The Cost of Conservation: Coordination-Memory Laws for Exact-Support Generation

arXiv:2609.26126v1 Announce Type: new Abstract: Many AI systems make decisions locally, even when every realized output must obey an additive conservation law, such as selecting exactly a fixed number of items. This constraint can be statistically invisible: small subsets of a balanced fixed-budget output look increasingly independent, yet communication-free coordinate-parallel generation needs exponentially many pre-shared plans, while a sequential exact sampler needs only logarithmic memory. We study product measures conditioned on additive conservation laws in the intermediate regime where one plan is selected before a fixed-order pass, every plan is a bounded-state stochastic executor whose support is entirely legal, and the mixture of plan laws approximates the target distribution in total variation. Our main result identifies the optimal asymptotic selector rate, up to constant factors, with the killed spectral profile of the conservation-difference walk. The resulting coordination cost decreases as an inverse power of live-state width, with an exponent determined by intrinsic conservation rank rather than alphabet size; the law extends to noncentral budgets and heterogeneous local scores. The converse is driven by a state-versus-resource-sum obstruction, while a rate-matching construction compiles discrepancy control into exact-support finite-state plans. Complementary results characterize block-parallel plan complexity and the benefit of programmable output order. Together, these results show precisely how online memory substitutes for front-loaded coordination in exact-support generation.