The Koide formula (Q = 2/3) has resisted derivation from first principles since its discovery in1981. We show that within the Fibonacci-Tetrahedral Lattice (FTL) framework, the Koide constant2/3 is a geometric identity of the fundamental tetrahedral node angle: −2 cos(109.471◦) = 2/3.The visible lepton generations are identified as orientational modes of a tetrahedral topologicaldisclination, governed by a universal Fibonacci gate selection rule (m ∝ ϕn). We extend thisframework to the quark sector and the neutrino sector, deriving a zero-parameter master equation forthe entire fermion spectrum. The extreme smallness of neutrino masses is shown to be a consequenceof recursive lattice frustration, with the heaviest neutrino predicted at m3 ≈ 0.048 eV. Quarkconfinement emerges as a mechanical lattice frustration, and the observed mass hierarchy is shownto be a dynamic outcome of impact-driven phase transitions. These results suggest that the matterspectrum is not a collection of arbitrary parameters but a geometric projection of the vacuum latticetopology.