[
  {
    "id": "ARR-2026-52B6MSS1W197W9T2-v1",
    "type": "article",
    "title": "Exact Memory of Finite Spectral Routing Tables: The Direct-Sum Occupancy Law and Its Singular Strata",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          13
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-52B6MSS1W197W9T2",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-52B6MSS1W197W9T2/versions/v1/",
    "note": "Research preprint; ARR-2026-52B6MSS1W197W9T2 v1. Not peer reviewed by AIRR.",
    "abstract": "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."
  }
]