Memory for Late Evidence: A Delay–Resolution Phase Law

Sequential state estimators are designed to forget: once a compact state is sufficient for predicting the future, the observations that produced it can be discarded. Late evidence exposes a fundamental limitation of this principle. A state may be sufficient for forward prediction yet insufficient to revise the present when an observation about a discarded timestamp arrives later. We characterize the minimum persistent memory required to support such revisions under arbitrary legal arrival orders. For a class of positive, invertible $K$-state models with bounded, locally rich on-time evidence and retrodictively observable late evidence, the optimal deterministic code satisfies $B_{ϵ}^{\star }(D)=\Theta \left((K-1)\min \{D+1,\log (1/ϵ)\}\log (1/ϵ)\right).$ Thus, correction-state memory is delay-limited only up to the point at which the required resolution dominates: beyond this horizon, memory becomes resolution-limited and no longer grows with physical delay. The upper bound stores a quantized cut belief together with the resolvable suffix, while a matching response-packing converse shows that every retained direction is necessary, ruling out an explanation based solely on truncating a mixing filter. Without bounded evidence, even delay one can preclude any finite uniformly accurate code. These results identify statistical resolution, rather than physical delay, as the relevant capacity variable for correctable state beyond the mixing horizon. A packed cut–suffix construction realizes the predicted memory and replay-work saturation, while a controlled recurrent bottleneck diagnostic reveals the forward–revision tension on prediction-equivalent histories. Overall, the theory provides a concrete criterion for sequential representations under late evidence: preserve the responses required for future corrections, not merely the information needed for prediction from the current state.