A direct proof of the complete Crouzeix inequality

For all positive integers $n$ and $m$, every $A \in M_n(\mathbb{C})$ and
every polynomial $P(z) = \sum_k B_k z^k$ with $B_k \in M_m(\mathbb{C})$, the
manuscript proves $\|P[A]\| \le 2 \max_{z \in W(A)} \|P(z)\|$, where
$P[A] = \sum_k A^k \otimes B_k$ and $W(A)$ is the numerical range. This is
the sharp constant-two numerical-range inequality for matrix-valued
polynomials in all finite base and coefficient dimensions, the matrix
formulation of the complete Crouzeix conjecture, and it includes matrices
whose numerical ranges are points or line segments. The proof places $W(A)$
inside an analytic convex domain and expands the Cauchy kernel in exterior
coordinates. A top singular pair of
$P[A]$ and the resolvent give coefficient functionals, and positivity of the
Hermitian part of the resolvent field compares their boundary norms, which
yields the constant two. Shrinking the domains to $W(A)$ completes the argument; a
nilpotent $2 \times 2$ matrix shows the constant is sharp.