Information Limits of Replay-Free Recurrent State Migration

When a recurrent model is replaced, its old state may not determine what the new model would compute from the same history. We characterize the information that must be retained before the replacement is known. For a time-invariant affine linear family, the minimum continuous causal memory for order-$k$ local migration equals the reachable dimension of a lifted coefficient system minus the information in the anchor state. A controllable $d$-state model with scalar input needs at most $kd$ additional real coordinates, even when arbitrary matrix replacement directions are revealed only after streaming. A single rank-one direction can already require all $kd$. Conditional singular values and auxiliary-bit bounds distinguish this local information law from recovery at a fixed tolerance. We also give invariant-subspace and future-output constructions. Controlled experiments compare the resulting representations. In a small language model, equal-memory cubic interpolation improves mean state fidelity over third-order coefficients but provides no meaningful predictive gain. An analytic history basis preserves useful adaptation more economically in a coupled identification task. With repeated slow poles, eight history coordinates meet normalized output error $10^{-4}$ across five fitted seeds despite substantial full-state error. Splitting the poles raises the smallest tested successful width to 64. These results separate necessary historical information from the cost and accuracy of a chosen representation.