[
  {
    "id": "ARR-2026-2Z86XAW5F88GQRZ2-v1",
    "type": "article",
    "title": "The inverse self-commutator cost at inertia $(m,3)$: tiling certificates, 3×3-block shifts and a padding phase",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          14
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-2Z86XAW5F88GQRZ2",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-2Z86XAW5F88GQRZ2/versions/v1/",
    "note": "Research preprint; ARR-2026-2Z86XAW5F88GQRZ2 v1. Not peer reviewed by AIRR.",
    "abstract": "For a traceless Hermitian $F\\in M_d(\\mathbb C)$ let $\\kappa_d(F)=\\tfrac12\\min\\{\\|C\\|_{HS}^2: CC^*-C^*C=2F\\}$. On the inertia-$(m,3)$ stratum ($m$ positive eigenvalues, three negative ones, $z$ zeros) we determine $\\kappa_d$ as the maximum of an explicit finite set $S(m,z)$ of linear forms for $m=3,\\dots,9$ and $z\\le2$, by exact Horn \"tiling\" certificates below and $3\\times3$-block weighted shifts above, the latter verified at every vertex of every chamber together with a convexity argument. The forms are costs of layerings; for the ordered family A the lower bound is proved for all $m$ and $z$ by Lidskii–Wielandt tail inequalities. Unlike inertia $(m,2)$, the value is not padding-invariant: for $m\\equiv1,2\\pmod3$ exactly one extra exposed form exists at $z=0$, so $\\kappa_{m+3}>\\kappa_{m+4}$ on an open set within the stated finite range. At an exact $m=4$ example, padding raises the minimum optimal rank from $4$ to $5$ while $\\kappa_7=74/61>\\kappa_8=73/61$. Six combinatorial families are conjectured to describe $S(m,z\\ge1)$ for all $m$, with quadratic growth; they are tested numerically out of sample for $m=10,11$."
  }
]