Across physics, chemistry, biology, and engineered systems, the operationally significant questionis often not whether a system will eventually reach a particular state, but whether it can be broughtthere within the time available. This paper establishes a single structural necessity: when causalresponse propagates at finite speed, there exist states that are theoretically admissible but practicallyunreachable within any finite time horizon. We formalize this as the causal accessibility horizon—ageometric boundary determined solely by propagation speed and actuation geometry, beyond whichno control action can have effect by a given time T. This constraint is categorical: it arises fromthe hyperbolic structure of finite-speed dynamics and is logically independent of dissipation, whichgoverns amplitude decay within the accessible region but does not determine its boundary. Theresult reframes questions of control, safety, and stabilization as finite-time reachability problemssubject to irreducible geometric limits.