This paper studies reachability constraints in a minimal finite deterministic dynamical system with discrete internal structure. The system consists of a binary internal variable coupled to a cyclic phase variable, evolving under two simple deterministic update rules. We investigate coarse-grained residual observables measured over a fixed time window, which record deviations of the phase and the internal binary state relative to the initial condition. Although the underlying dynamics is fully deterministic and the state space is finite, we show that not all coarse-grained residual configurations are reachable from a given initial state. For a fixed observation window of ten steps, we prove that only fourteen out of sixteen possible compressed residual outcomes can occur. The remaining two are structurally forbidden due to a number-theoretic coupling between the cyclic phase evolution and the parity of internal state transitions. The main result is established by an exact analytic proof and independently verified by exhaustive enumeration of all possible trajectories. This work demonstrates that strict reachability obstructions can arise in finite deterministic systems purely from arithmetic structure, without invoking stochasticity, noise, or asymptotic limits.