{
  "abstract": "Radchenko and Wheeler (RW) recently proved a finite pentagon relation for real quadratic special values of the modular quantum dilogarithm and used it to establish the rank-$1$ twisted convolution identity conjectured by the current authors. RW gave an explicit argument in the principal case and remarked that their proof holds for all rank-$1$ admissible tuples. We extend their proof to all rank-$r$ admissible tuples and provide an explicit dictionary between the modular quantum dilogarithm and the Shintani-Faddeev modular cocycle conventions in the respective papers. Thus, we establish that, if $d,r$ are positive integers such that $r<\\frac{d-1}{2}$ and $\\frac{d^2-1}{r(d-r)} \\in \\mathbb{Z}$, then there exist ghost $r$-SICs: i.e., configurations of $d^2$ rank-$r$ subspaces in $\\mathbb{C}^d$ that satisfy a non-Hermitian equichordal condition. Under the Stark conjecture, these configurations are Galois conjugate to Hermitian equichordal configurations called $r$-SICs (or rank-$r$ SIC-POVMs).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.39192",
  "author_total": 3,
  "authors": [
    {
      "name": "Marcus Appleby"
    },
    {
      "name": "Steven T. Flammia"
    },
    {
      "name": "Gene S. Kopp"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fss2qbzk02vdy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.39192",
  "fulltext_url": null,
  "id": "fss2qbzk02vdy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.39192",
  "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.MG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
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  "openalex": "W7215072206",
  "pdf_url": "https://arxiv.org/pdf/2609.39192",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fss2qbzk02vdy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.39192",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.39192"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "The twisted convolution identity and ghost $r$-SICs from finite quantum dilogarithms",
  "title_text": "The twisted convolution identity and ghost r-SICs from finite quantum dilogarithms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/fss2qbzk02vdy",
  "venue": "arXiv",
  "versions": []
}