Geometry and information allocation for finite-resolution final-state predictability

Final-state predictability in multistable dynamics cannot be reduced to short-term trajectory divergence. At finite resolution, a deterministic terminal map induces a local probability law over outcomes, while a limited query budget constrains its reconstruction. The key question is therefore not only where terminal behavior is uncertain, but where additional observations are most valuable. To address this problem, we develop a framework linking finite-resolution terminal laws, basin-boundary geometry, finite-time pullback geometry, action-dependent predictive risk, and finite-budget allocation. Boundary geometry identifies terminal structure that remains difficult to resolve. In turn, pullback geometry describes how local uncertainty is deformed along different directions before terminal outcomes separate. These descriptors are then used to estimate the risk changes associated with discrete query actions, allowing the global budget to be allocated by marginal predictive value rather than uncertainty alone. Experiments across continuous, discrete, and physically derived systems show that action-specific risk modeling is central to effective allocation, while pullback information adds decision-relevant structure beyond initial labels and scalar instability. Under family-held-out transfer without target-response retraining, anchoring the response to a common allocation baseline further reduces cross-mechanism downside. Overall, finite-resolution final-state predictability is a coupled geometric and information-allocation problem.