On the Flavor Yukawa coupling for the Lepton Mass Hierarchy and the Koide Relation

Abstract

This article proposes a geometric flavor mechanism to address the charged lepton mass hierarchy and formally derive the empirical Koide relation. Governed by a topological condensation $U(3)\to U(1)\times {\mathbb{Z}}_3$, the flavor vacuum couples a continuous isotropic background (the $U(1)$ singlet) to a discrete generation-splitting network (the ${\mathbb{Z}}_3$ octet). Mechanical stability imposes a strict stress-free condition on this vacuum, requiring the expansive energy of the continuous sector to perfectly balance the cohesive topological tension of the discrete lattice. Furthermore, the geometric projection of the continuous internal space onto the discrete nodes generates a topological action phase that rigorously locks the internal flavor angle at ${\theta }_F=-2/9$. By integrating these geometric constraints into the Yukawa Lagrangian, this framework perfectly recovers the Koide mass spectrum ($Q=2/3$) of the charged leptons (electron, muon, tau).