Primordial Gravitational Waves from Rotational‑Order Geometry: Tensor Perturbations and Spectral Index

We derive the primordial‑gravitational‑wave spectrum generated during inflation within the rotational‑order geometric unification framework. Drawing upon the 15‑dimensional Kaluza‑Klein reduction results from Papers 44 and 45, we analyse tensor perturbations of the four‑dimensional effective metric. The internal S³ orientation field modifies the propagation of tensor modes, shifting the effective sound speed and altering the tensor spectral index n_T. The standard inflationary consistency relation receives geometric corrections. We explore the parameter‑space regimes where this signal becomes detectable by next‑generation experiments including CMB‑S4 and the space‑borne interferometer LISA. For benchmark suppressed‑coupling parameters the correction is tiny; in the unsuppressed high‑energy regime the effect reaches observational sensitivity, furnishing a falsifiable prediction for the framework.