Make SQLite database self-create on serverless hosts

Vercel's serverless functions only allow writes to /tmp, and it starts
empty on every cold start. Point the database there in production and
create the User table on first request if it doesn't exist yet, so the
app can be deployed for a live preview without manual database setup.

Note: this is a demo-friendly stand-in — data does not persist across
cold starts on Vercel. A real persistent database is a later phase.

Co-Authored-By: Claude Sonnet 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01VL5YUEf2B9LG4xdVUQwcSg

CLAUDE.md

@@ -0,0 +1 @@
+@AGENTS.md