Physical Time Unified Physics: Global Regularity and Field Continuity in the ρₜ Medium

Companion paper to The ρₜ–k Physical Time Framework (DOI 10.5281/zenodo.17479344). This manuscript formalizes the mathematical and physical foundations of the unified ρₜ medium through a global-regularity proof of the coupled Navier–Stokes–Brinkman equations. It demonstrates that the ρₜ field obeys well-posed, divergence-free continuity across all density domains, linking quantum-scale coherence to cosmic expansion. The analysis confirms that irreversible Time consumption (k = −1) yields stable equilibrium under compressive and diffusive regimes, completing the theoretical structure of the Physical Time Framework.