

definition OAI.HilbertCrouzeix.dunfordIntegral line 58

def dunfordIntegral (A : H →L[ℂ] H) (γ : ℝ → ℂ) (f : ℂ → ℂ) : H →L[ℂ] H :=  (2 * (Real.pi : ℂ) * Complex.I)⁻¹ •    contourIntegral γ (fun z => f z • resolvent A z)

definition OAI.HilbertCrouzeix.holomorphicCalculus line 62

def holomorphicCalculus (A : H →L[ℂ] H) (U : Set ℂ) (f : ℂ → ℂ) : H →L[ℂ] H :=  dunfordIntegral A (chosenContour (numericalClosure A) U) f

definition OAI.HilbertCrouzeix.matrixUnit line 65

def matrixUnit {m : ℕ} (i j : Fin m) : Coeff m := Matrix.single i j 1

definition OAI.HilbertCrouzeix.holomorphicEval line 67

def holomorphicEval {m : ℕ} (A : H →L[ℂ] H) (U : Set ℂ) (F : ℂ → Coeff m) :    Amplification H m →L[ℂ] Amplification H m :=  ∑ i : Fin m, ∑ j : Fin m,    tensorOp (holomorphicCalculus A U (fun z => F z i j)) (matrixUnit i j)

abbreviation OAI.HilbertCrouzeix.RationalMatrix line 71

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

definition OAI.HilbertCrouzeix.poleFreeSet line 73

def poleFreeSet {m : ℕ} (R : RationalMatrix m) : Set ℂ :=  {z | ∀ i j, (R i j).denom.eval z ≠ 0}

definition OAI.HilbertCrouzeix.rationalFunction line 76

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

definition OAI.HilbertCrouzeix.rationalCalculus line 79

def rationalCalculus (A : H →L[ℂ] H) (r : RatFunc ℂ) : H →L[ℂ] H :=  Polynomial.aeval A r.num * Ring.inverse (Polynomial.aeval A r.denom)

definition OAI.HilbertCrouzeix.rationalEval line 82

def rationalEval {m : ℕ} (A : H →L[ℂ] H) (R : RationalMatrix m) :    Amplification H m →L[ℂ] Amplification H m :=  ∑ i : Fin m, ∑ j : Fin m, tensorOp (rationalCalculus A (R i j)) (matrixUnit i j)

definition OAI.HilbertCrouzeix.ZeroConclusion line 85

def ZeroConclusion (A : H →L[ℂ] H) : Prop :=

Show the remaining 10 lines

  numericalRange A = ∅ ∧  (∀ (m : ℕ) (F : ℂ → Coeff m), normSup (numericalRange A) F = 0) ∧  ∀ (m : ℕ), 0 < m →    (∀ T : Amplification H m →L[ℂ] Amplification H m, T = 0) ∧    (∀ (d : ℕ) (B : Fin (d + 1) → Coeff m), polynomialEval A B = 0 ∧      ‖polynomialEval A B‖ ≤ 2 * normSup (numericalRange A) (matrixPolynomial B)) ∧    (∀ (U : Set ℂ) (F : ℂ → Coeff m), holomorphicEval A U F = 0 ∧      ‖holomorphicEval A U F‖ ≤ 2 * normSup (numericalRange A) F) ∧    (∀ (R : RationalMatrix m), rationalEval A R = 0 ∧      ‖rationalEval A R‖ ≤ 2 * normSup (numericalRange A) (rationalFunction R))

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: tensor, range, calculus, zero, numerical, units, complex, rational and ten more.

Corollary 6.3 of The complete Crouzeix theorem: optimal similarity and a common positive boundary representation

Corollary (The sharp complete inequality for bounded operators)

Let \mathcal H be a complex Hilbert space, with no separability assumption, and let A\in\mathcal B(\mathcal H). Suppose first that \mathcal H\ne\{0\}, and put

W(A)=\{\langle x,Ax\rangle:x\in\mathcal H,\ \|x\|=1\}, \qquad K=\overline{W(A)}.

Then K is compact and convex and contains \spec(A). For every positive integer m, every nonnegative integer d, and every matrix polynomial P(z)=\sum_{k=0}^d B_kz^k with B_k\in M_m(\C), the operator P[A]=\sum_{k=0}^d A^k\otimes B_k on \mathcal H\otimes\C^m satisfies

\|P[A]\|\le 2\sup_{z\in W(A)}\|P(z)\| =2\max_{z\in K}\|P(z)\|.

More generally, if U is an open neighborhood of K and F=(f_{ij}):U\to M_m(\C) is holomorphic, its entrywise holomorphic evaluation, in the same base-first tensor convention, satisfies

F[A]:=\sum_{i,j=1}^m f_{ij}(A)\otimes E_{ij}, \qquad \|F[A]\|\le2\max_{z\in K}\|F(z)\|.

Here E_{ij} are the standard matrix units and f_{ij}(A) is given by the holomorphic functional calculus. In particular, K is a complete 2-spectral set for A: the same estimate holds at every finite coefficient size for every matrix-valued rational function whose poles lie outside K, with evaluation by the rational functional calculus. The constant two is optimal uniformly over all these choices.

If \mathcal H=\{0\}, every evaluation is the zero operator. With the supremum over the empty numerical range defined to be zero, the corresponding supremum-norm inequalities are 0\le0; the compact spectral-set assertion above is stated for nonzero \mathcal H.

Section 6, From admissible domains to bounded Hilbert-space operators Read it in the paper TeX source, line 1,175

Show the TeX

Let $\mathcal H$ be a complex Hilbert space, with no separability assumption,
and let $A\in\mathcal B(\mathcal H)$. Suppose first that
$\mathcal H\ne\{0\}$, and put
\[
 W(A)=\{\langle x,Ax\rangle:x\in\mathcal H,\ \|x\|=1\},
 \qquad K=\overline{W(A)}.
\]
Then $K$ is compact and convex and contains $\spec(A)$. For every positive
integer $m$, every nonnegative integer $d$, and every matrix polynomial
$P(z)=\sum_{k=0}^d B_kz^k$ with $B_k\in M_m(\C)$, the operator
$P[A]=\sum_{k=0}^d A^k\otimes B_k$ on $\mathcal H\otimes\C^m$ satisfies
\begin{equation}
 \|P[A]\|\le 2\sup_{z\in W(A)}\|P(z)\|
             =2\max_{z\in K}\|P(z)\|.
\end{equation}
More generally, if $U$ is an open neighborhood of $K$ and
$F=(f_{ij}):U\to M_m(\C)$ is holomorphic, its entrywise holomorphic
evaluation, in the same base-first tensor convention, satisfies
\begin{equation}
 F[A]:=\sum_{i,j=1}^m f_{ij}(A)\otimes E_{ij},
 \qquad \|F[A]\|\le2\max_{z\in K}\|F(z)\|.
\end{equation}
Here $E_{ij}$ are the standard matrix units and $f_{ij}(A)$ is given by the
holomorphic functional calculus. In particular, $K$ is a complete
$2$-spectral set for $A$: the same estimate holds at every finite coefficient
size for every matrix-valued rational function whose poles lie outside $K$,
with evaluation by the rational functional calculus. The constant two is
optimal uniformly over all these choices.

If $\mathcal H=\{0\}$, every evaluation is the zero operator. With the
supremum over the empty numerical range defined to be zero, the corresponding
supremum-norm inequalities are $0\le0$; the compact spectral-set assertion
above is stated for nonzero $\mathcal H$.

Other possible paper matches, closest wording first

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: integral, pole, product, real, compact, convex and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: including.
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, convex, integers and polynomial.
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, finite, matrix, holomorphic and polynomial.
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: chosen, function, holomorphic and integer.
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: product, map, resolvent, finite, matrix, holomorphic and one more.
Proposition 2.2 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: rational, function and holomorphic.

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 151 s; peak memory 6.3 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 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 linked supporting statements include numerical-range geometry and finite-compression identities.”
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

143 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: 78 to its own declarations, 82 to Mathlib’s documentation. 30 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 numericalClosure_geometry 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 23 of the file’s 26 other declarations; with the theorem itself they span 79 lines. 6 of them are named in the statement itself.

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

abbrev Coeffline 14in the statementlibrary
abbrev CoeffSpaceline 15in the statementlibrary
def numericalRangeline 19library
def numericalClosureline 22in the statementlibrary
abbrev Amplificationline 24library
def tensorOpline 27library
def matrixPolynomialline 31library
def polynomialEvalline 34library
def normSupline 38library
def contourIntegralline 40library
def contourIndexline 44library
def IsCauchyContourline 47library
def chosenContourline 53library
def dunfordIntegralline 58library
def holomorphicCalculusline 62library
def matrixUnitline 65library
def holomorphicEvalline 67library
abbrev RationalMatrixline 71in the statementlibrary
def poleFreeSetline 73in the statementlibrary
def rationalFunctionline 76library
def rationalCalculusline 79library
def rationalEvalline 82library
def ZeroConclusionline 85in the statementlibrary
theorem numericalClosure_geometryline 115selected theoremlibrary
theorem exists_finite_compressionline 121selected theoremlibrary
theorem rational_denominator_isUnitline 131selected theoremlibrary
theorem zeroConclusionline 136selected theoremlibrary

3 declarations the statement does not reach by name

These definitions and lemmas were not found by following names in the statement. Lean may still use them through rules such as instances or macros.

def compression line 100
def completedIsometry line 102
def tensorInclusion line 109

import Mathlibnamespace OAInoncomputable sectionopen Complex Setopen scoped TensorProduct Matrix.Norms.L2Operator Classical ComplexConjugatenamespace HilbertCrouzeixuniverse uabbrev Coeff (m : ℕ) := Matrix (Fin m) (Fin m) ℂabbrev CoeffSpace (m : ℕ) := EuclideanSpace ℂ (Fin m)variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]def numericalRange (A : H →L[ℂ] H) : Set ℂ :=  {z | ∃ x : H, ‖x‖ = 1 ∧ inner ℂ x (A x) = z}def numericalClosure (A : H →L[ℂ] H) : Set ℂ := closure (numericalRange A)abbrev Amplification (H : Type u) [NormedAddCommGroup H] [InnerProductSpace ℂ H]    (m : ℕ) := UniformSpace.Completion (H ⊗[ℂ] CoeffSpace m)def tensorOp {m : ℕ} (A : H →L[ℂ] H) (B : Coeff m) :    Amplification H m →L[ℂ] Amplification H m :=  (TensorProduct.mapL A (Matrix.toEuclideanCLM (n := Fin m) (𝕜 := ℂ) B)).completiondef matrixPolynomial {m d : ℕ} (B : Fin (d + 1) → Coeff m) (z : ℂ) : Coeff m :=  ∑ k : Fin (d + 1), z ^ k.val • B kdef polynomialEval {m d : ℕ} (A : H →L[ℂ] H) (B : Fin (d + 1) → Coeff m) :    Amplification H m →L[ℂ] Amplification H m :=  ∑ k : Fin (d + 1), tensorOp (A ^ k.val) (B k)def normSup {E : Type*} [Norm E] (S : Set ℂ) (f : ℂ → E) : ℝ :=  if S.Nonempty then sSup ((fun z => ‖f z‖) '' S) else 0def contourIntegral {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]    (γ : ℝ → ℂ) (f : ℂ → E) : E :=  ∫ t : ℝ in (0 : ℝ)..1, deriv γ t • f (γ t)def contourIndex (γ : ℝ → ℂ) (z : ℂ) : ℂ :=  (2 * (Real.pi : ℂ) * Complex.I)⁻¹ * contourIntegral γ (fun w => (w - z)⁻¹)def IsCauchyContour (K U : Set ℂ) (γ : ℝ → ℂ) : Prop :=  ContDiff ℝ 1 γ ∧ γ 0 = γ 1 ∧  (∀ t ∈ Set.Icc (0 : ℝ) 1, γ t ∈ U \ K) ∧  (∀ z ∈ K, contourIndex γ z = 1) ∧  (∀ z ∉ U, contourIndex γ z = 0)def chosenContour (K U : Set ℂ) : ℝ → ℂ :=  Classical.epsilon (IsCauchyContour K U)variable [CompleteSpace H]def dunfordIntegral (A : H →L[ℂ] H) (γ : ℝ → ℂ) (f : ℂ → ℂ) : H →L[ℂ] H :=  (2 * (Real.pi : ℂ) * Complex.I)⁻¹ •    contourIntegral γ (fun z => f z • resolvent A z)def holomorphicCalculus (A : H →L[ℂ] H) (U : Set ℂ) (f : ℂ → ℂ) : H →L[ℂ] H :=  dunfordIntegral A (chosenContour (numericalClosure A) U) fdef matrixUnit {m : ℕ} (i j : Fin m) : Coeff m := Matrix.single i j 1def holomorphicEval {m : ℕ} (A : H →L[ℂ] H) (U : Set ℂ) (F : ℂ → Coeff m) :    Amplification H m →L[ℂ] Amplification H m :=  ∑ i : Fin m, ∑ j : Fin m,    tensorOp (holomorphicCalculus A U (fun z => F z i j)) (matrixUnit i j)abbrev RationalMatrix (m : ℕ) := Matrix (Fin m) (Fin m) (RatFunc ℂ)def poleFreeSet {m : ℕ} (R : RationalMatrix m) : Set ℂ :=  {z | ∀ i j, (R i j).denom.eval z ≠ 0}def rationalFunction {m : ℕ} (R : RationalMatrix m) (z : ℂ) : Coeff m :=  fun i j => RatFunc.eval (RingHom.id ℂ) z (R i j)def rationalCalculus (A : H →L[ℂ] H) (r : RatFunc ℂ) : H →L[ℂ] H :=  Polynomial.aeval A r.num * Ring.inverse (Polynomial.aeval A r.denom)def rationalEval {m : ℕ} (A : H →L[ℂ] H) (R : RationalMatrix m) :    Amplification H m →L[ℂ] Amplification H m :=  ∑ i : Fin m, ∑ j : Fin m, tensorOp (rationalCalculus A (R i j)) (matrixUnit i j)def ZeroConclusion (A : H →L[ℂ] H) : Prop :=  numericalRange A = ∅ ∧  (∀ (m : ℕ) (F : ℂ → Coeff m), normSup (numericalRange A) F = 0) ∧  ∀ (m : ℕ), 0 < m →    (∀ T : Amplification H m →L[ℂ] Amplification H m, T = 0) ∧    (∀ (d : ℕ) (B : Fin (d + 1) → Coeff m), polynomialEval A B = 0 ∧      ‖polynomialEval A B‖ ≤ 2 * normSup (numericalRange A) (matrixPolynomial B)) ∧    (∀ (U : Set ℂ) (F : ℂ → Coeff m), holomorphicEval A U F = 0 ∧      ‖holomorphicEval A U F‖ ≤ 2 * normSup (numericalRange A) F) ∧    (∀ (R : RationalMatrix m), rationalEval A R = 0 ∧      ‖rationalEval A R‖ ≤ 2 * normSup (numericalRange A) (rationalFunction R))section CompressionDefinitionsomit [CompleteSpace H]def compression (A : H →L[ℂ] H) (E : Submodule ℂ H) [E.HasOrthogonalProjection] :    E →L[ℂ] E := E.orthogonalProjectionOnto.comp (A.comp E.subtypeL)def completedIsometry {V : Type*} {W : Type*}    [NormedAddCommGroup V] [NormedAddCommGroup W] [NormedSpace ℂ V] [NormedSpace ℂ W]    (f : V →ₗᵢ[ℂ] W) : UniformSpace.Completion V →ₗᵢ[ℂ] UniformSpace.Completion W :=  { f.toContinuousLinearMap.completion.toLinearMap with    norm_map' := f.isometry.completion_map.norm_map_of_map_zero      (f.toContinuousLinearMap.completion.map_zero) }def tensorInclusion (E : Submodule ℂ H) (m : ℕ) :    Amplification E m →ₗᵢ[ℂ] Amplification H m :=  completedIsometry (TensorProduct.mapIsometry E.subtypeₗᵢ (.id : CoeffSpace m →ₗᵢ[ℂ] CoeffSpace m))end CompressionDefinitionstheorem numericalClosure_geometry (A : H →L[ℂ] H) :    IsCompact (numericalClosure A) ∧ Convex ℝ (numericalClosure A) ∧      spectrum ℂ A ⊆ numericalClosure A := by  sorryomit [CompleteSpace H] intheorem exists_finite_compression {m d : ℕ} (A : H →L[ℂ] H)    (B : Fin (d + 1) → Coeff m) (ξ : H ⊗[ℂ] CoeffSpace m) :    ∃ (E : Submodule ℂ H) (hE : FiniteDimensional ℂ E),      letI : FiniteDimensional ℂ E := hE      ∃ η : Amplification E m,        tensorInclusion E m η = (↑ξ : Amplification H m) ∧        tensorInclusion E m (polynomialEval (compression A E) B η) =          polynomialEval A B (↑ξ) := by  sorrytheorem rational_denominator_isUnit [Nontrivial H] (A : H →L[ℂ] H)    {m : ℕ} (R : RationalMatrix m) (hR : numericalClosure A ⊆ poleFreeSet R)    (i j : Fin m) : IsUnit (Polynomial.aeval A (R i j).denom) := by  sorrytheorem zeroConclusion [Subsingleton H] (A : H →L[ℂ] H) : ZeroConclusion A := by  sorryend HilbertCrouzeixendend 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/HilbertCrouzeix.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.HilbertCrouzeix.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: 26 local vocabulary declarations
Lines of Lean, without comments or blank lines: 92
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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