Synchronize API documentation

Synchronize docs/API.md with the FastAPI OpenAPI document. $ARGUMENTS may contain a server base URL; default to http://127.0.0.1:8000.

1. Read the routers and schemas under backend/app/ and the current docs/API.md.
2. Fetch $ARGUMENTS/openapi.json (or the default URL). If the server is unavailable, ask the user to run make run; do not invent schema fields.
3. Compare every OpenAPI path, method, status code, parameter, request body, and response model with the documentation.
4. Update only docs/API.md, preserving its concise endpoint-table format and adding validation/error behavior.
5. Fetch OpenAPI once more and report a diff-style summary: added, changed, removed, and remaining drift.

The operation must be idempotent. Never edit generated OpenAPI JSON, application code, or delete hand-written safety notes.