🤖 AI Summary
Emergent phenomena in complex systems lack predictable, universal regularities.
Method: We propose and rigorously validate the “Mesoscale Peak Principle of Causal Efficacy”: causal efficacy—quantified via effective information (EI)—peaks not at microscopic or macroscopic scales, but at a characteristic mesoscale, revealing an intrinsic scale-selection mechanism by which local interactions generate global behavior. Our framework integrates maximum-entropy interventions with mutual information to construct a multiscale causal measure, and combines statistical model selection to identify causal structure.
Results: Robust, non-monotonic peaks in causal efficacy are consistently observed across both the Ising model and multi-agent collective-behavior models, confirming the principle’s universality. This work establishes a foundational, computationally tractable basis for modeling emergence, constructing effective theories, and performing cross-scale causal inference.
📝 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.