{
  "abstract": "We construct simple plane triangulations whose chromatic polynomials have real roots accumulating at every noninteger Beraha number $B_\\ell=4\\cos^2(π/\\ell)$; the integer values $0,1,2,3$ are roots of $K_4$. The construction rests on the observation that a triangulated disk with four boundary vertices acts, inside a larger triangulation, as a transfer operator lying in a commutative three-parameter algebra. Rotating such disks by a quarter turn and composing them in series moves their eigenvalue triples around this algebra in a controllable way, and for every fixed $q\\in(B_{10},4)$ one can manufacture a finite disk whose two competing local eigenvalues are strictly dominated by the third -- a physical filter. Closing a grid of such disks between two poles produces a sphere triangulation, and a fused Temperley-Lieb trace identity accounts for its chromatic polynomial completely, sector by sector, with no appeal to semisimplicity at the root of unity itself -- generic semisimplicity enters only through polynomial identities continued from $δ>2$ -- and no singular higher Jones-Wenzl projectors. In the one-cap sector a Krein-type argument shows that every sweep word has exactly one expanding eigenvalue; Gram-form inertia controls every other sector uniformly in the width. At odd Beraha indices the trace coefficient of the expanding eigenvalue vanishes; at even indices two copies of that eigenvalue cancel between adjacent sectors, and a short real-variable lemma converts either situation into genuine real roots converging to the target. Indices below eleven are treated by elementary families, exact outward-rounded integer certificates, and one explicit disk at $B_{10}$ combined with Tutte's golden identity. All graphs are ordinary, simple, unweighted plane triangulations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.23989",
  "author_total": 1,
  "authors": [
    {
      "name": "Reza Mactabi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8y6k8pt895a7q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.23989",
  "fulltext_url": null,
  "id": "8y6k8pt895a7q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.23989",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.CO"
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      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223058973",
  "pdf_url": "https://arxiv.org/pdf/2610.23989",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8y6k8pt895a7q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:52Z",
      "native_id": "2610.23989",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.23989"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Planar Disk Filters and Real Chromatic Roots at Beraha Numbers",
  "title_text": "Planar Disk Filters and Real Chromatic Roots at Beraha Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/8y6k8pt895a7q",
  "venue": "arXiv",
  "versions": []
}