[
  {
    "id": "ARR-2026-68BH5JDJCQ8XW8GT-v1",
    "type": "article",
    "title": "One Fold, Unique Contact and the Complete Two-Piece Rate–Distortion Function of Rank-One Complex-Projective Born Prediction",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          9
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-68BH5JDJCQ8XW8GT",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-68BH5JDJCQ8XW8GT/versions/v1/",
    "note": "Research preprint; ARR-2026-68BH5JDJCQ8XW8GT v1. Not peer reviewed by AIRR.",
    "abstract": "For a Haar-random pure state on $\\mathbb{CP}^{d-1}$ with squared-Frobenius distortion on density-matrix reproductions, the exact rate–distortion function $R_d(D)$ is known (ARR-2026-1D2QYXPCVY9H7ANB) as a one-dimensional envelope over a radius $b\\in[0,1]$ and a multiplier $\\lambda$; first contact was proved there, but uniqueness of the positive branch and absence of later branch exchange were left open (its Remark 3.2). We close that remark for every finite $d\\ge3$, in a note written to be read without the earlier record (only its envelope theorem is imported, with the elementary facts it rests on reproved here). The tool is the first-order equation $\\kappa M'=(\\kappa-d+1)M+(d-1)$ for $M={}_1F_1(1;d;\\kappa)$, which reduces the fold numerator $K_d'-\\kappa K_d''$ to the sign of $\\kappa-\\Psi_d(M)$ with $\\Psi_d$ an explicit rational function; a transversality argument in the $(M,\\kappa)$ plane then shows the stationary multiplier $\\lambda(\\kappa)$ has exactly one fold. This yields a unique coexistence multiplier $\\lambda_{c,d}$, no reentrance, and $R_d=$ a linear face glued at $D_{c,d}$ to an explicit complex-Bingham curve. A computer-assisted theorem gives the one-change law for the fold-numerator coefficients; $d=2$ recovers the Dytso–Cardone sphere result."
  }
]