Compute What the Task Needs: Operational Decision Frontiers Beyond Raw Scale

Scaling laws quantify gains along particular model--data--compute paths; they do not identify the resource intrinsically required by a decision problem. We separate three objects that are often conflated: decision value, the finite interface needed to preserve it, and the computation needed to extract that interface from a large input. For a budgeted pipeline class, the familiar log-loss identity gives an exact population frontier: optimal risk equals $H(Y)$ minus the largest attainable $I(Y;Z)$. We do not claim this identity, nor the classical conditional-mutual-information excess-risk bound, as new. Our first main result prices a reusable decision representation. If one finite representation must support every bounded downstream loss, essentially uniformly over the declared context law, then its minimum fixed-length bit budget is sandwiched between packing and covering entropies of the support of the posterior random variable $P(Y\mid X)$ in total variation. At zero tolerance the characterization is exact: one codeword is necessary and sufficient for each distinct posterior state. Our main computational result gives an exact routed-task separation: with $D=2^r+r$ input bits, adaptive inference preserves every decision using $r+1=\Theta(\log D)$ reads, while every nonadaptive exact rule must read all $D$ coordinates. Finally, for a declared finite family of discrete pipelines, we prove a finite-sample theorem that estimates its restricted task-information frontier simultaneously over all nonempty budgets and selects a near-frontier representation with high probability. The resulting message is positive rather than scarcity-driven: resource limits need not imply weaker decisions when computation follows posterior task structure.