feat(stage14): implement REST API for research lifecycle

- POST /v1/research — start research (non-blocking, background pipeline)
- GET  /v1/research/{id} — research metadata
- GET  /v1/research/{id}/status — detailed state machine status
- GET  /v1/research/{id}/sources — sources list
- GET  /v1/research/{id}/claims — claims list
- GET  /v1/research/{id}/evidence — evidence scores
- GET  /v1/research/{id}/report — research report
- DELETE /v1/research/{id} — delete research (non-completed)
- GET  /v1/research — paginated list of all research runs

Depth budgets (quick/normal/deep) control only resource limits.
In-memory store for now, to be replaced with PostgreSQL later.
Router mounted in main.py as tag "research-api".

src/nsct/api/main.py

@@ -111,6 +111,10 @@ def create_app() -> FastAPI:
     from nsct.api.audio import router as audio_router
     app.include_router(audio_router, tags=["audio"])
 
+    # Mount REST API router (Stage 14)
+    from nsct.api.rest_research import router as rest_research_router
+    app.include_router(rest_research_router, tags=["research"])
+
     return app