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/app/db/__init__.py

@@ -0,0 +1 @@
+"""Database engine, session factory, and declarative base."""

backend/app/db/base.py

@@ -0,0 +1,16 @@
+"""Declarative base for ORM models.
+
+No model classes are defined in this project yet, by design: the database
+schema is introduced deliberately through Alembic migrations rather than being
+created implicitly at import time.
+"""
+
+from sqlalchemy.orm import DeclarativeBase
+
+
+class Base(DeclarativeBase):
+    """Common base class for every ORM model.
+
+    Subclass this to declare a model. New models must also be imported by
+    ``app.models`` so that ``alembic revision --autogenerate`` can see them.
+    """

backend/app/db/session.py

@@ -0,0 +1,35 @@
+"""Database engine, session factory, and the FastAPI session dependency."""
+
+from collections.abc import Generator
+
+from sqlalchemy import create_engine
+from sqlalchemy.orm import Session, sessionmaker
+
+from app.core.config import get_settings
+
+settings = get_settings()
+
+engine = create_engine(
+    settings.sqlalchemy_url,
+    echo=settings.db_echo,
+    pool_pre_ping=settings.db_pool_pre_ping,
+    pool_recycle=settings.db_pool_recycle,
+    future=True,
+)
+
+SessionLocal = sessionmaker(
+    bind=engine,
+    class_=Session,
+    autoflush=False,
+    autocommit=False,
+    expire_on_commit=False,
+)
+
+
+def get_db() -> Generator[Session, None, None]:
+    """Yield a session per request and always close it."""
+    db = SessionLocal()
+    try:
+        yield db
+    finally:
+        db.close()