Topological Derivation of the CKM Matrix from Spinᶜ Holonomy on T²/Z₂ : Paper II in MEF Series.

We demonstrate that the complete Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix, including all four independent physical parameters (three mixing angles and one CP-violating phase), can be derived from the topology of a single compact eight-dimensional manifold K₈ = [(CP² × S²) ×w (T²/Z₂)]Spinᶜ, which forms the internal space of the twelve-dimensional Master Equation framework M₁₂ = (S¹ × S³) × K₈. The derivation proceeds with zero continuous free parameters. Every input is either a topological invariant (χ(K₈) = 12, χ(S²) = 2), a discrete quantum number (Spinᶜ charges \-1, -1, +2\), or a ratio fixed by the manifold geometry (aspect ratio τ₂ = L_ω/L_ψ). The CKM matrix elements emerge from wavefunction overlap integrals on the T²/Z₂ orbifold (the "pillowcase"), with three distinct geometric mechanisms at work: (i) the rectangular aspect ratio of the pillowcase encodes the Cabibbo angle via λC = sin²θW = 1/(1 + τ₂²); (ii) the Spinᶜ holonomy eiπ/2 generates the CP-violating phase; and (iii) a newly established geodesic path interference mechanism—whereby the Z₂ orbifold involution on the Spinᶜ bundle introduces destructive interference between the two distinct geodesics connecting diagonal fixed points—resolves the previously outstanding |Vub| and |Vtd| predictions. The resulting ten zero-parameter predictions, eight of which agree with Particle Data Group measurements to better than 6%, constitute the most overconstrained derivation of CKM observables from geometry in the literature.