"The Geometric Higgs Mechanism from K₈ Compactification. " Paper VI in MEF Series.

We demonstrate that the complete electroweak symmetry breaking mechanism of the Standard Model — the Higgs vacuum expectation value (v), the Higgs boson mass (mH), and the Mexican hat potential V (H) = -²|H|² + |H|⁴ — can be derived from the topology and geometry of the S² fibre within the eight-dimensional internal manifold K₈ = (CP² S²) w (T²/Z₂) ^Spinᶜ. The derivation proceeds through four stages. First, we prove via the Gauss–Bonnet theorem that the integrated S² Ricci scalar is a strict topological invariant, rigorously forbidding the naïve Kaluza–Klein origin of the Higgs mass term. Second, we identify the true mechanism: the competition between bulk geometric tension (which resists S² deformation) and topologically quantised Spinᶜ monopole flux (whose energy density decreases with area expansion), which together generate the Mexican hat potential with explicit analytic expressions for ² and in terms of the internal geometry. Third, we derive the effective 4D potential V (Rc, Rw) by integrating the full 12D Master Equation over the internal manifold, examine the stability of the radion Hessian, and show that the aspect ratio ₂ = Rw/Rc 1. 861 is dynamically locked by a composite three-component derivation (Goldberger–Wise mechanism, constraint algebra, flux polynomial) anchored on Borel–Weil–Bott representation theory of the Spinᶜ structure. Fourth, we extract the absolute moduli radii via Kaluza–Klein gauge coupling reduction, yielding Rw 13. 26, ₏₋, and compute the physical Higgs parameters. The resulting predictions — v 246, GeV and mH 125, GeV — follow from the geometry with one empirical input (ₛ to set the absolute length scale) and the global warped hierarchy projection factor e^-12. The Higgs field is not a fundamental scalar; it is identically the volume-preserving ellipticity of the S² weak-sector manifold.