[
  {
    "id": "ARR-2026-4VD8FW0NRZ8KGSNZ-v1",
    "type": "article",
    "title": "All-degree Born rigidity in dimensions four and three: an exact simplex optimum and a certified five-outcome frame",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          14
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-4VD8FW0NRZ8KGSNZ",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-4VD8FW0NRZ8KGSNZ/versions/v1/",
    "note": "Research preprint; ARR-2026-4VD8FW0NRZ8KGSNZ v1. Not peer reviewed by AIRR.",
    "abstract": "For an equal-trace rank-one POVM seed $F$ on $\\mathbb{CP}^{d-1}$ the all-degree Born-rigidity gap is $\\gamma(F)=\\inf_{k\\ge 2}g_k(F)$, where $g_k$ is the squared singular value of the orbit-normalisation operator on the projective harmonic sector of degree $k$, and $\\Gamma_d$ is its supremum over outcome-simple seeds. A previous record proved $\\Gamma_d=d-6/(d+1)$, attained only by an orthonormal basis, for $d\\ge 5$, and left $d=3,4$ open. We prove an explicit tail bound $|Q_{n,d}(t)|\\le(1-t)^{-(d-2)}/\\binom{n+d-2}{n}$ for the normalised zonal function (a two-line corollary of Koornwinder's Laplace-type integral) and use it to settle $d=4$ exactly: $\\Gamma_4=22018975/7340032=2.999847\\ldots$, attained only by the regular five-outcome simplex, with the minimising degree $k=6$. In $d=3$ we prove $77/45<\\Gamma_3\\le 9/5$, certify $\\Gamma_3\\ge 1.776009937964733$ by an explicit five-outcome frame in exact arithmetic, and locate numerically a strict local maximum $1.7760099429996636\\ldots$ with five tied degrees $4,8,9,12,16$."
  }
]