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 ($\mathcal{C}$, $\mathcal{A}$, $\mathcal{B}$, $\mathcal{M}$), 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.