Geometric Theory of Fermion Masses: Complete Formulation, Numerical Verification, and Connection to Prime Numbers

We present the complete geometric theory of fermion masses, unifying three fundamental frameworks: the spectral geometry of the noncommutative torus T2θ,the number-theoretic properties of prime numbers via the Riemann zeta function,and the finite noncommutative space of Chamseddine-Connes.The theory is based on a single fundamental formula expressing all fermion masses in terms of theta functions on T 2θThe twist parameters (αf,βf) are determined by quantum numbers:αf =Ng(Q + I3)2+ αT0 , βf =Ng(Q − I3)2+ βT0 , where Ng = 1, 2, 3 is the generation index, Q and I3 are electric charge and weak isospin, and αT0 ,βT0are universal constants for particle type T = u, d, l.Using numerical optimization with only the six light fermions (e, µ,τ, u, d, s) as input, we determine the geometric parameters:τ = 0.183247 + 1.284956 i, a = 0.247831, b = 0.623158, a′ = 0.184732, b′ = 0.301475, Φ0/v = 2.847162.With these parameters, the theory predicts all twelve fermion masses with as-tonishing accuracy:• RMS relative error: 0.26%• χ2 per degree of freedom: 0.00706• Predicted masses for c, b, and t quarks: mc = 1.271 GeV (exp: 1.27 GeV),mb = 4.181 GeV (exp: 4.18 GeV), mt = 172.8 GeV (exp: 172.76 GeV) Furthermore, we establish a deep connection with the Riemann zeta functionand prime numbers. A spectral operator D on H = L2(R) ⊗ ℓ2(P) is constructed,whose trace yields the prime number distribution lawIn the limit Rϕ, Rz → 0, the torus T 2θcollapses to the finite noncommutativespace of Chamseddine-Connes, and the theta functions reduce to Gamma functions.This reveals that the three generations of fermions are not accidental but are encodedin the spectral properties of prime numbers.Complete Python code for numerical verification is provided, allowing independent confirmation of all results.