This paper presents a minimal, domain-agnostic core structure for irreversible dynamics under constraints. We show that when a system is characterized only by connectivity, metric, and curvature, the introduction of irreversible cost (wear) necessarily leads to contraction of the reachable region, hysteresis, and eventual cyclic stabilization. The framework deliberately avoids domain-specific assumptions, optimization objectives, or formalization choices, allowing projection onto physical systems, information processing, and semantic fields. This work is intended as a foundational core fixing structural priority, with formalization choices, empirical validation, and domain-specific applications presented in subsequent publications.