When is it a Catholic’s duty to recognize a Pope?

Thank you for your question. He who is elected as the Bishop of Rome – by the clergy of Rome in the first millennium, and by the College of Cardinals in the second millennium (or, in rare cases, by a pope appointing his successor) – must always be accepted as the pope unless there is clear evidence that the election was invalid or that the man “elected” is a manifest heretic (as taught by Pope Paul IV). In the case of the Vatican II “popes,” they are undeniable manifest heretics and have presided over a new religion and a new gospel which contradicts what all the true popes have taught from St. Peter on. It is definite, therefore, that they cannot be accepted as true popes based on the very teaching of the validly elected popes themselves.