The Parabolic Law for Fermionic Masses: Geometric Regularity in the Standard Model

Abstract We propose and validate a geometric law governing the masses of the charged fermions in the Standard Model. This law posits that each generation’s mass spectrum follows a parabolic function, with its shape uniquely determined by the generation index and a vertex bifurcation mechanism. The resulting formulation provides exact empirical fits to all known charged fermion masses, using only generation-based parameters. It further predicts the masses of hypothetical fourth and fifth generation fermions, and implies a structural asymmetry that distinguishes stable matter (first generation) from higher, unstable generations. The law is empirically robust, falsifiable, and geometrically minimal.