[
  {
    "id": "ARR-2026-4Q1WFSE5DW9728QY-v1",
    "type": "article",
    "title": "The inverse self-commutator cost at inertia $(m,n)$: ordered-layering bounds, interlacing chains, and the Horn tiles that certify them",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          14
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-4Q1WFSE5DW9728QY",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-4Q1WFSE5DW9728QY/versions/v1/",
    "note": "Research preprint; ARR-2026-4Q1WFSE5DW9728QY 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 Not peer reviewed by AIRR.",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-$(m,n)$ stratum ($m$ positive and $n$ negative eigenvalues, $z$ zeros) we prove three results valid for all $m,n,z$: a dimension-free lower bound $\\kappa_d\\ge\\operatorname{cost}(L)$ for every ordered full-capacity layering $L$, certified by Lidskii–Wielandt and Weyl inequalities alone; an exact additive formula on the aligned-spike cone, where an optimizer is a direct sum of weighted shifts; and the closed form $\\kappa_d=\\operatorname{cost}(L_k)$ on explicit full-dimensional regions $R_k$, $1\\le k\\le m-n+1$, obtained from one rank-one interlacing step. Open question 1 of the inertia-$(m,2)$ paper is refuted by an exact weak-duality certificate: at inertia $(5,4)$ a Horn tile with $n=4$ boxes is necessary, and one valid form cannot be certified using Pieri-shaped tiles alone. The $d=8$ rank-onset example is explained by a late negative. Conjectures for general $(m,n)$ are stated with their numerical evidence."
  }
]