Internal-Time Viability Theory (V1)

This paper establishes a structural theory for viability in dynamical systems subject to moving safety constraints. Classical viability and barrier formulations implicitly assume that admissible sets evolve in a manner compatible with forward invariance when indexed by external clock time. However, for a class of systems in which admissible regions evolve non-monotonically or in a state-contingent manner, no external-time formulation admits classical invariance conditions. We show that, in such settings, viability requires the introduction of an endogenous internal time acting as a governing variable rather than a modeling convenience. Safety constraints are indexed by an internal phase whose progression follows a non-uniform, state-dependent tempo, and forward invariance is imposed with respect to internal-time evolution. A central necessity result establishes that when external-time invariance fails, internal-time indexing is required to recover viability. A minimal example demonstrates that the resulting mechanism cannot be reduced to time reparameterization or barrier scheduling in external time. This work builds on the prior foundational preprint:https://doi.org/10.5281/zenodo.18209944