Lossless Calibration Is Stored Memory: A Topological McMillan-Degree and Wigner—Smith Law for Passive Quantum Networks

How many internal passive modes are required to transmit prescribed quantum noise channels without loss while suppressing all signal directions elsewhere? We prove an architecture-independent answer for finite-dimensional rational networks. Let a causal rational inner scattering matrix S have McMillan degree n, signal block G, and complementary loss block C. At distinct regular boundary frequencies, suppose that G is isometric on input subspaces of dimensions k_j. If G is a strict contraction at one other frequency, then sum_j k_j ≤ n.Every lossless direction lies in the kernel of C; a nonzero maximal minor of C therefore has a zero of multiplicity at least k_j. Exterior powers of a minimal Blaschke—Potapov factorization show that no minor can have more than n zeros. The same factorization yields the exact topological delay identity(1/2π) ∫ tr Q(θ) dθ = deg_McM S = n,where Q = −i S* ∂θS is positive semidefinite. Thus lossless calibration multiplicity is bounded by integrated Wigner—Smith delay. The bound is sharp in every dimension. We establish robustness under analytic passive perturbations, prove why unstructured approximate samples cannot imply a degree bound, and show that a previously constructed six-port reservoir filter is universally optimal up to six internal states. Proofs and numerical certificates are reproduced by the linked public repository.