{
  "abstract": "We conjecture a positive four-shift identity for rank-two Nahm sums with denominator steps $(1,31)$, giving a mixed-base realization of the Monster McKay--Thompson series $T_{31A}$ through the classical Rogers--Ramanujan function $U(1,31)$. A theta dissection directly derives the modular target. For the underlying quadratic exponent $3r^2+31rs+93s^2$, we prove nonmodularity of every individual sum with rational linear terms and normalization, together with an exclusion from finite Fricke systems with meromorphic finite-width expansions at infinity. Exact agreement through $q^{2000}$ and six vanishing radial corrections support the proposed combination; a five-term dilogarithm certificate and an exact amplitude calculation prove its leading asymptotic behavior. We further propose a three-component Fricke law for a dual combination, with an explicit icosahedral matrix involving fifth-root constants, and a quadratic identity linking its components to $T_{31A}$. The dual translation law and matrix algebra are proved; its Fricke transformation and moonshine identity remain conjectural, supported respectively by numerical evaluations and exact finite coefficient checks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28509",
  "author_total": 1,
  "authors": [
    {
      "name": "Cetin Hakimoglu-Brown"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7te4hsn6jt9zp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28509",
  "fulltext_url": null,
  "id": "7te4hsn6jt9zp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28509",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214317775",
  "pdf_url": "https://arxiv.org/pdf/2609.28509",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7te4hsn6jt9zp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28509",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28509"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Rank-two Nahm sums, modularity obstructions, and the McKay--Thompson series $T_{31A}$",
  "title_text": "Rank-two Nahm sums, modularity obstructions, and the McKay--Thompson series T_31A",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/7te4hsn6jt9zp",
  "venue": "arXiv",
  "versions": []
}