{
  "abstract": "We prove that every trace-zero matrix $A\\in M_n(\\mathbb{C})$ admits a representation $A=BC-CB$ with $B,C\\in M_n(\\mathbb{C})$ and $\\lVert B\\rVert\\lVert C\\rVert\\le K\\lVert A\\rVert$, where $K$ is an absolute constant independent of $n$, and $\\lVert\\cdot\\rVert$ denotes the operator norm. For a fixed $t>0$, the proof splits according to whether $\\lVert\\operatorname{Re}(e^{\\mathrm{i}θ}A)\\rVert_1\\ge tn\\lVert A\\rVert$ holds for all $θ\\in\\mathbb{R}$, where $\\lVert\\cdot\\rVert_1$ denotes the trace norm. When this lower bound holds, we construct a commutator representation directly. Otherwise, the vector-selection theorem of Marcus, Spielman, and Srivastava yields smaller trace-zero compressions whose norms are small enough for the induction to close. We also construct an explicit family of zero-diagonal Hermitian unitaries that forces a lower bound of order $\\sqrt{\\log n}$ for $\\lVert B\\rVert\\lVert C\\rVert$ when either factor is required to be diagonal in the prescribed basis. The same family admits $\\varepsilon$-pavings with fewer than $2\\varepsilon^{-2}$ blocks and representations by two normal factors with optimal norm product $1/2$. This establishes a distinction between unrestricted commutator bounds and bounds under a prescribed diagonal restriction. The main results and their essential inputs are formalized in Lean 4 using Mathlib. The development also includes a formal derivation of the Kadison-Singer state-extension theorem from the same vector-selection theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09938",
  "author_total": 3,
  "authors": [
    {
      "name": "Hao Shen"
    },
    {
      "name": "Jiaqi Wang"
    },
    {
      "name": "Lihong Zhi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fxhfg0zfsdg3k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09938",
  "fulltext_url": null,
  "id": "fxhfg0zfsdg3k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09938",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7212147195",
  "pdf_url": "https://arxiv.org/pdf/2609.09938",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fxhfg0zfsdg3k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09938",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09938"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Dimension-Independent Commutator Bound",
  "title_text": "A Dimension-Independent Commutator Bound",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/fxhfg0zfsdg3k",
  "venue": "arXiv",
  "versions": []
}