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/alembic/script.py.mako

@@ -0,0 +1,26 @@
+"""${message}
+
+Revision ID: ${up_revision}
+Revises: ${down_revision | comma,n}
+Create Date: ${create_date}
+
+"""
+from collections.abc import Sequence
+
+from alembic import op
+import sqlalchemy as sa
+${imports if imports else ""}
+
+# revision identifiers, used by Alembic.
+revision: str = ${repr(up_revision)}
+down_revision: str | None = ${repr(down_revision)}
+branch_labels: str | Sequence[str] | None = ${repr(branch_labels)}
+depends_on: str | Sequence[str] | None = ${repr(depends_on)}
+
+
+def upgrade() -> None:
+    ${upgrades if upgrades else "pass"}
+
+
+def downgrade() -> None:
+    ${downgrades if downgrades else "pass"}