[loop-cycle-535] perf: cache Timmy agent at startup (#471) (#476)

## What
Cache the Timmy agent instance at app startup (in lifespan) instead of creating a new one per `/serve/chat` request.

## Changes
- `src/timmy_serve/app.py`: Create agent in lifespan, store in `app.state.timmy`
- `tests/timmy/test_timmy_serve_app.py`: Updated tests for lifespan-based caching, added `test_agent_cached_at_startup`

2085 unit tests pass. 2102 pre-push tests pass. 78.5% coverage.

Closes #471

Co-authored-by: Timmy <timmy@timmytime.ai>
Reviewed-on: http://localhost:3000/rockachopa/Timmy-time-dashboard/pulls/476
Co-authored-by: Timmy Time <timmy@Alexanderwhitestone.ai>
Co-committed-by: Timmy Time <timmy@Alexanderwhitestone.ai>

src/timmy_serve/app.py

@@ -75,6 +75,8 @@ def create_timmy_serve_app() -> FastAPI:
     @asynccontextmanager
     async def lifespan(app: FastAPI):
         logger.info("Timmy Serve starting")
+        app.state.timmy = create_timmy()
+        logger.info("Timmy agent cached in app state")
         yield
         logger.info("Timmy Serve shutting down")
 
@@ -101,7 +103,7 @@ def create_timmy_serve_app() -> FastAPI:
     async def serve_chat(request: Request, body: ChatRequest):
         """Process a chat request."""
         try:
-            timmy = create_timmy()
+            timmy = request.app.state.timmy
             result = timmy.run(body.message, stream=False)
             response_text = result.content if hasattr(result, "content") else str(result)