Under load the CoderAI broker WebSocket was being torn down with
"1011 keepalive ping timeout" and reconnecting (surfaced as 502s by the
fronting proxy). Root cause: synchronous work run directly on the asyncio
event loop starves the WebSocket keepalive so PONGs miss the 20s window.

Remove the blockers:

- analytics: dispatch the blocking token_usage / per-user INSERTs to a
  worker thread (fire-and-forget when a loop is running, inline otherwise)
  instead of calling them inline from async request handlers.
- database: add a MySQL connection pool (ping+reconnect, direct-connect
  fallback) and return connections to the pool on with-block exit instead
  of opening a fresh TCP+auth connection per query and leaking it to GC.
- utils: cache the tiktoken encoding and memoize count_messages_tokens by
  content, collapsing the several CPU-bound BPE passes per request into one.
- coderai_broker: run touch()/publish_response Redis I/O off the loop via
  asyncio.to_thread (awaited to preserve chunk order), matching the existing
  consume_request pattern.
- routes/coderai_broker: answer heartbeats immediately and process response
  frames on a dedicated FIFO worker so a slow publish can't delay the next
  heartbeat.
Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>