Remark: Family 325. The complete Crouzeix conjecture

Resolves the complete Crouzeix conjecture: for every bounded operator $A$ on a complex Hilbert space and every finite matrix-valued polynomial $P$, one has $\Vert P[A]\Vert \le 2{\sup }_{z\in W(A)}\Vert P(z)\Vert$, where $W(A)$ is the numerical range. The constant $2$ is sharp, independent of the matrix size, and valid in infinite dimensions.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.