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

This paper develops a geometric and numerical framework for describing single-qubit (spin-1/2) kinematics using a real-valued overlap function on the three-sphere (SU(2)). The overlap kernel reproduces standard spinor landmarks: extinction at antipodal separation, a sign flip after 2π rotation, and full closure at 4π. These properties arise purely from geometry and do not require complex amplitudes as primitives. On the measurement sphere (S²), we construct a normalized quadratic baseline density using two complementary selectors. The first is a static symmetry-based construction restricted to the lowest non-constant angular sector, where weak anisotropy selects a unique axis. The second is a dynamic corridor selector defined by a killed Dirichlet Fokker–Planck evolution on smooth two-cap domains. The long-time object of interest is the mass-normalized principal Dirichlet mode (a quasi-stationary shape). Its proximity to the quadratic comparator profile is quantified in L² norm and exhibits empirically certified scaling proportional to corridor size squared plus drift strength under mesh refinement. A projection analysis clarifies that the Hopf fiber average of a single-channel squared overlap is affine in the axial coordinate and therefore distinct from the quadratic baseline density. The quadratic profile must arise from a selector acting on S² rather than direct fiber averaging, resolving potential circularity. Numerically, a multi-resolution spectral sweep confirms stabilization of the continuum-scaled Dirichlet gap and documents multi-decade exponential convergence toward the quasi-stationary selector shape. The observed relaxation rate is proportional to the first Dirichlet eigenvalue, and traces collapse when plotted in scaled time. Comparator template traces plateau as expected, serving as a negative control. All results are produced by a fully specified numerical protocol that records parameters, seeds, hashes, and output paths in a machine-readable manifest, enabling independent reruns. The paper therefore combines geometric structure, perturbative spectral control, and reproducible numerical verification in a unified framework.