Topological Winding Modes and the Mass Hierarchy: Collapsing the Standard Model Free Parameters across 1/R ↔ R Boundaries

We extend the 10D T 5 × S 5 geometric framework to derive the charged lepton mass hierar-chy (e, µ, τ ) and fundamental gauge interactions from first-principles topological winding modes.In standard Quantum Field Theory, the Three Families are treated as distinct point-like elemen-tary species requiring arbitrary empirical Yukawa couplings. In this work, we demonstrate thatthe electron, muon, and tau represent discrete topological winding cycles (w = 1, 2, 3) of a singlecontinuous 1D string-like excitation wrapped around the compactified T 5 spatial torus. Evaluat-ing the elastic strain energy of higher-order winding loops at the self-dual turnaround boundary(R = 1.02143 lP ) yields exact rest-mass ratios without free parameters. Furthermore, we connectthis compact winding strain to O(5, 5) T -duality, demonstrating how localized 1/R micro-vorticesproject into macroscopic R-sphere geometries.(Dated: July 29, 2026)