Resonance Coherence in Gravitation: Effective Field-Theoretic Formalization of the Phase-Coherent Extension of General Relativity

This revised theoretical preprint formalizes resonance coherence, denoted Ω, as a retained macroscopic order parameter within a restricted gravitational effective field-theoretic framework. Building on the preceding resonance-coherence sequence, the paper clarifies the gravitational-domain status of Ω as a bounded diagnostic and structural variable associated with nonlinear phase-correlated systems under coarse-graining. The paper defines Ω as a macroscopic descriptor of retained phase organization, rather than as a new fundamental scalar field, microscopic interaction, hidden variable, materialization mechanism, or ultraviolet completion. It then embeds Ω within a minimal covariant scalar–tensor effective description, where the coupling function f(Ω), gradient contribution, and effective potential V(Ω) provide a controlled representation of coherence-sensitive gravitational response. The framework is explicitly constrained by general covariance, smoothness, slow variation, positivity of the effective coupling, stability against runaway behavior, recovery of general relativity in the appropriate limit, and compatibility with observational precision tests. Its contribution is formal and classificatory: it specifies the conditions under which resonance coherence may enter gravitational comparison as an effective order parameter, while separating this role from claims of empirical validation, completed modified gravity, prediction, operational control, or external realization. The paper also prepares, without completing, the transition toward a companion analysis of resonance-coherence order parameters in dissipative and informational systems. Broader applicability is treated as a separate domain-specific question requiring its own variables, observables, thresholds, failure modes, and admissibility conditions.