A Unified Golden-Ratio Formula for the Masses of Fundamental Fermions

ABSTRACT We present a unified formula for the masses of all fundamental fermions (charged leptons, neutrinos, and quarks) based on the golden ratio φ = (1+√5)/2. The formula extends the empirical Koide relation with a universal amplitude: r = √2 + φ^(2k)·δ δ = x⁴(1 − x⁵ − x¹⁰) x = 1/φ where k = 0 for leptons and neutrinos, k = 1 for down-quarks, and k = 1 with φ² enhancement for up-quarks. The formula reproduces charged lepton masses with accuracy 10⁻⁵, neutrino masses with accuracy 10⁻¹² (in the Koide angle), and quark masses with accuracy < 0.02%. We show that the Koide angle θ_K = 2π/3 + 2/9 = 132.73° follows from Z_N symmetry, and that only N = 3 generations yield all positive masses. The color-charge connection is established through the exact identity φ² + x² = N_c = 3. KEY RESULTS 1. Koide angle structure. We propose θ_K(N) = 2π/N + 2/N². For N = 3, this gives θ_K = 132.732395°, matching the experimental value 132.732331° to within 6.4 × 10⁻⁵ degrees. 2. Uniqueness of three generations. Only N = 3 yields all positive fermion masses. For N ≥ 4, at least one mass becomes negative, so nature's choice of three generations is forced by the physicality condition cos(θ_K + 2πi/N) > −1/√2. 3. Universal amplitude for quarks. The amplitude is r = √2 + φ^(2k)·δ with δ = x⁴(1 − x⁵ − x¹⁰) = 0.131556. This gives: - Leptons: r = √2 = 1.414214 (error 0.001%) - Down-quarks: r = √2 + δ = 1.545770 (error 0.018%) - Up-quarks: r = √2 + φ²δ = 1.758632 (error 0.011%) 4. Neutrino prediction. Using oscillation data (Δm²₂₁ = 7.53 × 10⁻⁵ eV², |Δm²₃₁| = 2.45 × 10⁻³ eV²), the Koide condition gives m₁ ≈ 0.019 meV, m₂ ≈ 8.68 meV, m₃ ≈ 49.50 meV, and Σm_ν ≈ 0.0582 eV — consistent with the Planck bound and testable by KATRIN, Project 8, and CMB-S4. 5. Color-charge identity. The exact algebraic identity φ² + x² = N_c = 3 connects the golden ratio to the number of colors in QCD. METHODOLOGY The analysis combines the empirical Koide relation (1981, 1983), PDG 2022 quark masses, neutrino oscillation data, algebraic identities involving the golden ratio, and numerical verification using Python. SCOPE The formula successfully describes charged leptons, neutrinos, up- and down-type quarks, and baryons (via T8CX). It does not describe mesons, the Higgs boson, or W/Z bosons. REPRODUCIBILITY All numerical results are reproducible using the Python code provided in Appendix B of the manuscript. AUTHORS MOBIDUR (Independent Researcher), DEEPSEEK (AI Research Assistant). LICENSE CC-BY-4.0.