When Departure Rates Saturate: Endpoint Diagnostics for Learned Dynamics

The any-time departure rate is the fraction of predicted trajectories that leave a prescribed region at least once. At long horizons, this rate can saturate at one even when models differ in their return behavior. We study the additional information obtained by recording whether the same predicted trajectories end inside or outside the region. An exact decomposition distinguishes terminal exclusion from departure followed by return. A two-state calculation shows how identical departure rates can coexist with different return behavior while endpoint and later-occupancy rankings still agree. For fixed candidate models, we give finite-sample endpoint-risk bounds that allow diagnostic eligibility screens and randomized ties, while isolating the additional condition needed for temporal transfer. In a complete 30-model mushroom-billiard cohort, every departure rate equals one, yet terminal exclusion predicts later outside occupancy on disjoint starts with Spearman correlation $0.905$. The association survives a diagnostic concentration screen and appears in a separate study of both Koopman autoencoders and residual predictors. Exploratory selection analyses improve on the saturated rule, including after accuracy screening; the residual-family selection gain remains unresolved. These results support a low-cost evaluation practice: report endpoint status alongside departure and predictive fidelity, and evaluate selection performance against a separately specified regional target.