[
  {
    "id": "ARR-2026-5QQF95VHTC9GABH8-v1",
    "type": "article",
    "title": "Sharp Onset and Unbounded Growth of Norm-Optimal Self-Commutator Rank",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          8,
          30
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-5QQF95VHTC9GABH8",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-5QQF95VHTC9GABH8/versions/v1/",
    "note": "Research preprint; ARR-2026-5QQF95VHTC9GABH8 v1. Not peer reviewed by AIRR.",
    "abstract": "For a traceless Hermitian matrix F in M_d(C), define kappa_d(F) = (1/2) min{||C||_HS^2 : CC^* - C^*C = 2F} and let r_*(F) be the least rank among its minimizers. Inertia gives the universal lower bound r_0(F) = max{n_+(F), n_-(F)}. We prove the exact low-dimensional classification r_*(F) = r_0(F) for d <= 7, including every singular spectrum. The bound fails sharply in dimension eight. For the entire explicit two-ray cone lambda(b,c) = (4c-3b,b,b,b,-c,-c,-c,-c), 0 <= b <= c, we determine the unrestricted and rank-four costs: their difference is min{b,c-b}, and every interior point has r_* = 5 > 4 = r_0. In particular, the integer spectrum (5,1,1,1,-2,-2,-2,-2) has unrestricted cost 13 and rank-four cost 14. Thus eight is the least dimension in which Hilbert--Schmidt norm optimality can force factor rank strictly above inertia. The proof is an exact computer-assisted Horn-polyhedral argument. Thirty-three sign--zero strata through dimension seven yield 813 exact face rays and 272 projected facets; every facet has a rational nonnegative Farkas certificate on the unrestricted cone. A separate exact replay verifies the dimension-eight phase and a two-valued dimension-nine witness. No floating point output is used as a proof object. Finally, for the target G_t with spectrum (5^(12t),-4^(15t)) in dimension 27t, we prove kappa_27t(G_t) = 87t and 17t+1 <= r_*(G_t) <= 18t. Since r_0(G_t) = 15t, the additive excess is at least 2t+1 and is therefore unbounded. The amplification is symbolic: a coarse-grid restriction of any fine hive transfers a frozen order-27 rank-17 lower certificate to every t by telescoping elementary rhombi."
  }
]