{
  "abstract": "Flavor mixing and CP violation in the Standard Model are conventionally parameterized by Euler angles that entangle the physical contributions of the up- and down-type quark sectors, obscuring the origin of CP violation itself. We present a non-perturbative geometric framework that resolves this entanglement through the commutativity condition $[\\mathbf{M}_R^2,\\mathbf{M}_I^2]=0$ for the Hermitian components of the mass-squared matrices, reducing the 18 real degrees of freedom of a general $3\\times3$ Yukawa matrix to five independent parameters and yielding an exact Cartesian-like parameterization. Explicit diagonalization directly gives fermion masses, CKM matrix elements, and the CP phase $δ$ in closed form, cleanly separating the distinct physical contributions of the up- and down-type sectors. Mapping flavor space onto a two-dimensional vector geometry, we derive the Jarlskog invariant $J$ in exact closed form and show that CP violation is governed by a discrete topological selection rule ($η_{\\text{relative}}\\in\\{0,\\pm1\\}$) together with the vector cross-product $\\mathbf{v}_u\\times\\mathbf{v}_d=xy'-x'y$. Across all 36 $S_3\\times S_3$ flavor topologies, this yields an exact $18/18$ classification into CP-violating ($J \\neq 0$) and CP-conserving ($J\\equiv 0$) cases, associated with two distinct CP-suppression mechanisms: geometric alignment through a vanishing vector cross-product and algebraic cancellation of imaginary contributions under permutation. At tree level, fitting to experimental CKM elements yields agreement at the $\\mathcal{O}(10^{-2})$ level, establishing this framework as a transparent analytical baseline for future non-commutative flavor extensions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.26681",
  "author_total": 1,
  "authors": [
    {
      "name": "Chilong Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cbewwsf8qwbt7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.26681",
  "fulltext_url": null,
  "id": "cbewwsf8qwbt7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.26681",
  "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",
      "hep-ph"
    ],
    [
      "arxiv",
      "hep-th"
    ],
    [
      "openalex",
      "3106|Particle physics theoretical and experimental studies"
    ]
  ],
  "openalex": "W7171838274",
  "pdf_url": "https://arxiv.org/pdf/2607.26681",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cbewwsf8qwbt7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:48Z",
      "native_id": "2607.26681",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.26681"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:hep-ph",
      "subfield": "high-energy"
    }
  ],
  "title": "An Exact Geometric Framework for Flavor Mixing and CP Violation in the Standard Model",
  "title_text": "An Exact Geometric Framework for Flavor Mixing and CP Violation in the Standard Model",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:48Z",
  "url": "https://research.draconitic.com/paper/cbewwsf8qwbt7",
  "venue": "arXiv",
  "versions": []
}