Task Entropy Must Be Paid: When Negative Curvature Helps Representation Learning

When does negative curvature become a useful representation resource rather than an architectural preference? A hierarchy alone does not decide this: the same input can induce a fine task that must preserve exponentially many distinctions or a coarse task that collapses most of them. We formalize this task-side demand through task entropy, the packing growth of a loss- or evaluation-induced pseudometric. For packings whose evaluation contains an $N$-way margin-identification restriction, we prove an exact capacity law: with unrestricted coordinatewise $L$-Lipschitz scores, a radius-$R$ representation supports exactly $P_{\text{M}}(R,m/L)$ identities at margin $m$. This yields a packing-restricted margin-error bottleneck, Euclidean resource tradeoffs, and a hyperbolic curvature threshold. On a regular tree, Euclidean obstruction and hyperbolic realization are governed by the same exponential leaf-growth rate, while a coarse task on the same hierarchy has zero task entropy and a bounded Euclidean realization. On a controlled 800-node WordNet intervention, Poincar\'e embeddings reduce hierarchy-task stress in all five seeds ($4.47\%\pm 1.80\%$); frozen-embedding margin-error curves provide a post-training identification diagnostic. Five real networks retain both positive and negative cases, probing task dependence and boundary cases highlighted by the capacity analysis rather than a universal optimizer-performance law.