feat: scaffold FastAPI backend and Vue 3 frontend

Backend (backend/):
- FastAPI app with layered app/{api,core,crud,db,models,schemas} structure
- TiDB connection via SQLAlchemy. The URL is built with URL.create rather
  than string formatting: the password contains '@', which would otherwise be
  parsed as the user/host separator and silently truncate the credential.
- Alembic environment wired to Base.metadata and the app settings, so
  backend/.env stays the single source of truth for credentials
- /api/health endpoint reporting database reachability
- models/ and crud/ are intentionally empty: no model classes are defined and
  no tables are created, at import time or otherwise

Frontend (frontend/):
- Vue 3 + Vite + TypeScript scaffold (create-vue, --bare)
- axios instance with a normalized error shape
- vue-router with a home route and a catch-all 404
- pinia + pinia-plugin-persistedstate; the app store persists selected keys
- Element Plus with its icons registered globally
- dev proxy forwards /api to the FastAPI service on port 8000

The database paper_doc was created in TiDB out of band; no table exists yet.

backend/app/api/routes/__init__.py

@@ -0,0 +1 @@
+"""Route modules, one per resource. Each exposes ``router``."""

backend/app/api/routes/health.py

@@ -0,0 +1,33 @@
+"""Health endpoint — also the smoke test for database connectivity."""
+
+from fastapi import APIRouter, Depends
+from sqlalchemy import text
+from sqlalchemy.orm import Session
+
+from app.core.config import get_settings
+from app.db.session import get_db
+from app.schemas.health import HealthResponse
+
+router = APIRouter(tags=["health"])
+
+
+@router.get("/health", response_model=HealthResponse, summary="Liveness and database probe")
+def health(db: Session = Depends(get_db)) -> HealthResponse:
+    """Report process liveness and whether TiDB answered a trivial query.
+
+    A database failure is reported in the body instead of raising, so that the
+    endpoint stays useful for diagnosing an unreachable database.
+    """
+    settings = get_settings()
+    try:
+        db.execute(text("SELECT 1"))
+        database = "ok"
+    except Exception:  # noqa: BLE001 - the failure mode is the payload
+        database = "unavailable"
+
+    return HealthResponse(
+        status="ok" if database == "ok" else "degraded",
+        app=settings.app_name,
+        database=database,
+        database_target=settings.safe_database_url,
+    )