[
  {
    "id": "ARR-2026-07EH3CZRD89YC8JN-v2",
    "type": "article",
    "title": "Lossless Calibration Is Stored Memory: A Topological McMillan-Degree and Wigner–Smith Law for Passive Quantum Networks",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          14
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-07EH3CZRD89YC8JN",
    "version": "v2",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-07EH3CZRD89YC8JN/versions/v2/",
    "note": "Research preprint; ARR-2026-07EH3CZRD89YC8JN v2. Not peer reviewed by AIRR.",
    "abstract": "How many internal state coordinates in a specified rational realization are required to transmit prescribed quantum noise channels without loss while suppressing all signal directions elsewhere? This paper synthesizes classical degree and phase-volume mechanisms in a protected signal/loss formulation for finite rational inner networks. For a rational inner scattering matrix of McMillan degree n, exact lossless calibration subspaces at distinct regular boundary frequencies have nodewise total dimension at most n if the signal block is a strict contraction at another boundary frequency. Local nullity forces multiplicity in a nonzero complementary loss minor; exterior powers of minimal Blaschke-Potapov factors bound its zeros by n. The same factorization identifies degree with integrated Wigner-Smith trace delay in the unit-circle/sample-delay coordinate; an explicit rotated Cayley map supplies a continuous-frequency convention and chain-rule delay conversion; a physical oscillator count is conditional on a minimal annihilation-only passive realization. An explicit finite inner family attains the bound. The Rouche extension preserves complex zero counts in disjoint pole-free closed regions for rational inner perturbed transfers holomorphic there and satisfying a strict contour margin for the same loss minor. Sampled approximate calibration alone cannot imply a degree bound. A previously constructed irreducible six-port reservoir filter has 3S exact calibration zeros and degree 3S+6, within six states of the optimum among finite rational inner completions with the same exact calibration data and a strict stop. Its exact integrated trace delay is 3S+6 and its peak grows exponentially. Under a specified stationary weak-coupling Markov model, the rate interpretation counts nodewise preserved input dimensions at distinct frequencies. Numerical degree entries are analytic formula values, not computed minimal realizations; quadrature and root audits test the explicit family. Sources, corrections and dated executions remain distinguished."
  }
]