Discrete S3 Flavor Symmetry, the Positivity Exclusion Horizon, and the Geometric Cutoff of Charged Lepton Generations
Abstract
The charged lepton mass hierarchy (e, μ, τ) is modeled by projecting an equilateral S₃ permutation symmetry onto three-dimensional square-root mass space ℝ³. We show that the empirical Koide invariant Q ≡ Σm / (Σ√m)² = 2/3 corresponds to maximal mixing between the isotropic vacuum singlet and the two-dimensional flavor doublet, fixing the relative geometric projection coefficient to √2.
Enforcing real, positive mass eigenvalues (√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 ≈ 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.
Furthermore, 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 → 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 (Γ ∝ m³), driving the quantum coherence parameter κ ≡ mc² / (2πΓ) strictly below unity above m ≈ 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_τ = 1776.97 MeV (0.91σ from experiment).
Related
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BibTeX
@misc{bray2026discrete,
title = {{Discrete S3 Flavor Symmetry, the Positivity Exclusion Horizon, and the Geometric Cutoff of Charged Lepton Generations}},
author = {Brett Bray},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23672731},
url = {https://zenodo.org/records/23672731},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Discrete S3 Flavor Symmetry, the Positivity Exclusion Horizon, and the Geometric Cutoff of Charged Lepton Generations
AU - Brett Bray
PY - 2026
DA - 2026/10/26
PB - Zenodo
DO - 10.5281/zenodo.23672731
UR - https://zenodo.org/records/23672731
AB - The charged lepton mass hierarchy (e, μ, τ) is modeled by projecting an equilateral S₃ permutation symmetry onto three-dimensional square-root mass space ℝ³. We show that the empirical Koide invariant Q ≡ Σm / (Σ√m)² = 2/3 corresponds to maximal mixing between the isotropic vacuum singlet and the two-dimensional flavor doublet, fixing the relative geometric projection coefficient to √2. Enforcing real, positive mass eigenvalues (√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 ≈ 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. Furthermore, 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 → 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 (Γ ∝ m³), driving the quantum coherence parameter κ ≡ mc² / (2πΓ) strictly below unity above m ≈ 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_τ = 1776.97 MeV (0.91σ from experiment).
ER -