Count the Groups, Not the Configurations: Exact Inference and Learning at Fixed Totals

A fixed total couples every variable in an integer allocation model. For interactions determined by group totals, we turn this constraint into a computational resource: sum local allocation weights into group-count tables, remove a redundant count coordinate, and sample exact configurations from the resulting small distribution. We reuse these tables to score whole batches of changes to group membership through exact partition-function ratios. For two groups, compilation costs $O(J{c}_{\max }r)$ for $J$ variables, maximum capacity $c_{\max }$, and total $r$, versus $O(J{c}_{\max }{r}^2)$ for an explicit total-and-signed-count recursion. A shared conditional allocation law also makes model comparison lossless at the count level. Controlled synthetic experiments with heterogeneous local weights yield $1.48$–$2.98\times$ more effective samples per CPU second than the best tested positive auxiliary-field mixtures, and count-compressed membership scoring is $1.75\times$ faster than its expanded implementation. Learning studies separate this computational benefit from the choice of statistical objective. Exact residual calculations further reveal how an omitted interaction can amplify rare configurations, damaging importance sampling before substantially changing the bulk distribution. The result is a reusable count representation for sampling, comparison, and discrete structure refinement.