## What happens

`AutoRouter.async_pre_routing_hook` (`litellm/router_strategy/auto_router/auto_router.py`) calls semantic-router's synchronous `__call__` — `routelayer(text=message_content)` — directly on the asyncio event loop, with no `await` / `asyncio.to_thread` / `run_in_executor` / async encoder. That call performs a **blocking HTTP embedding request**.

Additionally, the first request through the hook builds the entire route layer synchronously (`SemanticRouter(...)`, `auto_sync="local"`, embedding every configured utterance) inline on the loop, guarded only by a bare `if routelayer is None:` — no lock, so concurrent cold-start requests race and rebuild.

## Impact

Serially this is just added latency. Under concurrency, every routing embedding call freezes the whole worker: in-flight streams stall and requests serialize. It's very visible with clients that fan out parallel requests through an `auto_router/*` alias.

## The fix already exists next door

The complexity router solved exactly this: `complexity_router.py` builds its index under `self._semantic_routelayer_lock = asyncio.Lock()` via `await asyncio.to_thread(self._get_or_create_semantic_routelayer)` (added in #328…