Continuity and Conservation Laws Induced by the Coherence–Action Principle

We derive the continuity and conservation laws implied by the Coherence–Action Principle (CAP), the variational foundation of the Phase–Coupled Field (PCF) framework. Rather than postulating conservation a priori, the analysis shows how admissible histories constrained by CAP necessarily generate conserved currents and continuity relations. The resulting structure clarifies how coherence persistence and directed loss encode dynamical consistency across scales, and it provides the formal conditions required for coupling PCF to relativistic field theories. This work completes the structural scaffold of PCF by making explicit the conservation content implicit in its action-based formulation.