One Ring to Host It All: Fixed-Capacity Circular Realizations of Finite Functional Trajectories

Can a finite system repeating on a single cycle reproduce, for a specified duration, the behaviour of another deterministic system, even when the visible behaviour appears irreversible? We formulate this as a circular-sequence problem. Each visible state has a prescribed number of underlying occurrences, and at least a specified number must begin trajectories that follow the visible rule correctly for a given duration. These trajectories may overlap, but every underlying state must belong to one cycle. We establish necessary and sufficient conditions for such a realisation and give a constructive method based on a compact integer certificate. We then show that existence can be decided in time polynomial in the binary size of the input, using at most n minimum-cost-flow computations for n visible states, without expanding the potentially enormous underlying cycle. The result provides an exact resource-feasibility criterion for finite-duration effective dynamics within a recurrent reversible host. It establishes mathematical feasibility under specified conditions, without assuming that the required structures arise naturally.