[
  {
    "id": "ARR-2026-1VY960RN4T9E0RSC-v1",
    "type": "article",
    "title": "One Fold if and only if the Overlap Law is Right-Skewed: A Riccati Criterion for Tilted Beta Normalizers, with Consequences for Rank-One Projective Rate–Distortion",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          14
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-1VY960RN4T9E0RSC",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-1VY960RN4T9E0RSC/versions/v1/",
    "note": "Research preprint; ARR-2026-1VY960RN4T9E0RSC v1. Not peer reviewed by AIRR.",
    "abstract": "For $M={}_1F_1(a;c;\\kappa)$, the moment generating function of $X\\sim\\mathrm{Beta}(a,c-a)$, the stationary multiplier $\\lambda(\\kappa)=\\kappa/(2b(\\kappa))$ of the Curie–Weiss-type radial problem $\\max_{0\\le b\\le1}\\{K(2\\lambda b)-\\lambda R^2b^2\\}$, $K=\\log M-(a/c)\\kappa$, folds where $H=K'-\\kappa K''$ vanishes. We show that the logarithmic derivative $m=M'/M$ satisfies the Riccati equation $\\kappa m'=a-(c-\\kappa)m-\\kappa m^2$, so that $H=m(1-m)\\,(\\Psi(m)-\\kappa)$ with $\\Psi$ an explicit rational function, and a transversality argument in the plane $(m,\\kappa)$ proves: $H$ has exactly one positive zero if $c>2a$ (right-skewed overlap law) and none if $c\\le2a$. This settles the second open question of ARR-2026-68BH5JDJCQ8XW8GT, gives the complete scalar radial phase and two-piece envelope for every $(a,c)$ with explicit fold bounds for $a\\ge1$ (false for $a<1$), covers the Bessel family ${}_0F_1(;c;\\kappa^2/4)$ (no fold), and yields \"one fold iff $d\\ge3$\" for rank-one sources over $\\mathbb R$, $\\mathbb C$, $\\mathbb H$. For the oriented-two-plane normalizer, which is third order, we give an exact identity and numerics only."
  }
]