The Causal Accessibility Horizon: A Structural Limit on Finite-Time Reachability

Across physics, chemistry, biology, and engineered systems, the operationally significant question
is often not whether a system will eventually reach a particular state, but whether it can be brought
there within the time available. This paper establishes a single structural necessity: when causal
response propagates at finite speed, there exist states that are theoretically admissible but practically
unreachable within any finite time horizon. We formalize this as the causal accessibility horizon—a
geometric boundary determined solely by propagation speed and actuation geometry, beyond which
no control action can have effect by a given time T. This constraint is categorical: it arises from
the hyperbolic structure of finite-speed dynamics and is logically independent of dissipation, which
governs amplitude decay within the accessible region but does not determine its boundary. The
result reframes questions of control, safety, and stabilization as finite-time reachability problems
subject to irreducible geometric limits.