Paper 32 (Math): Real-Valued S3 Overlap Mathematical Structure and Spectral Consistency Framework

This paper develops a real-valued geometric framework for single-qubit (spin-1/2) kinematics based on the compact manifold S3≅SU(2)S^3 \cong SU(2)S3≅SU(2). A cosine overlap kernel cos⁡(Δ/2)\cos(\Delta/2)cos(Δ/2) is used to encode lifted spinor separation, reproducing spinor holonomy (sign reversal at 2π2\pi2π), overlap-extinction nodes at π\piπ and 3π3\pi3π, and 4π4\pi4π closure. The projection of squared overlap through the Hopf fibration is shown to produce an affine angular profile on S2S^2S2, clarifying that quadratic baseline densities on the measurement sphere do not arise directly from fiber averaging.

A baseline angular density is instead obtained by restricting to the lowest non-constant spherical harmonic sector and introducing weak anisotropy, which selects a unique axis through a quadratic form on the ℓ=1\ell = 1ℓ=1 space. Dynamic consistency is then examined using a killed Dirichlet Fokker–Planck operator on two-cap corridor domains. A perturbative spectral-gap bound establishes exponential convergence of renormalized solutions to the principal Dirichlet mode under weak drift. Mesh-refinement studies confirm stabilization of the Dirichlet gap and document multi-decade exponential convergence in representative runs.

The result is a self-contained geometric and analytical scaffold that separates kinematic overlap structure, projection behavior, symmetry-based mode selection, and diffusion-driven selector dynamics, with numerical verification under declared discretization conditions.