[
  {
    "id": "ARR-2026-4C11Z7N0BX8VP8P2-v1",
    "type": "article",
    "title": "Uniform tropical bounds for amplified determinants: finite phase witnesses and multiscale quantum noise",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          16
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-4C11Z7N0BX8VP8P2",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-4C11Z7N0BX8VP8P2/versions/v1/",
    "note": "Research preprint; ARR-2026-4C11Z7N0BX8VP8P2 v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "We study determinants of matrix pencils evaluated on auxiliary matrices under a normalized Hilbert–Schmidt energy constraint. For a fixed full noncommutative-rank pencil of size d, we show that its objective, regularized over all auxiliary sizes, is uniformly comparable on the entire positive coefficient orthant to the monomial envelope of its noncommutative Newton polytope. The lower bound uses auxiliary size max{1,d−1} and a single tuple whose entries vary only by scalar phases. A finite phase menu suffices. Thus every simultaneous coefficient reweighting has the same noise exponent at that fixed size as under unrestricted amplification. For mixtures of quantum channels this gives an exact piecewise-linear law for arbitrary simultaneous power-law noise rates, including the normalization of mixture weights. An explicit six-parameter qutrit pencil has five branches, one absent at scalar size; a single fixed size-two tuple approximates its regularized objective within a factor four for every positive parameter choice. The algebraic polytope and its stabilization are established results. Our contribution is a uniform analytic comparison and its finite-witness and noise-law consequences; we do not claim exact finite-size optimization or a coefficient-uniform approximation factor for general pencils."
  }
]