Flow-Based Computation in Closed State Spaces: Interacting Trajectories and Non-Terminating Dynamics

This preprint introduces a flow-based extension of metabolic computation in closed state spaces. Instead of modeling systems as sequences of discrete states, computation is described as the continuous interaction of trajectories evolving on a toroidal topology. Each trajectory represents a dynamic process governed by pressure, attention, and transformation intensity. In multi-agent settings, trajectories interact through shared fields, leading to coupling, deformation, synchronization, or divergence. The framework further formalizes the earlier qualitative notion of “metabolic climate” as a system of interacting flows. What was previously described phenomenologically is here expressed as a structured dynamical field. Computation is redefined as a non-terminating process. There is no final output state; instead, stable structures emerge as persistent patterns within continuous motion. Meaning is extended from the stability of states to the stability of trajectories. Structures persist or dissolve under dynamic pressure rather than resolving into results. This work builds on the previously published framework of metabolic computation and extends it toward a flow-based paradigm of cognition and computation.