{
  "abstract": "Stone conjectured in 1981 that every fully semimonotone $\\mathbf{Q}_0$-matrix is a $\\mathbf{P}_0$-matrix; Murthy and Parthasarathy (1995) proved this for orders up to $4$ and for several special classes. We study the principal minors of fully semimonotone matrices $A\\in\\Ezerof$ without using $\\Qzero$. Let $\\G(A)$ be the graph whose vertices are the index sets $α$ with $\\det A_{αα}\\neq 0$ (including $α=\\emptyset$), two vertices being adjacent when they differ in exactly one index. We prove that the sign of the principal minor is constant on every connected component of $\\G(A)$, and positive on the component of $\\emptyset$. Consequences: an $\\Ezerof$-matrix with no vanishing principal minor is a $\\mathbf{P}$-matrix, and any $\\Ezerof$-matrix satisfying a simple ``descending'' condition on its nonvanishing minors is a $\\mathbf{P}_0$-matrix. We show that a sign change across two steps through vanishing minors forces strictly positive coupling in the corresponding principal pivotal transform, and that a principal submatrix of minimal order with negative determinant is an almost-$\\mathbf{P}_0$ matrix whose inverse lies in $\\Ezerof\\cap\\Nzero$ and has zero diagonal. Explicit examples show that the component structure can contain negative components, that $\\det A>0$ does not force $A\\in\\Pzero$ in $\\Ezerof$, that $\\Ezerof$ is not invariant under $A\\mapsto A+\\varepsilon I$, and that $\\Ezerof\\cap\\Pzero$ contains matrices with $\\det A>0$ outside $\\Qzero$. Stone's conjecture itself remains open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22829",
  "author_total": 1,
  "authors": [
    {
      "name": "Sajal Ghosh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/draxxvfmjrpbh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22829",
  "fulltext_url": null,
  "id": "draxxvfmjrpbh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22829",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7204170188",
  "pdf_url": "https://arxiv.org/pdf/2608.22829",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/draxxvfmjrpbh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2608.22829",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22829"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Sign Propagation of Principal Minors in Fully Semimonotone Matrices and Stone's Conjecture",
  "title_text": "Sign Propagation of Principal Minors in Fully Semimonotone Matrices and Stone's Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/draxxvfmjrpbh",
  "venue": "arXiv",
  "versions": []
}