{
  "abstract": "We study exact moments and algebraic sampling of a reciprocal quadratic Mellin integral. A finite elementary formula determines every nonzero rational-weight first moment with specified branches. For $|b|<2\\sqrt{c}$, mixed half-weight moments admit a dilogarithmic formula, and all even half-weight moments follow from one polynomial recurrence with a geometric error bound. At zero Mellin weight with $b=0$, half-integer deformations form an explicit two-function elliptic space. At the singular boundary, a two-parameter reflection gives a radical integral equal to $π$ for every parameter, a cubic logarithmic integral in dilogarithms, and an elementary golden-ratio specialization. All odd zero-weight boundary moments also have explicit formulas. For a reduced rational weight $p/q$, the sampling function generates a field of degree $q^2$ and genus $(q-1)(q-2)/2$, outside the stated degeneracies. Explicit cubic maps and a Picard--Fuchs certificate fix its elliptic period normalization and yield CM companion series with polynomial-weight closure. At denominator four, we determine the three elliptic quotient maps and their differential pullbacks. For the original kernel $(c,b)=(5,-2)$, the Jacobian is isogenous to $E_{1728}\\times E_0^2$, and homological transfer gives gamma-valued periods of the sampling differential. Classical identities are credited and used as ingredients in these family-specific formulas.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20201",
  "author_total": 1,
  "authors": [
    {
      "name": "K. Srinivasa Raghava"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xsc64ywmtwc5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20201",
  "fulltext_url": null,
  "id": "0xsc64ywmtwc5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20201",
  "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"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7213605518",
  "pdf_url": "https://arxiv.org/pdf/2609.20201",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xsc64ywmtwc5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2609.20201",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20201"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Exact Moments and CM Periods of Reciprocal Quadratic Mellin Integrals",
  "title_text": "Exact Moments and CM Periods of Reciprocal Quadratic Mellin Integrals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/0xsc64ywmtwc5",
  "venue": "arXiv",
  "versions": []
}