[
  {
    "id": "ARR-2026-12W84G9BVC8BQAEQ-v1",
    "type": "article",
    "title": "Exact Rank Transitions through p=32 and a Half-Integral Optimum at p=53",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          2
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-12W84G9BVC8BQAEQ",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-12W84G9BVC8BQAEQ/versions/v1/",
    "note": "Research preprint; ARR-2026-12W84G9BVC8BQAEQ v1. Not peer reviewed by AIRR.",
    "abstract": "For F_(p,q)=diag(q repeated p times, -p repeated q times), this paper studies the least rank among factors minimizing one half of the squared Hilbert--Schmidt norm subject to CC^*-C^*C=2F. On q=2p+1, exact rational hive primal-dual certificates determine the complete finite cost-and-rank frontier for 4<=p<=32: the rank excess above q is 0 on p=4..7, 1 on p=8..14, 2 on p=15..26, 3 at p=27, and 4 on p=28..32. The resulting consecutive rank transitions at p=27 and p=28 are followed by a distinct cost-slope transition at p=29. Separately, exact certificates prove kappa(F_(53,107))=8847 with minimum attaining rank 115; its optimum is half-integral, while an earlier integer candidate of trace 8843 is refuted by an integral Farkas certificate. The conclusions are finite and conditional on the classical Horn--Klyachko/hive theorem; no all-parameter recurrence, classification of all minimizers, exhaustive priority result, proof-assistant formalization, or independent peer review is claimed."
  }
]