The Hydrosophic Master Equation (HME): A Domain-Agnostic Dynamical Core for Coherence, Load, and Regulation

The Hydrosophic Master Equation (HME) is a low-dimensional, continuous-time dynamical framework for modeling coherence dynamics in complex adaptive systems. The HME formalizes coherence as a bounded order parameter whose evolution is governed by structured interactions among dephasing, resonant load, energetic drag, phase alignment, and regulatory capacity. The formulation is explicitly domain-agnostic: it specifies latent state dynamics independent of substrate, scale, or measurement modality, allowing physiological, cognitive, relational, or environmental systems to be treated as instantiations of a shared state-space structure via domain-specific observation mappings. This working paper presents the core state variables, admissible coupling structure, boundedness and stability constraints, and an illustrative physiological instantiation demonstrating empirical plausibility without domain-specific optimization. The framework emphasizes trajectory-based behavior, asymmetric collapse and recovery dynamics, and early-warning signals preceding loss of coherence. The HME is not proposed as a universal physical law, but as a reusable dynamical scaffold intended to support falsification, cross-domain comparison, and extension through future instantiations and control-oriented analyses.