Sufficient to Act: What a Representation Must Reveal When Its Consumer Optimises

Representation learning usually defines sufficiency relative to prediction, but many systems consume representations by solving an optimisation problem and returning an action. We ask what information a representation must preserve when its consumer optimises. For a fixed downstream objective, query sufficiency and action sufficiency can separate sharply: there are input families for which answering all relevant queries requires $\Omega (n)$ bits while recovering the optimal action requires only $O(1)$. For constraint-generated minimum-cost hitting-set decisions, we then characterise universal-cost action sufficiency exactly: two inputs induce the same complete family of optimal actions under all strictly positive costs if and only if their hypergraphs have the same antichain of inclusion-minimal generators. Its union gives the exact action support, identifying candidates that can be removed with zero decision loss. We also show that pseudonymisation does not remove structure-determined nuisances and give exact conditions for quotienting the remaining. support while preserving the complete Argmin set. In a real-code study motivated by post-quantum cryptography (PQC) migration, the default model yields exact hypergraphs for 30 repositories; the action support retains a median 4.7% of graph vertices and the admissible quotient 3.5%, while pseudonymised full topology remains re-identifiable at about $19\times$ chance. On controlled synthetic instances, a graph-aware selector recovers the exact support on 37.0% of matched-budget cases but preserves the exact Argmin on 89.8% of sampled instance-cost pairs, showing that empirical objective preservation is weaker than universal action sufficiency.