Recursion as a Phase Transition in Causal Dependence

Recursion is often described as a structural feature of complex systems. A general dynamical criterion is presented in which recursion emerges as a shift in causal dependence. A system enters a recursive regime when its present state is more strongly predicted by its own recent past than by external input, expressed as P(S_t | S_(t-1), S_(t-2), …) > P(S_t | E_t). This transition occurs when internal stabilization exceeds environmental disruption, formalized as λ_self / λ_env > 1. Under this condition, activity persists across time, allowing feedback, temporal continuity, and self-referential organization to develop from the same underlying dynamics. The criterion is measurable using predictive models, mutual information, and time series analysis. It applies across physical, biological, neural, and computational systems, providing a unified account of the emergence of feedback and recursive organization grounded in dynamical systems and information theory. Structure is the observable form of persistent dynamics that have achieved sustained causal influence over their own evolution.