When State Information Pays: Cost Separation in Sequential Multi-Model Routing

Errors in agent trajectories can change the value of later model calls. We isolate this coupling in a two-state transmission model and compare state-conditioned policies with both deterministic and randomized open-loop schedules. Under capability ordering, the best schedule with a fixed number of strong calls has a strong suffix. This yields an exact deterministic cost in an explicit target window and reduces the randomized optimum to the lower convex envelope of K+1 suffix points. A feasible state policy then gives a matched-class cost-separation certificate. In a controlled instance, randomization reduces the open-loop cost from 12 to 5.32. The exact randomized state-history optimum costs 3.56, leaving a matched information gap of 1.76. A noisy-signal certificate vanishes under an uninformative signal, separating information from exogenous randomization. We evaluate a gold-free detector cascade on a real-execution data workflow. After calibrating a state-independent random trigger on costs alone, we freeze it before testing on 200 disjoint tasks with three fresh repetitions per policy (1,200 episodes). StopRoute reaches 79.50% success versus 74.50%, a 5.00-point paired gain on average, while the nominal-cost gap is 2.67%, inside the prespecified 5% point-estimate tolerance. Together, the exact model analysis and controlled online comparison isolate the value of where additional model calls are allocated, beyond randomization or heterogeneous model assignment.