Article 23: Lepton Masses, the Koide Relation, and the PEW Harmonic Simplex

The Primordial Energy Waves (PEW) framework addresses open problemPO-Leptons: derive the muon and tau masses from PEW geometry. This article establishes two results and documents a third. (i) HARMONIC LEVELS [B]: The three charged lepton masses correspond to harmonic levels n_e = 10.473, n_mu = 9.389, n_tau = 8.815 in the PEW table lambda_n = l_P * (1/alpha_em)^n. The span n_e - n_tau = 1.657 ≈ 1 + 1/phi = 1.618 (agreement 2.4%). (ii) KOIDE RELATION AS PEW SIMPLEX [A]: The empirical Koide relation K = (m_e + m_mu + m_tau)/(sqrt(m_e)+sqrt(m_mu)+sqrt(m_tau))^2 = 2/3 (9.2 ppm from CODATA) is identified as the leptonic simplex condition: the vectors x_i = sqrt(m_i)/sum(sqrt(m_j)) form a regularsimplex with sum x_i = 1 and sum x_i^2 = 2/3. The ratio 2/3 = C_F/2 connects to the SU(3) colour Casimir C_F = 4/3 (Article 20). This is the exact leptonic analogue of the proton quark charge simplex established in Article 22. (iii) CANONICAL PARAMETRISATION [A] : In the Brannen parametrisation sqrt(m_i) = A(1 + sqrt(2)*cos(theta_0 + 2*pi*(i-1)/3)),the correct amplitude is rho = 1, giving theta_0 = 2.3166 rad (132.73°), which exactly reproduces all three lepton masses.The phase theta_0 = 2.3166 rad has no known PEW expression to better than 10%; PO-Koide-phase remains open [D].