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/.gitignore

@@ -0,0 +1,39 @@
+# Logs
+logs
+*.log
+npm-debug.log*
+yarn-debug.log*
+yarn-error.log*
+pnpm-debug.log*
+lerna-debug.log*
+
+node_modules
+.DS_Store
+dist
+dist-ssr
+coverage
+*.local
+
+# Editor directories and files
+.vscode/*
+!.vscode/extensions.json
+.idea
+*.suo
+*.ntvs*
+*.njsproj
+*.sln
+*.sw?
+
+*.tsbuildinfo
+
+.eslintcache
+
+# Cypress
+/cypress/videos/
+/cypress/screenshots/
+
+# Vitest
+__screenshots__/
+
+# Vite
+*.timestamp-*-*.mjs