{
  "abstract": "We prove a structure theorem for the dimension towers~$\\{d_k(D)\\}_{k\\geq0}$ which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture over a real quadratic field~$K=\\Q(\\qD)$. If~$\\eps$ denotes the first totally positive power of a fundamental unit of~$K$ and $t_k = \\eps^k + \\eps^{-k}$ the trace of its $k$-th power, then the SIC dimension tower~$\\{d_k=1+t_k\\}_{k\\geq0}$ is the level~$m=3$ row of an infinite two-dimensional cyclotomic array~$\\{Ψ_m(t_k)\\}_{m\\geq1,k\\geq0}$ attached to~$K$, while the auxiliary factors~$(d_k+1)$ and $(d_k-3)$ are its ramified levels $m=2$ and $m=1$. Here~$Ψ_m$ denotes the minimal polynomial of~ $ζ_m+ζ_m^{-1} = 2\\cos{2π/m}$. This construction arose initially from an attempt to formulate relations among SIC dimensions in $q$-algebraic terms. The central object is a single closed composite norm relation for the two-parameter family $c_{m,k}=1-ζ_m\\eps^k$ over the cyclotomic field tower $\\{K(μ_m)\\}_{m\\geq1}$. This framework sheds new light on the mod-$p$ analogue of Leopoldt's conjecture, by placing the central 3-symmetry of Zauner's conjecture within a broader arithmetic context. Away from the primes dividing~$2mD$, the valuations $v_p(Ψ_m(t_k))$ at every fixed level~$m$ are described exactly in terms of a single local unit valuation, which is then related, through the~$p$-adic class number formula, to the corresponding $p$-adic $L$-value.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.04431",
  "author_total": 1,
  "authors": [
    {
      "name": "Gary McConnell"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6bzempvsjp7m7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.04431",
  "fulltext_url": null,
  "id": "6bzempvsjp7m7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.04431",
  "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.NT"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7211870530",
  "pdf_url": "https://arxiv.org/pdf/2609.04431",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6bzempvsjp7m7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:14Z",
      "native_id": "2609.04431",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.04431"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "SIC dimension towers via cyclotomic polynomials",
  "title_text": "SIC dimension towers via cyclotomic polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/6bzempvsjp7m7",
  "venue": "arXiv",
  "versions": []
}