Scalar extremizers for complete Crouzeix equality in dimension four

Assume the complete Crouzeix bound and the analytic Faber coefficient estimate stated in the paper. Let A be a complex matrix of order at most four, and let P be a matrix-valued polynomial of arbitrary finite degree and coefficient size. If ||P[A]|| = 2 maxW(A) ||P|| > 0, we prove that there are constant unit coefficient vectors u, v for which p = v* P u attains the same equality. Jin's scalar rigidity theorem then implies that W(A) is a disk, and the extracted scalar polynomial is a monomial after centering and scaling that disk. The main step converts zero slack in a positive boundary-field inequality into a rank-one condition on the coefficient factors of extremal singular vectors. An elementary Hermitian-pencil argument supplies the required regular boundary arc in dimension four.