Angular Momentum Transpace in Rotating First-Space Halos: Topological Quantization of Black-Hole Spin and Coupled Satellite Orbital Dynamics in the CSF-CMT Framework

Within the Coarse-Graining Spacetime-Field and Consciousness-Mind-Theory (CSF-CMT) framework, we derive the explicit mechanism of transpace angular momentum transport between the First Space (string-empowered vacuum) and the Third Space (emergent matter geometry). We establish the layered angular momentum structure of the First-Space Halo (FSH) surrounding galactic-center black holes, driven by the differential coarse-graining timescales of the six string-empowered states: gravitational (G), electromagnetic (EM), magnetic (M), weak (W), strong (S), and repulsive (R). A central result is the topological quantization of black-hole spin: the Kerr parameter a = Jc/GM^2 maps to an integer topological winding number n_{\rm EO} via a \approx n_{\rm EO}/2 in the macroscopic limit, with repulsion-state corrections regularizing the horizon. We derive the explicit angular-momentum pumping rate \Gamma_{\rm pump} from the cross product of gravitational and magnetic coarse-graining gradients, and construct the general solution for satellite orbital dynamics in a rotating FSH, yielding coupled equations for precession, nutation, and radial oscillation under CSF-CMT corrections. The expansion dynamics of Sgr A* are incorporated as a time-dependent boundary condition that slowly pushes the diffuse FSH layer outward. A cross-space, coarse-grained conservation law for angular momentum is formulated, replacing the standard \nabla_\mu T^{\mu\nu}=0 with an emergent source equation that accounts for topological pumping, gravitational-wave loss, and accretion. Cosmological consequences and observational predictions for the Event Horizon Telescope, X-ray spectroscopy, and gravitational-wave detectors are discussed.