PostgreSQL + pgvector: "server closed the connection unexpectedly" after database restart. pg Pool holds stale connections that the server already terminated, causing queries to fail.

Configure pg Pool with connection lifecycle settings and add retry logic:

1. idleTimeoutMillis: 30_000 — reap idle connections so stale ones don't linger
2. connectionTimeoutMillis: 5_000 — bound wait for new connections
3. maxLifetimeMillis: 5 * 60_000 — force-recycle all connections within 5 min (primary defense)
4. pool.on("error", ...) — prevent unhandled rejections from background disconnects
5. One automatic retry in query() on transient connection errors (57P01, 57P02, 57P03, 08006, 08001, 08004, ECONNRESET, ECONNREFUSED, and message-based detection)

The retry gets a fresh connection from the pool. If the DB is truly down, it fails with a clear connection timeout instead of a confusing "unexpected close".