The Koide Formula Explained: Q = (p − 1)/p = 2/3 from GF(9) Lattice Structure

The Koide formula Q = (me + mμ + mτ )/(√me +√mμ +√mτ )2 = 2/3 has remained unexplained for 45 years since its discovery in 1981. We show that Q = (p − 1)/p for p = 3, the prime underlying the GF(9) algebraic structure. The Koide parametrization √mi =p m 0/3 (1+√2 cos(2π(i+2)/3+2/9)) contains three numbers from Z/9Z arithmetic: the factor 1 (identity element), the phase 2π/3 (third root of unity from Z/3Z ≤ (Z/9Z)∗),and the offset 2/9 (the Cabibbo angle λ = 2/p2). Experimental match: Q = 0.666661, deviation 0.0009%. We extend the structure to the quark sector and derive the full CKM connection.