Phase Coherence as an Order Parameter in Coupled Dissipative Systems

This revised theoretical preprint develops phase coherence as an effective order parameter in coupled dissipative systems. Building on the preceding formalization of resonance coherence Ω and its conditional projection beyond the gravitational domain, the paper restricts the analysis to systems in which phase-like variables, nonlinear coupling, dissipation or irreversible reduction, perturbation, scale separation, and coarse-graining conditions can be explicitly declared. The paper treats synchronization as a coherence-supported regime of effective dynamics rather than as perfect phase locking, centralized timing, global control, or externally imposed temporal order. Within this framework, Ω classifies the degree to which phase organization remains macroscopically available after microscopic phase trajectories, local fluctuations, and fine-grained dynamical details have been averaged, dissipated, or integrated out. The analysis distinguishes coherent, partially coherent, metastable, intermittent, fragmented, and incoherent regimes by the behavior of Ω relative to declared thresholds, fluctuations, recovery times, and correlation structure. Candidate observable signatures include fluctuation amplification, critical slowing down, hysteresis, path dependence, cluster fragmentation, recovery-time increase, and loss of long-range phase correlation. These signatures are treated as empirical constraint regimes, not as direct prediction or control mechanisms. The resulting framework supplies a bounded diagnostic grammar for retained phase organization in coupled dissipative dynamics. It prepares later analyses of coherence-conditioned information encoding and coherence-conditioned temporal structure while preserving the distinction between synchronization dynamics, information retention, temporal organization, external realization, prediction, and control.