Gauge Theory on the Metric Bundle of a Spin 4-Manifold, V: The Generation Index, Fiber Regularization, Kaluza–Klein Spectrum, and Cosmological Perturbation Theory

We resolve the six open mathematical problems identified in Part IV of this series. (1) We prove that the Shiab-Yukawa coupling εˆΦ satisfies the Callias growth condition on the non-compact fiber Sym2+(R4) ∼= GL(4)/O(4), establishing indL2 (D/Y + εˆΦ) = 2 on the observerse Y 14 and rigorously fixing the fermion generation count. (2) We compute the regularized fiber volume using the Harish-Chandra c-function for GL(4)/O(4) with eigenvalue cutoffs, finding V reg ∼ e6Λ where Λ = ln(λmax/λmin); fiber this exponential suppression provides a natural geometric origin for a small cos- mological constant, with the observed value Λeff ≈ 10−122 M4 corresponding to Λ ≈ 47. (3) We solve the spectrum of the fiber Laplacian ∆V on the truncated fiber and show that the naive Kaluza–Klein scale lies far above the electroweak scale; the third fermion generation is an imposter: it carries the same gauge quantum numbers from the Cl(14) decomposition but acquires its mass through resonant mixing with the KK sector, while the two fundamental families receive only small seesaw corrections of order v2 |ε|2/MKK. (4) We develop the cosmological per- EW turbation theory for the torsion-modified Friedmann equations, obtaining scaling estimates for the primordial power spectra: nT > 0 (blue tilt), r ∼ 0.003, and flocal ≈ 2.1—all within current observational bounds and testable by CMB-S4 and NL LiteBIRD. (5) We extend the torsion model to galaxy clusters, finding that thermal suppression of the coherent spin density renders the torsion contribution negligi- ble (Mtorsion /Mtotal ∼ 10−22 ), consistent with the standard NFW description of cluster dynamics. Together with the MCMC rotation curve analysis released as a companion code package, these results close the principal gaps in the theoretical framework.