Online Compression of Controlled Memory by Age-graded Rebucketing

We construct finite-memory vector approximations of controlled memories in which each past input evolves through age-dependent linear contractions. Future controls arrive online and can induce noncommuting matrix actions, so discard- ing a past input must preserve its response to controls that have not yet been observed. Our algorithms maintain consecutive birth intervals on an increasing age grid and merge adjacent intervals without recovering discarded writes. Under a capped log-age modulus, one vector per interval gives uniform output error ϵ with $O({ϵ}^{-1})$ vector states. With additional second-order age smoothness, positive redistribution between two endpoints improves this sufficient budget to $O({ϵ}^{-\frac{1}{2}})$, counting both endpoint vectors. The bounds hold for every bounded signed-vector write stream and common external control path, uniformly over elapsed time and contraction lifetime when regularity constants are fixed. The analysis couples decay-weighted local error with reciprocal-width packing. For the endpoint rep- resentation, a mean-preserving virtual birth coordinate cancels the first variation of the complete chronological matrix product. Small operator-level comparisons use dual bounds on conditional worst-write error to examine the resulting finite- budget approximations without model training.