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The complete Crouzeix conjecture: the numerical range is a complete 2-spectral set

Shown: for every bounded operator $A$ on any complex Hilbert space (no separability assumed) and every matrix polynomial $P$ , $\|P[A]\|\le2\sup_{W(A)}\|P\|$ , with 2 sharp; the same bound holds for finite matrix-valued functions holomorphic near $\overline{W(A)}$ and for rational functions with poles outside it. For a matrix and an admissible convex domain containing $W(A)$ , an optimal similarity with condition number at most 2 makes the conformal image contractive, with one positive boundary density representing all matrix-valued analytic evaluations. It contains the scalar Crouzeix theorem as the case $m=1$ .

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The complete Crouzeix conjecture: the numerical range is a complete 2-spectral set · Complete Crouzeix conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: The complete Crouzeix conjecture: the numerical range is a complete 2-spectral set

specialist registry · passed

No independent mathematician has checked this yet. Checked here: the complete inequality in Corollaries 6.1 and 6.3 of the principal manuscript and Theorem 1.1 of the direct companion were read against the complete conjecture. lean/formalization.yaml lists main results for both manuscripts (CompleteCrouzeix, DirectCrouzeix, StructuralCrouzeix). The CompleteCrouzeix comparator was read here: for all $n,m\ge1$ , $d$ , $A\in M_n$ and $B_j\in M_m$ , $\|\sum_k A^k\otimes B_k\|\le2\sup_{W(A)}\|\sum_k z^kB_k\|$ , and 2 is the least such universal constant. This states the headline claim for matrices. The operator version, challenge CrouzeixHilbert, is not in the formalization catalogue; it is linked from lean/docs/325.md, its solution module exists at the pinned commit, and its statement (arbitrary complex Hilbert space, plus holomorphic and rational extensions and sharpness) was read here. Not rebuilt here.

Source ↗

The complete Crouzeix conjecture: the numerical range is a complete 2-spectral set

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

Source ↗

Companion: A direct proof of the complete Crouzeix inequality (Sept 26)

paper · pending

paper link supplied by the VibeMathed registry.

Source ↗

Lean proof (OAI.CompleteCrouzeix.main)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Comparator statement: CompleteCrouzeix.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Comparator statement: CrouzeixHilbert.lean (operators on any Hilbert space)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Lean proof of the operator version (OAI.CrouzeixHilbert.hilbert)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

Source ↗

Artifacts and verifiers

Lean proof (OAI.CompleteCrouzeix.main)

lean artifact · pending

Artifact ↗

Comparator statement: CompleteCrouzeix.lean

lean artifact · pending

Artifact ↗

Comparator statement: CrouzeixHilbert.lean (operators on any Hilbert space)

lean artifact · pending

Artifact ↗

Lean proof of the operator version (OAI.CrouzeixHilbert.hilbert)

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

Comparator statement: CompleteCrouzeix.lean evidence for this event

This event attributed to Unreleased internal OpenAI model

Companion: A direct proof of the complete Crouzeix inequality (Sept 26) evidence for this event

The complete Crouzeix conjecture: the numerical range is a complete 2-spectral set evidence for this event

Lean proof of the operator version (OAI.CrouzeixHilbert.hilbert) evidence for this event

The complete Crouzeix conjecture: the numerical range is a complete 2-spectral set parent of this event

Lean proof (OAI.CompleteCrouzeix.main) evidence for this event

VibeMathed record: The complete Crouzeix conjecture: the numerical range is a complete 2-spectral set evidence for this event

Comparator statement: CrouzeixHilbert.lean (operators on any Hilbert space) evidence for this event

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