Routing Without Learning: A Mechanism-Design Abstraction and the Limits of Capability Estimation

An agentic OS must decide which model executes which subtask, under uncertainty about how well any model will perform. We treat this as an allocation problem and propose an abstraction that splits the routing utility along a boundary that mirrors the ML/systems divide: a semantic term, estimated by a language model, and a resource term, computed deterministically from public benchmark scores and profiled per-role token consumption. The split confines all estimation uncertainty to one identifiable dimension of the utility tensor. Under this abstraction the allocation is an assignment problem solvable exactly in polynomial time, and we derive closed-form lower bounds on allocation quality from stated properties of the tensor. Without training, the resulting router reaches the cost-accuracy frontier at comparable accuracy on GAIA at three to ten times lower cost. The semantic half proves far weaker in practice than the formalism suggests. Four welfare objectives - utilitarian, Nash, envy-free, and a weighted hybrid -yield statistically indistinguishable allocations, and relevance estimates correlate only weakly with task success. In our deployment the dynamic range of the resource term substantially exceeds that of the semantic term; the imbalance is checkable in closed form before deployment and generalizes to any multiplicative router. Predicting model-subtask fit remains open, and current methods, ours included, do not yet answer it. Code is available at https://anonymous.4open.science/r/Routing-CE19.