Standard Model Parameters from a Twelve-Dimensional $\Spinc$ Compactification

A specific twelve-dimensional Kaluza–Klein compactification, $\mathscr{M}_{12} = (S^1 \times S^3) \times K_8$ with internal manifold $K_8 = [(\mathbb{CP}^2 \times S^2) \times_w (T^2 / \mathbb{Z}_2)]^{\text{Spin}^c}$, predicts four Standard Model parameters from a single topological invariant $\chi(K_8) = 12$ and zero continuous free parameters. The Weinberg angle follows from $\text{Spin}^c$ holonomy on the pillowcase orbifold and equals $1 / (1 + \tau_2^2) \approx 0.224$ at the compactification scale, where $\tau_2$ is the pillowcase aspect ratio fixed by the gauge coupling ratio $\alpha_w / \alpha_s$. Three fermion generations follow from the Dolbeault cohomology dimension $h^{0,2}(\mathbb{CP}^2, \mathcal{O}(-4)) = 3$. The Planck–weak hierarchy follows from the warp exponent $\zeta = 12\pi$ fixed by tadpole cancellation. The Cabibbo parameter $|V_{us}| \approx 0.225$ follows from D4 path holonomy on the same pillowcase, yielding the structural identity $|V_{us}| = \sin^2 \theta_W$ which is parameter-free and falsifiable. A discovered fourth fermion generation, or a measured $\sin^2 \theta_W$ deviating from $|V_{us}|$ at the compactification scale by more than RG running can account for, would falsify the framework. Full derivations are given in the comprehensive treatment, deposited separately.