Implement full MVP: backend, auth, AI sidebar, persistence, and tests

[kkondo3jp]

Summary

• FastAPI backend with SQLite persistence, board CRUD (GET/PUT /api/board/{username}), AI connectivity check, and structured AI board-action endpoint backed by OpenRouter
• Docker + startup scripts for one-command local development on Windows, macOS, and Linux
• AuthGate sign-in screen (dummy credentials user/password) gating the Kanban board
• AI assistant sidebar wired to the backend - sends board context to the AI and applies structured board updates in real time
• Next.js static export served by FastAPI; board state is fully persistent via the backend instead of local React state
• Full test coverage: 7 backend pytest tests, 12 frontend Vitest unit tests, 3 Playwright E2E tests - all passing
• CLAUDE.md with commands, architecture overview, and milestone reference

Test plan

• pytest backend/tests/ - 7/7 passed (persistence, AI board actions, AI connectivity, frontend serving)
• npx vitest run - 12/12 passed (kanban logic, AuthGate flows, KanbanBoard with mocked API and AI)
• Playwright E2E - 3/3 passed (loads board, adds card, moves card between columns)

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