When Distributed Memory Compresses Exactly: Sufficient Fusion of Heterogeneous Temporal Traces

Long-term agent systems increasingly distribute memory across components, modalities, devices, or cooperating agents. More memory sources, however, need not imply proportionally more decision-relevant memory. We isolate an exact statistical baseline for this question. A latent temporal class $t$ (for example, an age bin) is observed through independent Poisson memory traces. We call each trace-producing memory source a \emph{reporter}; its rate may vary with age. Conditional on the pooled event count, reporter identities are multinomial. If the age-dependent log-composition vectors lie in a $q$-dimensional affine subspace, then the entire $d$-reporter vector is exactly reducible, for inference about $t$, to the pooled count plus $q$ scalar linear summaries. Thus the number of sufficient fused coordinates depends on the dimension of temporal heterogeneity, not on the number of reporters. In a controlled $d=12$ synthetic check with a two-dimensional log-composition subspace and 32,000 Monte Carlo samples, the resulting three-scalar representation exactly matches every classifier decision of the full 12-count vector and has the same accuracy ($0.7596$). Because the pooled count is identically distributed across the eight equally likely classes, its Bayes accuracy is exactly $1/8=0.1250$. This is a deliberately restricted generative model, not a claim that arbitrary LLM or embodied memories admit such compression. Its role is to provide a falsifiable baseline for multi-agent and multimodal memory systems: measure whether additional memory sources contribute new age-dependent composition directions or merely replicate information already present in a smaller sufficient statistic.