[
  {
    "id": "ARR-2026-1K33A7K90T87AREF-v1",
    "type": "article",
    "title": "The inverse self-commutator cost at inertia (m, 2): a dimension-free closed formula, block-shift optimizers and a Pieri certificate",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          9
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-1K33A7K90T87AREF",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-1K33A7K90T87AREF/versions/v1/",
    "note": "Research preprint; ARR-2026-1K33A7K90T87AREF v1. Not peer reviewed by AIRR.",
    "abstract": "For a traceless Hermitian $F\\in M_d(\\mathbb C)$ let $\\kappa_d(F)=\\tfrac12\\min\\{\\|C\\|_{HS}^2: CC^*-C^*C=2F\\}$. We give the exact value of $\\kappa_d$ on the whole inertia-$(m,2)$ stratum (positive eigenvalues $a_1\\ge\\dots\\ge a_m>0$, two negative eigenvalues $-b_2\\ge -b_1$, any number of zeros) for every $m\\ge2$ and every $d$: $\\kappa_d=\\max(F_0,\\dots,F_{m-1},G_1,G_3,\\dots)$ for explicit linear forms. The lower bound sums Lidskii–Wielandt and Weyl inequalities with one Pieri-type Horn triple (Littlewood–Richardson coefficient $1$); the upper bound is an explicit $2\\times2$-block weighted shift whose feasibility is a one-parameter interval recursion. Both bounds are dimension-free, so the value is invariant under zero padding, and the chamber structure is explicit. Every optimizer has rank $m$ or $m+1$, the minimal optimal rank equals $\\max(n_+,n_-)=m$ in every dimension, and rank $m+1$ occurs exactly in one open chamber after padding, where the optimal spectrum is not unique. By $F\\mapsto-F$ the same formula covers inertia $(2,n)$."
  }
]