Self-Parallel Complete Dynamics: Three-Generation Mirror Geometry Confronts Gravitational Sirens, Migdal Effect, and the H0 Tension

The Self-Parallel Complete Dynamics (S-PCD) theory posits that all fundamental physics emerges from the self-parallel evolution of a three-generation mirror spacetime manifold $\M = (\mathbb{R}^4 \bowtie_{\mathbb{Z}_2} \mathbb{R}^4)^{\oplus 3}/\mathbb{Z}_3$. We demonstrate that this geometric framework provides a unified resolution to three pressing observational challenges: (i) the gravitational sirens measurement of the Hubble constant, where the mirror sector induces a redshift-dependent effective gravitational constant; (ii) the Migdal effect in direct detection experiments, naturally explained by dark photon exchange from the $\mathbb{Z}_2$-partner sector; and (iii) the $H_0$ tension, resolved through generation-dependent cosmic evolution histories encoded in the $\mathbb{Z}_3$ cyclic structure. The master equation $\nabla_{\dot{\mathcal{M}}}\dot{\mathcal{M}} = 0$ yields falsifiable predictions: $\Delta N_{\text{eff}} = 0.28 \pm 0.05$ at recombination, a Migdal cross-section $\sigma_n \sim 10^{-30}\,\text{cm}^2$ for $m_{\tilde{\gamma}} \sim 10\,\text{MeV}$, and a late-time asymptotic value $H_0 = 71.2 \pm 1.3\,\text{km/s/Mpc}$ that reconciles Planck CMB and SH0ES distance ladder measurements.