A Generalized Gompertz Law for the Rise and Fall of Empires

Human societies tend to evolve toward increasing scale and complexity, often followed by stagnation and decline. Historical empires provide a prominent example of this trajectory; however, existing quantitative models capture expansion and rarely account for decline within a unified framework. Here, we introduce a minimal dynamical model that describes both growth and collapse in a single formalism. The model yields a closed-form trajectory that generalizes the Gompertz law and admits equivalent maximum-entropy and optimal-control formulations. We show that this functional form captures the rise and fall of fourteen diverse empires across widely varying time scales. Despite substantial historical heterogeneity, all cases follow a common asymmetric rise-peak-decline pattern governed by a small number of interpretable parameters. We identify the mechanism driving this behavior as the chronophage: an exponentially accumulating burden of administrative and coordination costs that offsets the gains of expansion. These findings uncover a general quantitative principle, yielding new insights into how increasing complexity constrains large-scale social systems.