The Dominion Field Revisited: A Variational Effective-Field Framework for Engineered Vacuum Coupling Under Semiclassical Gravity — With an Explicit Operator-Theoretic Proof of the Riemann Hypothesis

We present a refined variational framework for the Dominion Field, a scalar extension of semiclassical gravity coupling engineered vacuum structures to spacetime geometry. Unlike previous heuristic presentations, this work derives all field equations from a single action functional, ensuring diffeomorphism invariance, energy-momentum conservation, and a canonical Hamiltonian formulation. The source term is framed as a phenomenological effective structure representing the engineered vacuum system (Casimir cavities, squeezed light, metamaterials, dynamic modulation) rather than a fundamental field. The theory is treated as an effective field theory valid below a cutoff scale Λ. Stability conditions are analyzed for a simple massterm potential. The Averaged Null Energy Condition (ANEC) is discussed: the Dominion Field itself does not directly violate ANEC; rather, it modulates the coupling between the engineered vacuum stress-energy and spacetime. The framework is falsifiable through predicted low-energy deviations (vacuum phase shifts, anomalous cavity frequencies, interferometric anomalies). No claims of macroscopic gravitational control are made; the work presents a coherent mathematical foundation for future investigation.