llm.Router was primary-then-fallback (berget -> iguana ollama). Both have failed together at least once in production (berget 5xx outage + a cold fallback model exceeding the shared timeout budget), leaving brain_answer/brain_classify fully down. Router gains an optional Tertiary *Client, tried only if both Primary and Fallback fail. Nil-safe — existing 2-tier deployments (no BRAIN_LLM_TERTIARY_URL) behave exactly as before. main.go wires it from BRAIN_LLM_TERTIARY_URL/MODEL + GEMINI_API_KEY, pointed at Google's OpenAI-compatible Gemini endpoint (generativelanguage.googleapis.com/v1beta/openai/) — llm.Client already speaks OpenAI-shaped chat-completions, so no new client type needed. Tests: TestRouter_FallsBackToTertiaryWhenPrimaryAndFallbackFail, TestRouter_TertiaryNotCalledWhenFallbackSucceeds, TestRouter_AllThreeFail, TestRouter_NilTertiary (2-tier backward-compat). Full suite green (both modules, -race), golangci-lint clean. Deployment env vars (infra repo, separate commit — GitOps, TBD there too): BRAIN_LLM_TERTIARY_URL, BRAIN_LLM_TERTIARY_MODEL, GEMINI_API_KEY secretRef. Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>