[
  {
    "id": "ARR-2026-6WX2JF38WE87GB2M-v1",
    "type": "article",
    "title": "Algebraic Query Support for Unitary Oracles: Exact Hilbert Laws, Harmonic Spectra, and General-Tester Bounds",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          14
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-6WX2JF38WE87GB2M",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-6WX2JF38WE87GB2M/versions/v1/",
    "note": "Research preprint; ARR-2026-6WX2JF38WE87GB2M v1. Not peer reviewed by AIRR.",
    "abstract": "The standard k-query dimension bound for an unknown d-dimensional unitary uses the entire degree-k symmetric tensor space and scales as k^(d^2-1). We refine that bound to the degree-k Hilbert function of the projective oracle variety and compute the resulting support exactly in two physically structured families. For a fixed-angle qubit rotation with unknown axis, the repeated-query span has dimension (k+1)^2 on the generic branch, binomial(k+2,2) on the traceless branch, and one on the central branch. A compact-orbit leverage argument inserts this exact support into the general-tester bound, applying to the parallel, sequential, and mathematically admissible indefinite-order strategies covered by the standard tester model. The associated spherical and planar-axis frames admit closed harmonic spectra, purity formulae, endpoint cascades, and a complete tightness classification. For d-level selective-phase oracles, the projective closure is a Segre variety and the exact support is binomial(k+d-1,d-1)^2, reducing the ambient exponent from d^2-1 to 2d-2. These results quantify geometry-dependent query support rather than physical memory or an achievable discrimination probability in every finite ensemble. The work excludes inverse-oracle access, controlled bypasses, noise, finite-sample estimation, and tester models beyond those explicitly stated."
  }
]