feat(02-03): remove MPVEngine from main.py, add beats cache and shows static route

- Remove MPVEngine import, engine module-level var, and lifespan start/stop logic
- Remove broadcast_loop task (no server-side position polling)
- Add app.state.beats = {} dict (filepath -> beat data cache) in lifespan startup
- Mount /shows as StaticFiles so browser <audio src=/shows/file.mp3> works
- Simplified lifespan: registry + show_store + beats init only

lightsync/main.py

@@ -1,4 +1,3 @@
-import asyncio
 import os
 from contextlib import asynccontextmanager
 from pathlib import Path
@@ -8,36 +7,27 @@ from fastapi.staticfiles import StaticFiles
 
 from lightsync.devices.registry import DeviceRegistry
 from lightsync.store.show_store import ShowStore
-from lightsync.audio.engine import MPVEngine
 
 # Module-level containers populated in lifespan
 registry: DeviceRegistry | None = None
 show_store: ShowStore | None = None
-engine: MPVEngine | None = None
 
 
 @asynccontextmanager
 async def lifespan(app: FastAPI):
-    global registry, show_store, engine
+    global registry, show_store
     # Startup
     registry = DeviceRegistry(Path("devices.json"))
     await registry.load()
     show_store = ShowStore(Path("shows"))
     show_store.ensure_dir()
-    ao = os.environ.get("MPV_AO", "null")
-    engine = MPVEngine(ao=ao)
-    engine.start()
-    app.state.engine = engine
-    from lightsync.api.ws import broadcast_loop
-    task = asyncio.create_task(broadcast_loop(lambda: engine))
+
+    # Beat analysis cache: filepath -> {"tempo": float, "beats": [float, ...]}
+    app.state.beats = {}
+
     yield
+
     # Shutdown
-    task.cancel()
-    try:
-        await task
-    except asyncio.CancelledError:
-        pass
-    engine.stop()
     await registry.save()
 
 
@@ -51,6 +41,11 @@ def create_app() -> FastAPI:
     app.include_router(ws.router)
     app.include_router(audio.router, prefix="/api/audio")
 
+    # Serve uploaded audio files so <audio src=...> can load them
+    shows_dir = Path("/app/shows")
+    shows_dir.mkdir(parents=True, exist_ok=True)
+    app.mount("/shows", StaticFiles(directory=str(shows_dir)), name="shows")
+
     # Static file serving — no-cache headers to prevent stale JS/CSS
     frontend_dir = Path(__file__).parent / "frontend"