[
  {
    "id": "ARR-2026-61Y0FFA39M8KMBJ5-v1",
    "type": "article",
    "title": "Complete Rank-Two Born-Prediction Rate–Distortion on Gr_C(2,4): All-Field Matrix–Bingham Rigidity and a Unique Coexistence Transition",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          13
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-61Y0FFA39M8KMBJ5",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-61Y0FFA39M8KMBJ5/versions/v1/",
    "note": "Research preprint; ARR-2026-61Y0FFA39M8KMBJ5 v1. Not peer reviewed by AIRR.",
    "abstract": "We solve a finite-dimensional external-spectrum optimization problem for the complex matrix–Bingham law and use it to determine an unrestricted Shannon rate–distortion function. If P is Haar on Gr_C(2,4) and A is traceless Hermitian with fixed Frobenius norm, then E exp[s tr(AP)] is, for every s>0, uniquely maximized, up to unitary conjugacy, by the balanced spectrum (R/2,R/2,-R/2,-R/2). The exceptional model Gr_C(2,4)=Gr_2^+(R^6) turns the orbital integral into a positive series whose coefficients share one simplex-vertex maximizer. For the Haar rank-two state source rho_P=P/2, conditional least squares places arbitrary reports in the full body {sigma: 0<=sigma<=I/2, tr sigma=1}. We obtain its exact classical rate–distortion function for every 0<=D<=1/4. Beyond a scalar dual representation, we prove the complete radial phase diagram: one fold, one positive coexistence contact, no reentrance, and an exact two-piece frontier consisting of one time-sharing segment and one matrix–Bingham branch. The proof reduces the fold derivative to a power series with exactly one negative coefficient followed by strictly positive coefficients. Covariant channels attain the frontier, and the same value is the source-universal worst-state capacity; arbitrary joint n-block memories cost exactly n times the one-letter frontier. The result concerns classical memory for calibrated Born-probability prediction, not quantum rate–distortion, click simulation, or a derivation of Born's rule."
  }
]