The Standard Model accommodates fermion masses and mixing through nineteen free Yukawa andmixing parameters, offering no structural explanation for their values. Here we derive the complete flavor structure from two principles: the quadratic (Born-rule) form of the mass Hamiltonian and a topological constraint from the Clifford torus in S3 that fixes the interference amplitude to A = √2. We prove that the number of generations N = 3 is the unique solution to the identity Q × N = 2, where Q = 2/3 is the Koide ratio and 2 = χ(S2) is the Euler characteristic of the Bloch sphere.A single geometric phase δ = 2/9 rad determines the entire lepton sector: the charged-lepton mass spectrum (predicting mτ = 1776.97 MeV versus the experimental 1776.86 ± 0.12 MeV), all three PMNS mixing angles via the exact relations θ23 = π/4, θ13 = 2δ/3, θ12 = π/4 − δ (derived from Z3 representation theory), and the Wolfenstein parameter λ = sin δ. The neutrino sector follows from a complementary phase shift Δδ = 3π/4, predicting absolute masses withP mν = 0.059 eV. The universal mass curve evaluated at θ = π yields a dark-sector state at 53.85 MeV whose π-harmonic at 17.14 MeV matches the Atomki X17 anomaly to 0.8% with zero free parameters. The Clifford torus geometry further predicts maximal CP violation (δCP = −π/2) and exact μ—τ reflection symmetry. In total, the model yields twenty experimentally testable predictions from two input masses and onefundamental constant.