{
  "abstract": "We conjecture an explicit evaluation of the pure fifth Bessel moment $\\int_0^\\infty K_0(t)^5\\,dt$ as a quadratic expression in the critical value $L(f,2)$ of the weight-three, level-60 newform $f$ (LMFDB orbit 60.3.b.a) identified in the twisted fifth-moment modularity theorem of Lim, Tu and Yu, with coefficients in $\\mathbb{Q}(\\sqrt{5})$ and the square taken before the real and imaginary parts. Directed interval computations, using no stored Bessel or $L$-values, bound the absolute discrepancy by $10^{-358}$. We prove three exact modular identities for $f$: the Petersson-norm formula $\\langle f,f\\rangle_{60} = \\frac{3(5-\\sqrt{5})}{2π^4}|L(f,2)|^2$, the coefficient-conjugation relation $L(f^σ,2) = κL(f,2)$ with explicit $κ\\in \\mathbb{Q}(\\sqrt{5},i)$, and the twisted symmetric-square evaluation $L(χ_{-4}\\mathrm{Sym}^2 f,2) = \\sqrt{15}\\,π^2 \\langle f,f\\rangle_{60}$, together with $L(χ_{-4}\\mathrm{Sym}^2 f,3) = π^4\\langle f,f\\rangle_{60}/8$, in the full Euler-factor normalization of Lim, Tu and Yu. The last identity shows that the companion norm conjecture $D_{5,\\mathrm{odd}} = \\frac{3\\sqrt{15}(5-\\sqrt{5})}{2}|L(f,2)|^2$ is equivalent to the symmetric-square conjecture $D_{5,\\mathrm{odd}} = π^2 L(χ_{-4}\\mathrm{Sym}^2 f,2)$ of Lim, Tu and Yu, while the exact relation $D_{5,\\mathrm{even}} = π^2 D_{5,\\mathrm{odd}}/(2\\sqrt{15})$ follows from Chuang's period formulas. Every Bessel-to-modular equality, including the individual-period formula, remains conjectural. Complete proofs, exact rational certificates and verification programs are included as ancillary files.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22318",
  "author_total": 1,
  "authors": [
    {
      "name": "Jonas Matuzas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c7grpht4ts7pm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22318",
  "fulltext_url": null,
  "id": "c7grpht4ts7pm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22318",
  "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.GM"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214051531",
  "pdf_url": "https://arxiv.org/pdf/2609.22318",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c7grpht4ts7pm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.22318",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22318"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2602|Advanced Mathematical Identities",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A quadratic critical-value conjecture for the fifth Bessel moment",
  "title_text": "A quadratic critical-value conjecture for the fifth Bessel moment",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/c7grpht4ts7pm",
  "venue": "arXiv",
  "versions": []
}