Large language model (LLM) routers are increasingly deployed to optimize quality, latency, and cost through dynamic model selection and request rewriting. However, a compromised router can cause ineligible selection on the control plane, unendorsed rewriting or unfaithful relay on the data plane, undermining LLM routing integrity. We present VeriRouter, a verifiable LLM routing framework. Under an endorsed public context binding a private semantic index and rewrite banks, VeriRouter combines ZK proofs with a provider-issued serving certificate to establish integrity across both planes while keeping the router's private routing assets hidden from clients. For model selection, we design an efficient ZK proof for semantic search. For rewriting, local checks enforce message preservation and additions from committed banks. On real-world workloads, VeriRouter's model-selection and rewrite proofs average 21.92~s and 4.24~s, yielding 1,307$\times$ and 101$\times$ speedups over the fastest completing baselines. Relative to serving without verification, VeriRouter adds only 4.08 s under per-turn selection and 0.11 s under session-pinned selection.

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