A Law of Emergence: Maximum Causal Power at the Mesoscale

TL;DR

This paper establishes a universal law of emergence showing that a system's causal power peaks at a specific mesoscale, supported by evidence from physical and agent-based models, advancing understanding of complex systems.

Contribution

The paper introduces and proves a law indicating that causal power at a certain scale peaks at a mesoscopic level, supported by empirical evidence from diverse systems.

Findings

Causal power (EI) exhibits a maximum at a mesoscopic scale.

The peak is confirmed in 2D Ising model near criticality.

Agent-based models also show a clear peak 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.