Functional Martingale Coherence and Amortized Uncertainty for Frozen Forecasters

Predictive resampling treats unobserved future data as a source of uncertainty, recursively imputing observations from a forecaster to induce a distribution over a target functional of interest. However, a generic pretrained forecaster need not be coherent under its own imputations. Sampled continuations can systematically shift the target, conflating future uncertainty with self-induced drift. We first develop a calibration method that constructs functionally coherent continuations while keeping the forecaster fixed. Exact one-step responses to candidate imputations enable a KL-minimal reweighting of the imputation distribution to remove self-induced drift when feasible. Same construction naturally extends to local covariance calibration, but local control alone does not determine uncertainty over an entire continuation. We derive an exact finite-horizon decomposition of target covariance into local variation and residual drift. Building on this, we introduce Amortized Martingale Coherence (AMC), which learns to estimate the resulting covariance target from current-state information without repeated future rollouts. Across tabular and language forecasters, functional repair reduces self-induced drift while preserving the frozen predictor. The resulting finite-horizon uncertainty captures future belief instability beyond local covariance, and AMC can predict much of this signal from the current state without future rollouts.