[
  {
    "id": "ARR-2026-77QM18J2KG9679B7-v1",
    "type": "article",
    "title": "Finite-Sample Spectral-Gap Falsification: Exact Weighted Visibility Minimax, Hidden-Atom LAN, and Honest Dependent Tests",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          13
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-77QM18J2KG9679B7",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-77QM18J2KG9679B7/versions/v1/",
    "note": "Research preprint; ARR-2026-77QM18J2KG9679B7 v1. Not peer reviewed by AIRR.",
    "abstract": "Finite Euclidean correlation matrices are routinely converted into spectral gaps by generalized eigenvalue, Prony, or Lanczos procedures. We determine a finite-sample boundary for what such data can falsify and what they cannot certify by non-rejection. Block Hankel pencils give monotone lower bounds on the visible transfer edge, while exact rank stabilization recovers the complete visible spectrum. An atom of weight w at any x*=exp(-g) changes every moment by at most w while forcing gap at most g; arbitrarily small strictly positive gaps therefore remain hidden even when an extra zero mode is forbidden. For a fixed nonsingular Gaussian experiment we compute the exact hidden-atom likelihood and derive its LAN information, sharp local power envelope, and powerless/local/consistent phase at the w sqrt(n) scale. Under a disclosed visibility floor, a fourth-kind Chebyshev filter solves the weighted localizer minimax exactly; a two-atom measure attains the bound, making the uniform sign threshold necessary and sufficient in the declared polynomial class. This margin feeds an exact-level two-coordinate Wishart test and a closed depth–sample resource law. A distribution-free companion uses paired differences and the exact quadratic range to handle bounded iid readouts with unknown mean and same-sample selection over a finite filter bank. A dependence-robust extension treats one stationary bounded beta-mixing trajectory: sparse pairing, Berbee coupling, and an explicit lag-covariance correction give finite-sample level, power, and total chain-horizon bounds under an externally certified mixing envelope. An interacting ANNNI-chain experiment through 2^16 states demonstrates exact symmetry blindness and multichannel recovery. Rejection falsifies an overstated gap; non-rejection alone is not a positive gap certificate."
  }
]