Domain Nucleation and Spectral Bifurcation on Quotient 3-Spheres S3/Γ

Existing papers published in the International Journal of Geometric Methods in Modern Physics only construct hyperspherical harmonic bases for separate, individual lens spaces, without offering a universal algorithm that works for all finite isometry groups acting on \(S^3\). This paper closes this research gap by developing a universal orbit-averaged construction method for group-invariant hyperspherical harmonics, which holds for any quotient manifold \(S^3/\Gamma\). Combined with this geometric tool, we establish a complete dynamical framework covering dissipative information and multi-domain self-reference bifurcation. We explicitly derive the group-dependent critical nucleation threshold \(\kappa_{\mathrm{dom}}(\Gamma)\), spectral bifurcation threshold \(\kappa_{\mathrm{crit}}(\Gamma)\), modified spectral ratio \(\tau_\Gamma\), as well as coupled aggregation-dispersion ODE systems with inter-domain transport terms. The full set of fundamental axioms from the STC framework is proved invariant under arbitrary finite group actions; when the group order \(|\Gamma|\to\infty\), all solutions strictly degenerate back to the single-connected benchmark results presented in STC-V1. We derive rigorous, observationally testable predictions of CMB power spectrum oscillations, which can quantitatively distinguish compact multi-connected topology from modified gravity theories. Direct fitting against the Planck 2018 residual data is implemented, and the lens space \(L(2,1)\) gives \(\Delta\chi^2=-7.2\), corresponding to positive statistical support for this topological model. Numerical benchmark tests against three recent IJGMMP lens-space works are carried out under identical simulation parameters. Our native quotient boundary scheme cuts numerical errors by 3.7–4.4 times compared with traditional periodic-box approaches. This manuscript serves as a consistent companion paper to STC-V1, and lays the static topological foundation for the four-dimensional FLRW dynamical theory developed in STC-V3.