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.

frontend/tsconfig.app.json

@@ -0,0 +1,18 @@
+{
+  "extends": "@vue/tsconfig/tsconfig.dom.json",
+  "include": ["env.d.ts", "src/**/*", "src/**/*.vue"],
+  "exclude": ["src/**/__tests__/*"],
+  "compilerOptions": {
+    // Extra safety for array and object lookups, but may have false positives.
+    "noUncheckedIndexedAccess": true,
+
+    // Path mapping for cleaner imports.
+    "paths": {
+      "@/*": ["./src/*"]
+    },
+
+    // `vue-tsc --build` produces a .tsbuildinfo file for incremental type-checking.
+    // Specified here to keep it out of the root directory.
+    "tsBuildInfoFile": "./node_modules/.tmp/tsconfig.app.tsbuildinfo"
+  }
+}