Relative Gravitational Matter–Energy Confinement (RGMEC): Mathematical Structure

Relative Gravitational Matter–Energy Confinement (RGMEC) provides a complementary structural interpretation of gravitational time dilation that remains mathematically identical to the Schwarzschild limit of General Relativity. Compactness C = GM/(R c²) is identified as the primary descriptor of gravitational confinement, and the relative temporal evolution factor Θ = √(1 − 2C) follows directly from the Schwarzschild geometry. RGMEC introduces no new physical field, force, free parameter, correction factor, or modification of Einstein’s equations. The monograph organizes the physical variables into a constrained configuration space and develops the associated fiber-bundle geometry, structural transport and curvature, sheaf-theoretic global organization, spectral geometry, persistent topology, variational principles, and observational correspondence. The framework carefully distinguishes established relativistic quantities from derived confinement quantities and exploratory structural descriptors, including the structural correlation length ξ, Structural Coherence Index SCI = ξ/R, and Structural Connectivity Index SCN = N_eff/N_total. Appendix A examines dynamical confinement and the trapped-regime boundary. Appendix B develops and stress-tests the Alhawarat–Tripathy candidate SCI closure through independent benchmarks, null tests, and sensitivity analyses. Appendix C introduces Operational Causal Accessibility in RGMEC and the Causal Accessibility Factor (CAF), providing an operational organization of established null-geodesic reachability structure without modifying General Relativity or introducing additional physical parameters. The organizing principle of the work is: Physical meaning precedes mathematical representation.