Toroidal Geometry of Fermion Generations: First-Principles Derivation of Lepton Mass Parameters

We present a fundamental derivation of lepton mass parameters from the toroidal geometry offermion generations. Building upon the topological distinction where fermions reside on tori (T 2,genus 1) while bosons on spheres (S2, genus 0), we show that the three lepton generations correspondto three fundamental cycles on a fermionic torus forming a closed triangle. The parameters κlep =0.10 and β(lep)X = 24qX /g2X (MP ) emerge from string quantization and modular geometry, yieldingβ(lep)I = 80.0, β(lep)J = 30.0 with 3.3% and 2.4% accuracy compared to fitted values 82.7 and 29.3.Remarkably, Koide’s relation RK = (me + mμ + mτ )/(√me + √mμ + √mτ )2 ≈ 2/3 emerges as adirect consequence of triangular closure on the torus. The number 24 originates from heterotic stringwinding quantization, while κ = 1/(2πτ2) reflects the torus aspect ratio. This provides a completefirst-principles geometric foundation for lepton masses and generations.