[
  {
    "id": "ARR-2026-1DGGW5KX2M9R4VJT-v1",
    "type": "article",
    "title": "A universal capacity bound for amplified determinants and polynomial witnesses in the Hermitian case",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          17
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-1DGGW5KX2M9R4VJT",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-1DGGW5KX2M9R4VJT/versions/v1/",
    "note": "Research preprint; ARR-2026-1DGGW5KX2M9R4VJT v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "For an arbitrary finite tuple of complex square matrices, we maximize the geometric mean of the squared singular values of its linear pencil over auxiliary matrices with unit normalized Hilbert-Schmidt energy and unrestricted size. We prove that this optimum lies between C/e and C, where C is the maximum, over coefficient covariances, of the dimension-normalized operator capacity. No Hermitian or positive-span assumption is imposed on this comparison. An invertible finite evaluation supplies a perturbation inside the original matrix space, subharmonicity selects its phase, and quantitative resolvent comparison, algebraic spectral regularity and Han’s least-singular-value theorem control the entire logarithmic tail. This transfers the Mai–Speicher free determinant formula to feasible finite witnesses. Separately, for Hermitian coefficients whose real span contains the identity, we give an explicit polynomial auxiliary-size bound and a Gaussian sampling guarantee. We do not assert an optimal constant or a general polynomial-time witness algorithm."
  }
]