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/main.py

@@ -0,0 +1,29 @@
+"""FastAPI application entry point.
+
+Run locally with::
+
+    uvicorn app.main:app --reload --port 8000
+"""
+
+from fastapi import FastAPI
+from fastapi.middleware.cors import CORSMiddleware
+
+from app.api.router import api_router
+from app.core.config import get_settings
+
+settings = get_settings()
+
+app = FastAPI(
+    title=settings.app_name,
+    debug=settings.debug,
+)
+
+app.add_middleware(
+    CORSMiddleware,
+    allow_origins=settings.cors_origins,
+    allow_credentials=True,
+    allow_methods=["*"],
+    allow_headers=["*"],
+)
+
+app.include_router(api_router, prefix=settings.api_prefix)