Recurrent Memory Repair: When Local Sufficiency Fails at Finite Budgets

We study how shared retention gates can improve prediction while limiting changes in the influence of past writes. For normalized recurrent memory with fixed candidate values and write strengths, we constrain changes in their allocation weights by Kullback–Leibler (KL) divergence or the operator norm of the sensitivity change. We construct a sigmoid-gate family with a shared bias whose best first-order loss decrease at the baseline matches that of freely adjustable allocations at equal Fisher cost. Yet we prove that its best loss decrease vanishes as sequences grow at a fixed sufficiently small positive budget. Adding the squared difference between consecutive candidates to the sigmoid input with one new shared coefficient yields a loss decrease bounded away from zero with length, at the same budget and from the same pre-update gates and allocation. For this construction, the fixed-task improvement survives all separately bounded candidate-value perturbations within a positive radius independent of length and budget, with pre- and post-update gate values held fixed. General results characterize how gate changes accumulate to consume the budget and which gate features realize prescribed allocation changes. In a 20-seed synthetic study with a frozen trained gate-generating network, quadratic augmentation attains larger calibration loss decreases in every seed than equally sized random augmentation at matched local task efficiency and mean KL cap.