Geometric Derivation of Lepton Generations and the Weak Mixing Angle via Topological Hopfion Resonances

This repository contains a theoretical manuscript and its accompanying numerical Python script, presenting a top-down geometric derivation of the Standard Model's lepton generations (Electron, Muon, Tau) and the weak mixing angle (Weinberg angle). By modeling the quantum vacuum as a continuous scalar field, fundamental particles are strictly defined as stable topological excitations (Hopfion knots). The generations are differentiated exclusively by their topological winding numbers ($W=2$, $W=7$, $W=29$) and geometric resonance radii ($R=16\xi_e$, $R=14\xi_e$) within the field geometry. This framework replaces arbitrary free parameters (such as Yukawa couplings) in the Standard Model with deterministic topological geometry, offering a formal mathematical justification for the existence of exactly three lepton generations based on the constraints of differential topology and exotic spheres.