Topological Origin of Lepton Mass Structure: The Koide Relation and Berry Phase on the Clifford Torus

Description: We present a geometric derivation of the charged lepton mass relations based on the topology of the 3-sphere (S³). We identify the empirical Koide formula (Q=2/3) as a geometric identity describing the projection of the mass vector onto the Clifford Torus, a minimal surface of maximal symmetry within S³. Furthermore, we demonstrate that the long-unexplained phase angle θ ≈ 2/9 corresponds to a quantized Berry Phase arising from the Z₃ topology of the generation fiber. This framework suggests that the lepton mass hierarchy is determined by the global topology of the vacuum manifold and is protected by topological invariants. Based on this strict topology, we predict the tau lepton mass to be 1776.97 MeV.