Topological Quantization of Fermion Masses in a Degenerate Double-Helical Vacuum Manifold

We formulate a noncommutative geometric framework in which Standard Model fermion mass ratios, flavor mixing parameters, and gauge couplings emerge as spectral invariants of a compact Riemannian 3-manifold with degenerate double-helical topology. The vacuum geometry is constructed as a vertically oriented torus with catenoidal minimal surfaces connecting asymptotically flat sheets; its pitch and transverse scale are rigidly fixed by Pogorelov-type embedding constraints derived from the Euclidean metric signature {√1, √2, √3}.

Starting from the real spectral triple, we compute the leading eigenvalues of the helical Dirac operator in the adiabatic approximation. The continuous geometric prediction for the muon-to-electron mass ratio matches experiment at the 1.31% level without free parameters. We demonstrate that this residual deviation arises from topological frustration: the incompatibility between the continuous geometric invariant and the requirement of integer winding numbers for physical fermionic eigenmodes. Applying the Călugăreanu-White theorem, we show that the system resolves this tension by deforming the helical axis, absorbing the deficit into geometric writhe, which acts as an effective gauge potential shifting Dirac eigenvalues. Matching to experiment fixes the physical twist number $Tw = 103$, yielding a mass ratio in agreement with data at the $2 \times 10^{-6}$ level.

Furthermore, we derive the inverse fine-structure constant as approximately 137.0370 from discrete angular holonomy, accurate to 8 ppm. The Cabibbo angle emerges as approximately 0.2246, matching experiment at 0.17%. The framework also generates sub-eV neutrino masses via a geometric seesaw mechanism and predicts a falsifiable Planck-scale Lorentz invariance violation testable in polarized beam experiments. All relations follow from rigidity constraints, topological quantization, and group-theoretic projections without phenomenological input.