Deriving the Koide Phase Parameter from Coupling Dimensionality: A Unified Mass Equation for Elementary Particles

The Koide formula, known since 1982, relates the masses of the three charged leptons with remarkable precision through a ratio Q = 2/3. Its parameter δ — the phase angle in the Brannen parameterization — has remained empirically determined without theoretical justification. We show that this parameter obeys a simple law: δ(N) = √(N−1) × π/2, where N is the number of spatial dimensions to which the particle couples through its temporal twin. This law gives δ = 0 for scalar particles (N=1), δ = π/2 for gauge bosons (N=2), and δ ≈ √2 × π/2 ≈ 127° for fermions (N=3), consistent with the measured value of 132.73° to within the family-dependent temporal correction. We propose a unified mass equation incorporating this derivation and discuss its physical interpretation. A testable prediction concerning the sum of neutrino masses follows: Σmν ≈ 0.0625 eV.