THE COHERENCE EQUATION - A Candidate Macroscopic Law for Noise-Driven Decay of Structure in Complex Systems

This paper proposes a candidate macroscopic law governing how coherent structure in complex systems degrades under increasing noise. Starting from six minimal axioms—nullification, permutation invariance, monotonicity, independent composition, relational substrate, and falsifiability—we show that any admissible coherence measure with separable structural and noise dependence must have a structural numerator proportional (up to normalization) to a geometric product Φ = A·I·D. We then introduce the Relational Loss Hypothesis: coherence is stored primarily in relations (edges) between components, not in the components themselves, and noise destroys these relations at a rate proportional to the square of the current relational density. Under a mean-field approximation, this yields a microscopic relational decay law dC/dE = -k·C² with solution C(E) = Φ / (1 + k·Φ·E). Defining a stability floor ε = 1/(k·Φ) gives the final form of the Coherence Equation C(E) = Φ / (E + ε). We provide explicit, cross-domain operationalizations for Alignment (A), Integration (I), Directionality (D), and Noise/Entropy (E) in four classes of systems: EEG/neural systems, swarm robotics, social opinion dynamics, and coupled oscillator networks. In each case, all quantities are normalized to the unit interval, enabling dimensionless comparison and model fitting. From the Coherence Equation we derive a set of falsifiable, cross-domain predictions: (1) hyperbolic heavy-tail decay of coherence with noise, in contrast to exponential disappearance; (2) low-noise hypersensitivity, particularly in high-Φ systems; (3) a Φ-dependent half-noise scale E₁/₂; (4) algebraic residual coherence at high noise; and (5) reciprocal linearity, where plots of 1/C versus E yield straight lines with slope k and intercept 1/Φ. We situate the Coherence Equation within a model-selection framework that compares it against exponential, stretched-exponential, and general power-law alternatives using likelihood-based criteria (AIC/BIC) and Bayes factors. We define explicit falsification criteria: the hyperbolic model is rejected if competing models achieve, for example, ΔAIC > 10 or comparable Bayes factors across multiple independent datasets. This paper makes no empirical claims. The Coherence Equation is proposed as a testable candidate law. Its scientific status depends entirely on how well it survives empirical tests against plausible alternatives in real and simulated systems.