Exact Memory of Finite Spectral Routing Tables: The Direct-Sum Occupancy Law and Its Singular Strata

We give a single exact state-counting theory for finite passive spectral routing tables. Fix distinct boundary frequencies, one input k-plane, and a word whose symbol a requests a target k-plane Y_a exactly n_a times. If the used targets are in direct-sum position—they need not be orthogonal and may have arbitrarily small principal angles—then the minimum McMillan degree among square finite rational-inner interpolants is k(L - min_a n_a). Thus the rarest target fixes generic passive memory, independently of node spacing and word order. The lower bound uses block-dual detectors: direct-sum geometry supplies a constant compression that is invertible on one target and annihilates every other target. Its determinant has k forced zeros at every wrong node, while a Blaschke–Potapov minor has no more zeros than the network has states. The upper bound is a matrix spectral compiler. Target-weighted node polynomials form a full-rank polynomial column; matrix Fejér–Riesz factorization normalizes it to a rational-inner column of degree at most the lower bound, and a degree-preserving lossless completion closes the network. Beyond the direct-sum locus we prove a detector-rank hierarchy, a maximum weighted hyperplane-occupancy bound for lines, and the complete three-line phase diagram. We also give a span sandwich, open-dense genericity, fail-closed noisy certification, an exact collision discontinuity, and the optimal integrated Wigner–Smith delay. Producer and independent verifiers audit nonorthogonal scalar and block tables, collision openings, and singular-incidence fixtures.