Topological Defect Flow Theory as an Effective Realization of Quantum Vortex Theory

We present a discrete framework describing the dynamics of topological defects on a phase lattice. The system exhibits a transport regime governed by dipolar defect motion and a stochastic source term responsible for topology-changing events. We show that these source events correspond to localized relaxations of topological stress and are statistically governed by local observables. This leads to a natural separation between transport and event regimes. The resulting Topological Defect Flow Theory (TDFT) provides an emergent effective description of the system, where field-like behavior arises from an underlying discrete and event-driven dynamics. The work establishes a bridge between discrete modular structures, stochastic processes, and effective field theories, and suggests connections to continuum limits, statistical formulations, and computational interpretations.