{
  "abstract": "We study the rank-twelve potential modular datum denoted $12^{39,406}_{4,1276.}$ in the low-rank classification tables. Its global dimension is $(1287+351\\sqrt{13})/2$, and its fusion ring agrees with that of the known rank-twelve Haagerup modular category. The datum satisfies the modular relations, has nonnegative integral Verlinde coefficients, and passes several familiar genus-one necessary tests. We prove, nevertheless, that it cannot be realized by a unitary modular tensor category. The obstruction already occurs in the representation of the three-strand braid group on $\\operatorname{Hom}(C_{[10]},B^{\\otimes3})$. Generalized Frobenius--Schur indicators determine the eigenvalues of the two-strand generator on this twelve-dimensional space. The central relation in $B_3$ then reduces the problem to a product of three prescribed unitary conjugacy classes. A sixfold symmetry reduces the $1302$ determinant-compatible cases to $217$ representatives. Rank-one inequalities leave only $17$ of these, and ten explicitly displayed degree-zero rank-two Schubert inequalities exclude the remaining cases. A violated inequality with gap $2/117$ is worked out explicitly. All calculations used in the obstruction are exact. The argument uses unitarity essentially and leaves non-unitary categorification open.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15792",
  "author_total": 2,
  "authors": [
    {
      "name": "Jiuhe Liu"
    },
    {
      "name": "Xuzhihan Ding"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8ttrr10852ac6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15792",
  "fulltext_url": null,
  "id": "8ttrr10852ac6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15792",
  "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.QA"
    ],
    [
      "openalex",
      "2608|Algebraic structures and combinatorial models"
    ]
  ],
  "openalex": "W7221793097",
  "pdf_url": "https://arxiv.org/pdf/2610.15792",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8ttrr10852ac6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15792",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15792"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.QA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A Multiplicative Horn Obstruction to the Unitary Categorification of a Rank-Twelve Modular Datum",
  "title_text": "A Multiplicative Horn Obstruction to the Unitary Categorification of a Rank-Twelve Modular Datum",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/8ttrr10852ac6",
  "venue": "arXiv",
  "versions": []
}