[
  {
    "id": "ARR-2026-33BE6K3HW78F4T7F-v1",
    "type": "article",
    "title": "Exact Memory of Direct-Sum Spectral Fan-Out: Generic Maximality, Colliding Targets, and the Three-Line Phase Diagram",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          11
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-33BE6K3HW78F4T7F",
    "version": "v1",
    "genre": "Archived manuscript",
    "URL": "https://airr.science/papers/ARR-2026-33BE6K3HW78F4T7F/versions/v1/",
    "note": "Archived manuscript; ARR-2026-33BE6K3HW78F4T7F v1. Not peer reviewed by AIRR. Historical import; not assessed by AIRR.",
    "abstract": "At L boundary frequencies, a passive lossless network must route one fixed k-dimensional input channel to prescribed output k-planes. For targets with joint span dimension r, every square finite rational-inner interpolant satisfies the sharp universal sandwich r-k <= d_min <= k(L-1). Direct-sum targets therefore have exact minimum McMillan degree k(L-1); when N >= Lk this maximal-memory locus is open and dense. The paper gives an explicit positive Pick completion, a realization- and inertia-based lower bound, a family of targets that collide while retaining maximal exact degree for every nonzero opening, an exact quadratic certification margin, and a fail-closed noisy projector test. For three scalar target lines it also proves the complete phase diagram: degree zero for one common line, degree one exactly for three distinct coplanar lines, and degree two otherwise. Deterministic certificates and an independent implementation accompany the proofs."
  }
]