Commutative Merge Does Not Ensure Order-Invariant Agent Memory

An associative, commutative, and idempotent merge guarantees order-independent combination of fixed artifacts. In language-model memory, however, the artifacts are generated from batches of observations, and arrival order can change those batches. We test the gap between these two operations. In a controlled 4B study, all 300 world–arm cells produce multiple states under free arrival schedules; the same 300 cells produce a single state when extracted sub-batches are held fixed and only their processing order changes. The control localizes variability before fixed-artifact merging; the relative contributions of changed context and generation remain unresolved. A deterministic counterexample explains why merge algebra alone is insufficient, and canonical replacement gives agreement under fixed extraction outputs without requiring event-wise decomposition. We then measure what agreement establishes about utility. Historical design effects vary across scales and substrates. With atoms fixed, grouping and latest-value cues have positive interactions of 12.87 and 11.11 percentage points under two instructions, both surviving six-test Holm correction under the stated sign-flip reference assumption. In a 12-world online comparison, per-event extraction agrees across the evaluated arrival orders at 23.86% of canonical trie's summed request time, while trie's question-level gains and losses are mixed. Order-invariant merging leaves the production of its inputs outside its guarantee; answer quality and construction cost remain separate empirical outcomes.