A Law of Emergence: Maximum Causal Power at the Mesoscale

Abstract: Complex systems universally exhibit emergence, where macroscopic dynamics arise from local interactions, but a predictive law governing this process has been absent. We establish and verify such a law. We define a system's causal power at a spatial scale, $\ell$, as its Effective Information (EI$\ell$), measured by the mutual information between a targeted, maximum-entropy intervention and its outcome. From this, we derive and prove a Middle-Scale Peak Theorem: for a broad class of systems with local interactions, EI$\ell$ is not monotonic but exhibits a strict maximum at a mesoscopic scale $\ell^*$. This peak is a necessary consequence of a fundamental trade-off between noise-averaging at small scales and locality-limited response at large scales. We provide quantitative, reproducible evidence for this law in two distinct domains: a 2D Ising model near criticality and a model of agent-based collective behavior. In both systems, the predicted unimodal peak is decisively confirmed by statistical model selection. Our work establishes a falsifiable, first-principles law that identifies the natural scale of emergence, providing a quantitative foundation for the discovery of effective theories.