Average Action Efficiency Rises Monotonically in Self-Organizing Systems via Stochastic Least-Action Dynamics

Abstract: Self-organizing systems convert noisy motion into efficient structure, yet a universal, dimensionless measure of this transformation is lacking. We derive the Average Action Efficiency (AAE) - events per total action - from a stochastic path-integral least-action principle. A Lyapunov identity links its monotonic rise to the action variance and the rate of noise reduction, defining growth, saturation, and decay regimes. Agent-based ant foraging and single-molecule ATP-synthase data confirm the predicted sigmoidal rise and plateau. Because AAE needs only an event count and an integrated action, it offers a lightweight metric and design rule for feedback-controlled self-organization across physics, chemistry, biology, and active matter.