{
  "abstract": "Several apparently distinct formulations of the charged-lepton mass relation $p_2/e_1^2 = 2/3$ share a common algebraic object: the ratio of flavor-breaking to flavor-symmetric amplitude components on an internal family space. We show that (i) the relation expressed as a Fourier-mode constraint on Shulga's compact cycle, (ii) the same relation as a Descartes curvature condition in the sense of Kocik, and (iii) the Soddy completion of the resulting circle configuration are related by exact algebra, with no approximations. The Soddy completion defines a companion scale $F^2 = α^2 μ_\\star$ with $α= \\sqrt{3/2} - 1$ and $μ_\\star = \\sum_\\ell m_\\ell$; identifying this scale with $\\bar{m}^{\\rm MS}_s(μ_\\star)$ yields a light-quark mass cascade with the ratio $m_d/m_s = 5/2 - \\sqrt{6}$. Combined with the leading Gatto-Sartori-Tonin relation, this gives the down-sector Cabibbo seed $\\sqrt{m_d/m_s} = \\sqrt{3/2} - 1 = 0.22474\\ldots$, numerically close to the measured $|V_{us}| = 0.22503 \\pm 0.00068$ ($0.4σ$). A full CKM evaluation requires the up-sector phase correction, which we defer. The correspondence is testable by future lattice QCD+QED determinations of $m_d/m_s$; explicit failure modes are listed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Modern Physics Letters A (2026)",
  "arxiv": "2610.36960",
  "author_total": 1,
  "authors": [
    {
      "name": "Andrew M. Brilliant"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/af026gghdmwr7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36960",
  "fulltext_url": null,
  "id": "af026gghdmwr7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36960",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "physics.gen-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.36960",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/af026gghdmwr7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:50Z",
      "native_id": "2610.36960",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36960"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.gen-ph",
      "subfield": "general"
    }
  ],
  "title": "Descartes Curvature Geometry, Compact-Cycle Amplitudes, and the GST Cabibbo Seed",
  "title_text": "Descartes Curvature Geometry, Compact-Cycle Amplitudes, and the GST Cabibbo Seed",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:50Z",
  "url": "https://research.draconitic.com/paper/af026gghdmwr7",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1142/s021773232650272x",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}