Complete Detectable Spacetime Geometry: Closure Sector Origin of the Galaxy Dynamics and the Hubble Tension

A unified description of gravitational phenomena across galactic and cosmological scales remains incomplete within the standard framework of General Relativity supplemented by dark matter and dark energy. In particular, the empirical regularity of galaxy rotation curves and the persistent, structured discrepancy between early-and late-universe determinations of the Hubble constant indicate the presence of underlying geometric constraints not captured by existing formulations. This work develops the framework of Complete Detectable Spacetime Geometry (CDSG), in which physically admissible configurations are restricted by the requirement of closure-complete spacetime action. The resulting formulation extends the variational principle through a Unified Variational Closure Principle (UVCP), introducing a closure field Kµ that encodes spacetime action flow and generates an additional, geometrically induced stress-energy contribution. The framework preserves General Relativity in the high-acceleration limit while producing controlled modifications in the low-acceleration and multi-source regimes. At galactic scales, the local closure sector yields a re-circulation law. This relation is derived as the unique leading-order asymptotic solution of the nonlinear closure field equation under closure-recursion and no-hidden-defect conditions, rather than introduced phenomenologically. It reproduces the observed radial acceleration relation without invoking dark matter halos and provides competitive fits to the SPARC dataset. At cluster scales, non-local cross-sector interactions generate environment-dependent enhancements of the gravitational response, addressing the known limitations of purely local modifications. At cosmological scales, the sector decomposition of the closure field implies that inferred observables depend on the closure sector probed by the measurement process. This leads to a quantitative relation for the inferred Hubble parameter which produces a systematic uplift of late-time measurements relative to the global value. Using independently constrained closure-sector parameters, the resulting prediction aligns with Cepheid-calibrated determinations while preserving the lower expansion rate inferred from early-universe probes, thereby providing a leading-order closure-sector mechanism for resolving the Hubble tension without modifying early-time cosmology. The CDSG framework yields a unified structural interpretation in which galaxy dynamics, environmental effects, and cosmological discrepancies arise from the same closure-complete organization of spacetime action. It further generates falsifiable predictions, including environmental dependence of inferred expansion rates and controlled deviations from universality in the radial acceleration relation. These results suggest that gravitational phenomena across scales may be understood as manifestations of a single geometric constraint, rather than requiring separate dynamical components. The framework therefore provides a unified geometric mechanism linking local gravitational phenomena and cosmological inference without introducing additional matter components or modifying early-universe physics.