PRIVILEGED FORESIGHT DISTILLATION: AMORTIZED FUTURE CORRECTION FOR WORLD ACTION MODELS

World action models jointly predict future video and action during training, raising an open question about what role the future-prediction branch actually plays. A recent finding shows that this branch can be removed at inference with little to no loss on common manipulation benchmarks, suggesting that future information may act merely as a regularizer on the shared visual backbone. We propose instead that joint training induces an action-conditioned correction that privileged future observations impose on action denoising, and that current-only policies capture this correction only partially. Making the account precise, we formulate privileged foresight as a residual in the action-denoising direction—the difference between what a model predicts given the true future and what it predicts given only the current frame—and introduce Privileged Foresight Distillation (PFD), which transfers this residual from a training-time teacher into a small adapter on a current-only student. The teacher and student share the same backbone and differ only in the attention mask over video tokens; future video is never generated at inference. Controlled experiments support that this gain reflects a future-conditioned correction rather than added capacity: with the total teacher weight held fixed, routing it directly rather than as a residual drops the policy below the undistilled baseline. Empirically, PFD improves over Fast-WAM on LIBERO and RoboTwin manipulation benchmarks while preserving the current- only inference interface with only a slight adapter-induced latency overhead. This view reframes the role of future information in world action models: not as a target to predict, nor as a branch to discard, but as a compressible correction to be distilled—one that the privileged path carries even when it is the worse predictor.