LFIS-31 applies the Light Frame Cadence Theory (LFCT) closure architecture to the nuclear, compact-object, and gravitational-coupling domains. Building on the framework infrastructure established in LFIS-24 and the Core trilogy, the volume develops nuclear binding structure from cadence closure, shell-topology realizability, closure-rank geometry, heavy-nucleus architecture, compact-object transport, gravitational-wave prefactor structure, Hawking-temperature carrier families, gravitational coupling, and neutron-star radius scaling. The central organizing principle is the closure-rank scaffold (D(m)=2m+3), whose realizations appear across multiple physical domains. Nuclear stability, shell architecture, compact-object behavior, gravitational-wave prefactors, Hawking-temperature carriers, the gravitational coupling (\alpha_G), and neutron-star radius structure are presented as different observable faces of one closure-based representational architecture. The volume introduces no new axioms; all results are derived from or articulated through the cadence framework established in the Foundation Axioms, LFIS-24, and the Core trilogy. LFIS-31 serves as the primary application volume for closure-rank physics within the LFIS series. It collects the framework's nuclear-binding program, compact-object transport architecture, gravitational-coupling identifications, and associated cross-domain structural correspondences while maintaining explicit tier discipline between theorem-tier results, structural identifications, and candidate derivations.