The Roman state, in part, has been in continous operation for 2,770 years.
Someone has ruled over Rome since it started, usually, but not always in Rome (Avignon). Constantinople had the better case as the successor State of Rome until 1453, and I would put a claim for the Vatican on part with the Russian claim to be the third Rome.
Continuity isn’t necessary for a successful insurrection—indeed, an interruption of continuity is generally a prerequisite.
Someone has ruled over pretty much everywhere, but that's not the point. The Russian claim is not the same thing. I'm not talking about a hypothetcial legacy. I mean literally, the Catholic Church is the unbroken continuation of part of the Roman state all the way back to the republic. That's a fact. It is the same organization with a continuing sequence of office holders. Even the still used term "pontiff" predates Christianity. Only the gods have been swapped out. When the political components of the Roman state were wiped out the religious component survived and still exists. The Roman state never entirely went away.