{
  "abstract": "Kirkland and Šmigoc constructed a family of stochastic matrices realising the Type III boundary polynomials in the Karpelevič region and conjectured that, conversely, every stochastic realisation of such a polynomial must come from their construction. We prove this conjecture for the full nonzero parameter range $0<α\\le1$, for genuine Type III reduced Ito polynomials of order $n$, $f_α(x)=x^y(x^q-(1-α))^d-α^d$, where $n=qd+y$. For $0<α<1$, the proof first reduces every realisation to a two-shift cyclic normal form using the Dmitriev--Dynkin boundary theorem. The remaining argument is finite and combinatorial: Coates' coefficient formula and the equality case of a weighted Turán theorem force the $q$-cycles associated with the backward edges to split into $d$ complete multipartite classes of equal total weight. A circular-arc telescoping argument then converts this additive equality into the product condition required by Kirkland and Šmigoc. The endpoint $α=1$ is treated separately. We also explain why the closed endpoint $α=0$ is degenerate: the literal extension to this endpoint fails, because reducible realisations with closed $q$-cycles and transient states need not contain the global $n$-cycle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.27219",
  "author_total": 2,
  "authors": [
    {
      "name": "Brecht F. Verbeken"
    },
    {
      "name": "Vincent Ginis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2qpjh0hz14zxh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.27219",
  "fulltext_url": null,
  "id": "2qpjh0hz14zxh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.27219",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "arxiv",
      "math.SP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2607.27219",
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2qpjh0hz14zxh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2607.27219",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.27219"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The Type III realisation conjecture of Kirkland and Šmigoc",
  "title_text": "The Type III realisation conjecture of Kirkland and Šmigoc",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/2qpjh0hz14zxh",
  "venue": "arXiv",
  "versions": []
}