Admissibility in Continuous-Time Dynamical Systems

This paper develops a continuous-time formulation of admissibility as a restriction on the space of realizable trajectories, extending prior discrete-time results in the Symbolic Modular Field Theory (SMFT) framework. Admissibility is treated as a structural constraint rather than as a stabilization, control, or regularization mechanism. The analysis is formulated at the level of continuous semigroups on metric spaces. Under restriction to a closed, forward-invariant admissible set, it is shown that exponential contraction of the semigroup implies the existence of a unique globally attracting equilibrium. Stability follows from restriction of admissible trajectories; no modification of the underlying dynamics is introduced. An ordinary differential equation realization is included as a corollary to record that the abstract hypotheses may be satisfied in standard continuous-time settings. No claim is made that admissibility holds generically or universally. Failure modes and falsifiability are addressed in earlier papers in the series and are referenced explicitly. This work is intended as a structural component of a larger theoretical program. It makes minimal claims, states assumptions explicitly, and isolates admissibility from dynamical enforcement mechanisms. The result is applicable wherever the stated conditions are met and is not tied to any specific physical interpretation.