Geometric Theory of Global Helical Vector Fields on the Three-Sphere: Conformal Deformation Coupling & Spectral Matching

This paper develops a complete geometric theory of helical vector fields on the three-sphere S³, built rigorously upon the preceding S-1 foundations: global non-singular Riemannian geometry, SO (4) isometry decomposition, and hyperspherical harmonic eigenspectrum of the Laplace–Beltrami operator. The theory reuses selected tools from the T-series: the conformal scaling family gₐ=a²g₀, the spectral entropy functional Sₒ₏₄₂, the tensor perturbation amplitude, and the two-dimensional phase diagram. Three irreducible geometric facts are established. First, on S³ there exists at most one globally smooth helical vector field; its three-component orthogonal decomposition is unique, and no second independent global helical field can exist. Second, the flow generated by this helical field automatically induces the one-parameter conformal metric family gₐ=a²g₀, and the closed-form coupling between the helical strength parameter and the critical scale a₂ₑ₈ₓ (k) is derived. Third, the spectral entropy Sₒ₏₄₂ evolves monotonically in below a unique threshold ₂ₑ₈ₓ and decreases above it; this threshold coincides with the geometric bifurcation of the a correspondence. The original two-dimensional (a, ) phase space is extended to a full three-dimensional phase diagram (a, , ), with helical strength as an independent control parameter. The paper is predominantly pure differential geometry and spectral mathematics; only Chapter 7 provides a minimal (<5% of the main text) purely geometric analogy to spacetime structures, without constructing any dynamical field equations or particle models. The global helical field constructed herein provides the unique geometric carrier medium for the forthcoming S-3 domain nucleation dynamics.