Rotational Substrate Field Theory Section 15: Particle Mass Spectrum and the Koide Formula

I provide a technical derivation of the lepton mass spectrum by linking particle mass directly to the rotation frequency of a vortex soliton within my framework. I demonstrate that the standard mass-energy relation follows naturally from the kinetic energy term of my model's Lagrangian and the quantization of the lattice. My analysis of the electron vortex confirms this relationship with high precision, showing an exact correspondence between the expected angular frequency and the observed mass. I also show how the empirical Koide formula for leptons is not an arbitrary coincidence but a geometric consequence of the permutation symmetry found in the three color planes of the cuboctahedral lattice. By using the structure of diagonal generators within the specific Lie algebra, I derive a phase angle that allows me to predict the mass ratios of the electron, muon, and tau leptons to five or six significant figures without needing extra parameters. While I identify the absolute mass scale of the electron as the starting point for these calculations, I note that calculating this value directly from substrate constants remains a focus for future work. This document also explores how these geometric principles might eventually extend to quark and neutrino masses through similar symmetry arguments.