The visibility of the Roman Catholic Church is tied,
(i) To the Roman Pontiff when there is one, and the Roman Church in an interregnum, namely the clergy incardinated into the diocese of Rome and
(ii) To the hierarchy throughout the world, namely the bishops who have succession from the Apostles, with both orders and jurisdiction.
Apostolic succession is an external mark and is critical to the question of visibility. It is like genealogical succession from King David the Messiah was to have and which Christ Our Lord demonstrated.
Bishops without the power of jurisdiction do not have formal Apostolic succession. Now, individual bishops who do not have ordinary jurisdiction pose no doctrinal problem to an explanation of the crisis, but if your explanation leads you to the conclusion that there are no such bishops in the whole Church, then you need to rethink or modify some aspect of your explanation, because that conclusion is heretical. The same First Vatican Council says,
Just as he sent apostles, whom he chose out of the world , even as he had been sent by the Father, in like manner it was his will that in his Church there should be shepherds and teachers until the end of time.
In order, then, that the episcopal office should be one and undivided and that, by the union of the clergy, the whole multitude of believers should be held together in the unity of faith and communion, he set blessed Peter over the rest of the apostles and instituted in him the permanent principle of both unities and their visible foundation.
Now, as is evident here, and as any theology manual will tell you in any case, the shepherds and teachers who are sent are those who have full and formal succession from the Apostles, including the power of jurisdiction.
This also provides an answer as to how and where the perpetual Petrine sucession comes into it. The Apostolic sucession itself is closely interconnected with the Petrine succession, so that if the one indefinitely ceases, the other will as well. The conclusion of an indefinitely extended interregnum, then, would be that the Catholic Church would cease to be Apostolic, which is an inadmissible conclusion.
If you take cuм Ex as your guide, by the way, this is evident, for that very bull says clearly, as is obvious, that a heretic non-Pope would grant neither stability nor right to anyone, so that the bishops he appoints would have no office or jurisdiction at all.
Dear Jerry, for the record, I agree with you and with St. Robert on the specific question of whether the Pope who becomes a heretic will lose his jurisdiction. But the other opinion, that of Suarez, Cajetan, John of St. Thomas and others, that the Pope would lose his authority only at that moment when the sentence is passed on him by the Church, is also a permitted theological opinion.
But Cardinal Billot isn't talking about that, (nor is St. Alphonsus contradicting himself) he is introducing a superior principle worthy of consideration, hardly taken into account by sedevacantists today, with some exceptions.
"Finally, what one may think of the possibility or the impossibility of an heretical pope, there is at least one point absolutely clear which no one can put in doubt, and it is that the acceptance, the adherence, of the Universal Church to a pope will always be, by itself, the infallible sign of the legitimacy of such-and-such a pontiff; and consequently of the existence of all the conditions required for legitimacy."
For the record, I don't think most sedevacantists are schismatic, just mistaken about something they have in most cases not thought through sufficiently in good faith.
But I'm afraid your statement "The magisterium of the Church has never taught that there must be a certain number of bishops or the faithful for the Church to exist. As long as there are at least one priest or bishop and at least a few faithful , the Church and the hierarchy are alive and visible" is not correct, since the power of jurisdiction given to the Apostles being handed down is essential to Apostolic succession.
Dear Cantarella, some specifics aside (like that about Pope Honorius), I agree with the gist of what you say. May God send us a holy Pope, a shepherd after His own heart.
“The most evident mark of God’s anger and the most terrible castigation He can inflict upon the world are manifested when He permits His people to fall into the hands of clergy who are priests more in name than in deed, priests who practice the cruelty of ravening wolves rather than charity and affection of devoted shepherds ...
“When God permits such things, it is a very positive proof that He is thoroughly angry with His people, and is visiting His most dreadful anger upon them. That is why He cries unceasingly to Christians, ‘Return O ye revolting children ... and I will give you pastors according to My own heart’. (Jer. 3:14,15)