{
  "abstract": "For positive exponents $α_1,\\ldots,α_4$ we evaluate the cyclic four-electron Slater integral, the topology recorded as case V in Bonham's 1965 classification, in a finite real expression of weight at most three. The general answer is an explicit sum of eighteen blocks involving only logarithms, dilogarithms and trilogarithms of rational functions of the exponents; all logarithm arguments are positive and all nonconstant polylogarithm arguments may be placed in $(-1,1)$, so no radical, generalized polylogarithm, contour prescription or unevaluated limit is required. At equal exponents the value collapses to $I(λ,λ,λ,λ)=(4π)^4λ^{-4}[\\mathrm{Li}_3(1/4)/2-7ζ(3)/24+π^2\\log 2/12-2\\log^3 2/3]$, a weight-three period over $\\mathbb{Q}$ ramified only at 2 and 3. Three independently implemented numerical routes agree at four exponent tuples to at least 32 decimal places, an exact-dyadic evaluation of the classical formula gives eight 100-place certificates, and three published four-electron calibration values are reproduced to every printed digit. We also prove the small- and large-exponent limit laws with closed-form coefficients, dihedral symmetry, and a strict rearrangement law for the three inequivalent cyclic placements of four exponents. The formula applies to the four-cycle and does not evaluate the two chorded topologies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.00036",
  "author_total": 1,
  "authors": [
    {
      "name": "Jonas Matuzas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c4sy2vjwg7nxf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.00036",
  "fulltext_url": null,
  "id": "c4sy2vjwg7nxf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.00036",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "physics.atom-ph"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7215797507",
  "pdf_url": "https://arxiv.org/pdf/2610.00036",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c4sy2vjwg7nxf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:03Z",
      "native_id": "2610.00036",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.00036"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.atom-ph",
      "subfield": "amo"
    }
  ],
  "title": "Closed-form evaluation of the cyclic four-electron atomic correlation integral",
  "title_text": "Closed-form evaluation of the cyclic four-electron atomic correlation integral",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/c4sy2vjwg7nxf",
  "venue": "arXiv",
  "versions": []
}