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/api/client.ts

@@ -0,0 +1,48 @@
+import axios, { AxiosError, type AxiosInstance } from 'axios'
+
+/**
+ * Shared axios instance.
+ *
+ * `baseURL` defaults to the same-origin `/api` prefix, which Vite proxies to
+ * the FastAPI service in development (see vite.config.ts). Point
+ * `VITE_API_BASE_URL` at an absolute URL to bypass the proxy.
+ */
+export const http: AxiosInstance = axios.create({
+  baseURL: import.meta.env.VITE_API_BASE_URL ?? '/api',
+  timeout: 15000,
+  headers: {
+    'Content-Type': 'application/json',
+  },
+})
+
+/** Normalized failure shape, so views never have to inspect AxiosError. */
+export interface ApiError {
+  message: string
+  status?: number
+  detail?: unknown
+}
+
+function normalizeError(error: AxiosError): ApiError {
+  const status = error.response?.status
+  const payload = error.response?.data as { detail?: unknown } | undefined
+
+  let message: string
+  if (payload?.detail && typeof payload.detail === 'string') {
+    message = payload.detail
+  } else if (error.code === 'ECONNABORTED') {
+    message = 'The request timed out.'
+  } else if (!error.response) {
+    message = 'Could not reach the API. Is the backend running on port 8000?'
+  } else {
+    message = error.message
+  }
+
+  return { message, status, detail: payload?.detail }
+}
+
+http.interceptors.response.use(
+  (response) => response,
+  (error: AxiosError) => Promise.reject(normalizeError(error)),
+)
+
+export default http

frontend/src/api/health.ts

@@ -0,0 +1,15 @@
+import http from './client'
+
+/** Mirrors `app.schemas.health.HealthResponse` on the backend. */
+export interface HealthResponse {
+  status: string
+  app: string
+  database: string
+  database_target: string
+}
+
+/** Probe the backend, including its TiDB connection. */
+export async function fetchHealth(): Promise<HealthResponse> {
+  const { data } = await http.get<HealthResponse>('/health')
+  return data
+}