{
  "abstract": "Which vertex-transitive graphs of prime order have quantum symmetry? The question of Banica, Bichon and Chenevier is open in the dense regime of Paley graphs, where coherent-algebra methods give no information. To each such graph we attach a two-basepoint Terwilliger algebra of its cyclotomic scheme and study the module it generates from the basepoints: fullness forces the quantum permutation algebra to be commutative, and the module admits no intermediate state, containing either exactly two point masses or all $p$ of them. One point mass, captured at any depth, therefore suffices, and Chassaniol's orbital criterion is the depth-one case. Three consequences follow. A sharp counting argument replaces the Banica--Bichon--Chenevier threshold $p>6^{φ(k)}$ by the quadratic bound $p>(k-1)(k-2)+2$, where $k$ is the type. Four certificates, each a short list of additions modulo $p$, settle $C_{31}(2,4,8,15)$ and $C_{41}(4,10,16,18)$, the two graphs left open by Chassaniol, and complete the classification for type at most $10$ without machine assistance. An exact computation extends the dichotomy ``quantum symmetry if and only if complete or empty'' to all prime orders $p\\le250$, settling the Paley graphs $P_{p}$ with $p\\le241$, the first beyond $P_{17}$. What remains is the capture of a single explicit vector: the midpoint $2^{-1}$ of the two basepoints.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00462",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohammad F. Marashdeh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2r4akjzpxw0bf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00462",
  "fulltext_url": null,
  "id": "2r4akjzpxw0bf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00462",
  "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.CO"
    ],
    [
      "arxiv",
      "math.OA"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7206188706",
  "pdf_url": "https://arxiv.org/pdf/2609.00462",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2r4akjzpxw0bf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00462",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00462"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Two-basepoint Terwilliger algebras and the quantum symmetry of prime-order circulants",
  "title_text": "Two-basepoint Terwilliger algebras and the quantum symmetry of prime-order circulants",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/2r4akjzpxw0bf",
  "venue": "arXiv",
  "versions": []
}