{
  "abstract": "The charged lepton mass hierarchy (e, μ, &tau;) is modeled by projecting an equilateral S₃ permutation symmetry onto three-dimensional square-root mass space ℝ&sup3;. We show that the empirical Koide invariant Q &equiv; &Sigma;m / (&Sigma;&radic;m)&sup2; = 2/3 corresponds to maximal mixing between the isotropic vacuum singlet and the two-dimensional flavor doublet, fixing the relative geometric projection coefficient to &radic;2.\n\nEnforcing real, positive mass eigenvalues (&radic;m_k > 0) establishes a hard geometric exclusion horizon (135° < ϕ < 225°). By the Step-Size Theorem, the 120° trilateral ℤ₃ cyclic partition uniquely leaps over this 90° forbidden window, with the electron resting at ϕ_e &asymp; 132.74° (2.26° shy of the zero-mass cliff). Conversely, any symmetric partition N ≥ 4 inevitably traps at least one phase node within the unphysical negative-amplitude domain.\n\nFurthermore, the discrete ℤ₃ orthogonality of the ground-state eigenmodes ensures the diagonality of the electromagnetic current, preserving individual lepton flavor numbers and forbidding tree-level flavor-violating decays such as μ &rarr; eγ. The global chiral phase shift δ enforces exact CP-invariance with vanishing electric dipole moments (d_e = 0). Finally, incorporating the Standard Model longitudinal electroweak gauge coupling reveals that heavy lepton decay widths scale cubically with mass (&Gamma; &prop; m&sup3;), driving the quantum coherence parameter κ &equiv; mc&sup2; / (2π&Gamma;) strictly below unity above m &asymp; 700–750 GeV. The charged lepton spectrum terminates at exactly three generations by joint geometric and electroweak necessity, cleanly predicting a pole tau mass of m_&tau; = 1776.97 MeV (0.91σ from experiment).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Brett Bray"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2jes2hg8rj85a.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23672731",
  "fulltext_url": null,
  "id": "2jes2hg8rj85a",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23672731",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23672731/files/Bray26-S3-Lepton-Hierarchy.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2jes2hg8rj85a.ris",
  "sources": [
    {
      "first_seen": "2026-11-09T12:20:44Z",
      "native_id": "23672731",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23672731"
    }
  ],
  "subjects": [],
  "title": "Discrete S3 Flavor Symmetry, the Positivity Exclusion Horizon, and the Geometric Cutoff of Charged Lepton Generations",
  "title_text": "Discrete S3 Flavor Symmetry, the Positivity Exclusion Horizon, and the Geometric Cutoff of Charged Lepton Generations",
  "type": "preprint",
  "updated_at": "2026-11-09T12:20:44Z",
  "url": "https://research.draconitic.com/paper/2jes2hg8rj85a",
  "venue": "Zenodo",
  "versions": []
}