The Agent Loop Is a Routing Decision: Measuring the Switching Tax in Multi-Turn LLM Routing

Multi-turn LLM routers choose a model at every step of an agent loop, but they price each step as if billing had no memory. Providers bill statefully: a model reads at a discount only the prefix it has cached. We measure this on live APIs with 211 SWE-bench agent trajectories and 3,473 billed calls, and replay their switch schedules on four more deployments. A switch to a model not yet used in the session re-bills the whole prefix at the write rate (315 of 322 such switches read nothing), while a return re-reads most of the prefix that model last cached. The switching tax is therefore $(p^{\mathrm{cw}}-p^{\mathrm{cr}})(c_{t-1}-q\cdot c_{\mathrm{seen}})$, where q is the effective cache retention of the deployment. Across six screened deployments q runs from 0.81 to 0.99, and one model pair has effective retention 0.97 on its vendor's API but 0.81 and 0.83 on two re-hosts. With each deployment's cache block and minimum cacheable prefix, 95% of predicted trajectory bills land within ±10% of the billed cost, against 11% under full rebuild. From the tax we derive a closed-form break-even horizon and an output-aware diagnostic Ψ(q) that chooses between committing to one model and following demand, and hands cases inside a band to a tax-greedy rule. In live replays the prescribed policy is the cheaper one at every horizon outside the band, and inside it tax-greedy stays within 1.05× of the better fixed policy. The price sheet alone picks the costlier policy on one of these pools at 9 of 10 horizons. We prove a K-independent downgrade threshold for ordered tiers and worst-case linear-in-T regret for per-step learners when q < 1. Re-billing 1,785 production sessions with the router's own decisions, the switching tax is 20% of the median session's bill.