What Must Survive? Exact Task-Information--State Frontiers Under Partial Task Revelation

A state may need to be compressed before its exact downstream task is known. For a finite family of linear task operators Tuu∈U, we characterize exactly how much state must survive when only one of K coarse task messages is available at compression time: p^*(K)= min_substackPcal partition of U|Pcal|≤ K maxC∈Pcalrank(TC), where TC stacks the tasks that remain unresolved within cell C. Thus task information reduces retained state precisely by separating tasks whose joint task-visible subspace is expensive. We derive the dual bit frontier, an irreducible common-core floor, and a sharp singular-value characterization for nonzero tolerance. For overlapping task families, a cumulative overlap deficit places the exact frontier within an additive Delta of weighted load balancing. Optimal partitioning is strongly NP-hard, yet direct-sum families admit classical approximation guarantees and a PTAS. We stress-test the theory on learned task operators from CIFAR-10, Fashion-MNIST, SVHN, and Burgers observables. For Burgers with K=2 and eps=0.35, a constructive partition attains p=4, while balanced random partitions require median p=6, with held-out worst standardized RMSE 0.0368. The results distinguish uniform task sufficiency from distributional and decision-level performance.