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/src/App.vue

@@ -0,0 +1,47 @@
+<script setup lang="ts">
+import { RouterLink, RouterView } from 'vue-router'
+import { useAppStore } from '@/stores/app'
+
+const appStore = useAppStore()
+</script>
+
+<template>
+  <el-container class="app-shell">
+    <el-header class="app-header">
+      <div class="brand">paper-doc</div>
+      <el-menu mode="horizontal" :ellipsis="false" router class="app-nav">
+        <el-menu-item index="/">Home</el-menu-item>
+      </el-menu>
+      <el-button link @click="appStore.toggleSidebar()">
+        {{ appStore.sidebarCollapsed ? 'Expand' : 'Collapse' }}
+      </el-button>
+    </el-header>
+
+    <el-main>
+      <RouterView />
+    </el-main>
+  </el-container>
+</template>
+
+<style scoped>
+.app-shell {
+  min-height: 100vh;
+}
+
+.app-header {
+  display: flex;
+  align-items: center;
+  gap: 24px;
+  border-bottom: 1px solid var(--el-border-color);
+}
+
+.brand {
+  font-weight: 600;
+  font-size: 18px;
+}
+
+.app-nav {
+  flex: 1;
+  border-bottom: none;
+}
+</style>