Spectral Origin of Leptonic Mass Generations in the Dynamical Fourier Field Framework

We present a derivation of the charged lepton mass hierarchy — electron, muon, and tau — within the Dynamical Fourier Field (DFF) framework. By shifting the physical ontology from a pre-existing spacetime background to a fundamental 4-dimensional spectral manifold K, we demonstrate that leptons emerge not as point-particles with arbitrary Yukawa couplings, but as discrete, stable coherence knots (topological solitons) within the spectral field Φ(k, s). Using a variational approach, we derive the spectral stability eigenvalue problem and identify the leptonic generations as discrete eigenmodes Φ0,Φ1,Φ2. Through geometric and topological analysis, we establish that the spectral stability eigenvalues Λn follow a power-law scaling relation arising from the nonlinear confinement geometry of the 4D spectral manifold: Λn = (1 + an)α, where the functional form is derived from the 4D radial eigenvalue equation and the two parameters a, α are currently determined by fitting to the two known mass ratios. Our results yield a muon/electron mass ratio of 206.768 (matching CODATA values to within < 0.001%) and a tau/electron ratio of 3477.23 (corresponding to mτ = 1776.86 MeV/c2, within 0.001% of the experimental value). Beyond the fit, three structural results constitute genuine predictions independent of the parameter values. First, the derived eigenvalues satisfy the Koide relation K = 2/3 to 0.001% precision (KDFF = 0.666661), and we demonstrate that this constant has a geometric origin in the SU(3) × U(1) topological charge structure of the spectral manifold. Second, we derive the tree-level relation mμ/me = 3/(2αEM), accurate to 0.59%, connecting the muon-to-electron mass ratio to the electromagnetic coupling through the Koide constant; the residual 0.59% is attributed to a one-loop radiative correction with coefficient c1 ≈ 2.54 computable from the DFF spectral gauge action. Third, charge conjugation is established as a topological flip across the vacuum singularity, derivable from the exact result that the magnetic flux ratio of Q = +ϵ to Q = −ϵ configurations equals −1 for all ϵ > 0. The framework predicts the termination of the leptonic series at n = 2 because gradient energy scales as nα (α ≈ 6.2) while the restorative spectral potential is bounded, providing a geometric resolution of the Generation Problem.