Finite-Dictionary Tests of Predictive Adequacy for State--History Models

A history-based predictor may outperform a fitted state model because relevant information was omitted, because the state model was poorly fitted, or because uncertainty was underestimated. We define predictive adequacy by a prespecified bound on excess population risk relative to a finite dictionary of complete predictors. Established empirical-Bernstein inequalities yield certificates for training-selected predictors and for selection within a finite class during evaluation. The guarantees require independent evaluation groups and permit dependence within groups. Pre-evaluation bounds on each pair's loss-contrast range improve on a generic bounded-loss baseline. Cosmology-motivated synthetic references expose underfitting, history censoring and false precision from counting correlated rows as independent. At Brier tolerance $0.01$, with 70 state candidates and nominal error $0.05$, a specified truncated-history pair's variance-inclusive population-moment planning curve crosses the tolerance at 13,464 independent evaluation groups with its verified range, versus 24,388 generically. These crossings are not power guarantees or universal sample-size requirements. With 16,384 total groups and 512 used for training, the range refinement raises evaluation-time class certification from 0 to 86 of 100 trials, while training empirical-risk minimization gives 99 of 100. Pair-envelope class selection reaches 100 of 100 at all tested allocations of 32,768 total groups. Gains depend on the contrasts that determine the bound, not merely on dictionary size. In a separate exact two-predictor reference, null-boundary error is approximately $0.029%$ at 1,024 groups, below nominal $5%$, quantifying conservatism rather than worst-case sharpness. These synthetic results concern specified predictors and observation laws, not autonomous dynamical closure or nonlocal gravitational physics.