feat(06-02): add feature extraction and segmentation backend modules + API

- Create lightsync/audio/features.py with extract_features and extract_features_async
- Create lightsync/audio/segments.py with detect_segments via agglomerative clustering
- Add /api/audio/segments and /api/audio/features endpoints to audio.py
- Initialize app.state.segments cache in main.py lifespan

lightsync/api/audio.py

@@ -10,6 +10,8 @@ from fastapi.responses import StreamingResponse
 from pydantic import BaseModel
 from lightsync.audio.waveform import extract_peaks
 from lightsync.audio.beats import analyze_async
+from lightsync.audio.segments import detect_segments_async
+from lightsync.audio.features import extract_features_async
 
 router = APIRouter()
 
@@ -190,3 +192,39 @@ async def fetch_youtube(request: Request, body: YouTubeFetchRequest):
             yield f"data: {json.dumps({'type': 'error', 'msg': str(e)})}\n\n"
 
     return StreamingResponse(event_stream(), media_type="text/event-stream")
+
+
+@router.get("/segments")
+async def get_segments(request: Request, path: str, n_segments: int | None = None):
+    """Return structural segments for an audio file. Cached in app.state.segments."""
+    segments_cache: dict = getattr(request.app.state, "segments", None) or {}
+
+    cache_key = f"{path}:{n_segments or 'auto'}"
+    if cache_key not in segments_cache:
+        p = Path(path)
+        if not p.exists():
+            raise HTTPException(status_code=404, detail=f"File not found: {path}")
+        try:
+            segments = await detect_segments_async(path, n_segments)
+            if not hasattr(request.app.state, "segments") or request.app.state.segments is None:
+                request.app.state.segments = {}
+            request.app.state.segments[cache_key] = segments
+        except Exception as exc:
+            raise HTTPException(status_code=500, detail=f"Segmentation failed: {exc}") from exc
+    else:
+        segments = segments_cache[cache_key]
+
+    return {"segments": segments, "count": len(segments), "file": path}
+
+
+@router.get("/features")
+async def get_features(request: Request, path: str):
+    """Return extracted audio features (onsets, chroma, RMS) for a file."""
+    p = Path(path)
+    if not p.exists():
+        raise HTTPException(status_code=404, detail=f"File not found: {path}")
+    try:
+        features = await extract_features_async(path)
+    except Exception as exc:
+        raise HTTPException(status_code=500, detail=f"Feature extraction failed: {exc}") from exc
+    return {**features, "file": path}