On Emergent Phenomena in a Constraint Network Dynamical System

This paper studies a discrete dynamical system termed the Constraint Network Dynamical System. The system is rigorously defined by three axioms and four operators (C, A, B, M), describing the motion, interaction, and state transitions of fundamental units with binary directions and energy magnitudes in three-dimensional space. This paper addresses the core question of whether this system necessarily gives rise to hierarchical stable structures and provides a rigorous mathematical proof. The main results are as follows: on an open dense subset of the state space, the long-term evolution of the system necessarily converges to an attractor state characterized by sealed-state nodes and dynamic chains; there exists a set of emergent invariants whose values are determined by the operator structure and independent of the initial state; all observables can be strictly classified into three categories. This paper aims to provide the dynamical systems theory with a new object of study possessing emergence, invariance, and classifiability.