TRIM: A Theory of Retrieval with Incremental Memory -- Defect, Benefit, and Critical Horizon

Most theories of in-context learning (ICL) treat the prompt as fixed. This fixed-prompt view misses the key difficulty in memory-updated retrieval systems such as Memory Mosaics. An inference-time write can help later queries, but it can also shift softmax weights and create interference. The question is when the benefit exceeds the later cost. We introduce TRIM (Theory of Retrieval with Incremental Memory), a theorem-level theory of append-only softmax retrieval under a write-independence abstraction. The five-theorem spine links retrieval-weight concentration, local write-induced defect, recurrence-level accumulation, ideal squared-loss benefit, and a benefit-cost comparison with critical-horizon regimes. TRIM places helpful memory and harmful interference within a single frame, so the harmful length scale is derived rather than tuned. We record a first-order dynamic correction for state-dependent writes separately, and it is not part of this spine. We give a theorem-facing empirical protocol. Exact synthetic checks test the spine identities and inequalities. Learned base-MM analyses measure calibrated analogues of concentration, local defect, horizon worsening, and realized benefit. These analyses test whether TRIM observables organize learned systems along the same routes.