Phase Dynamics in Flow-Based Computation: Interference and Emergent Structure in Closed Systems

This preprint extends flow-based computation in closed state spaces by introducing phase as a fundamental component of trajectory dynamics. Building on the previously established framework of interacting flows on a toroidal topology, this work augments each trajectory with a phase variable, enabling interference-like behavior. Trajectories may reinforce or suppress each other depending on their phase relationships, leading to resonance, synchronization, and emergent structure formation. Computation is further generalized: from state stability, to flow persistence, and now to phase coherence. Meaning is no longer defined solely by stable configurations or trajectories, but by coherent interference patterns sustained under continuous interaction. The framework demonstrates how quantum-like interference phenomena can emerge from classical flow dynamics without reliance on quantum mechanical formalism. This work represents the third layer in a developing theoretical framework of computation, following prior work on metabolic computation and flow-based dynamics.