Paper XIII: Geometric Representation of the Higgs Vacuum Manifold from Quaternionic S³ Wavefront Geometry

We formulate a geometric representation of the Higgs vacuum manifold within a quaternionic S³ wavefront model. The input data are the unit-quaternion hypersurface S³ ⊂ ℍ, the complex-vector-space identification ℍ ≅ ℂ², and the Hopf fibration. After separating the spatial radius R_U(t) from the Higgs field-space radius v/√2, the normalized spatial wavefront and the normalized Higgs vacuum manifold are identified as the same three-sphere. The explicit map H(q) = (v/√2)(q⁰ + i q¹, q² + i q³)^T is norm-preserving and SU(2)_L-equivariant. In this representation, the three Maurer–Cartan tangent directions on S³ ≅ SU(2) correspond to the electroweak Goldstone directions, while the radial direction is represented by the physical Higgs mode. The Higgs hypercharge Y_H = 1/2 is obtained from the Hopf U(1)_Y action and is checked against Yukawa gauge invariance. The electroweak scale used in the paper is treated as a conditional topological input from the instanton sector; determinant-level prefactors, Yukawa matrices, and the cutoff scale remain open calculations.