P29: The Three Particle Generations as Resonances of the 3+e Condensation - A Geometric Origin of the Koide Relation and the Fermion Mass Hierarchy

eighteen independent mass values, as unexplained experimental facts. The Koide formula, an empirical relation among the three charged lepton masses accurate to better than 0.002%, has likewise remained without an accepted physical origin since its proposal in 1981. This paper derives both results from the BFUT condensation functional and the 3+e proton topology established in companion papers P16 and P17. The proton is shown to be the ground-state quantum resonance of the condensation functional, recovering the proton mass to 0.003% with a single scale calibration and no further free parameters. Second and third generation particles are identified as higher resonances of the same structure rather than independent entities. The three-fold (D3h) symmetry of the three-quark, one-interstitial-unit topology is shown to require the Koide relation Q=2/3 for the charged leptons, recovered numerically to 0.0011%, and the explicit coupling matrix responsible for this relation is constructed. The total lepton mass scale is independently derived from the W boson mass and a structural bifurcation count already fixed in P16, removing one further free quantity. The electron mass, separately derived in P16, is connected to this matrix structure by an exact algebraic identity. Several quantities required to complete the derivation, including the angle separating the muon and tau eigenvalues, the absolute normalization of the coupling constant, and the corresponding matrix for the quark sector, remain undetermined and are stated explicitly as open problems on which work continues.