Take a session-level lock with pg_try_advisory_lock on a dedicated, unpooled connection, keyed by a stable hash of the graph name. The winner runs the tick; every other replica gets false at once and returns.
A LangGraph thread id is the key the checkpointer stores a run's state under, so derive it from what the run is about — graph, mode, period, subject — and "run this" becomes "make sure this has run": a retry resumes the existing thread instead of starting a second.
Move PostgresSaver.setup() into a migration you commit, review and apply as the schema owner before deploy. Then let the application read the schema version at boot and refuse to start when the database is behind.