Let this be said in passing against those who, trying to justify certain attempts at schism made in the time of Alexander VI, allege that its promoter broadcast that he had most certain proofs, which he would reveal to a General Council, of the heresy of Alexander. Putting aside here other reasons with which one could easily be able to refute such an opinion, it is enough to remember this: it is certain that when Savonarola was writing his letters to the Princes, all of Christendom adhered to Alexander VI and obeyed him as the true Pontiff.
For this very reason, Alexander VI was not a false Pope, but a legitimate one. Therefore he was not a heretic at least in that sense in which the fact of being a heretic takes away one’s membership in the Church and in consequence deprives one, by the very nature of things, of the pontifical power and of any other ordinary jurisdiction.”
So the very verified fact that Pope Alexander VI had universal acceptance at the moment when he was alleged to be a heretic suffices to show, from the faith, that the accusation was false. It is not even necessary, says Cardinal Billot, to examine very closely the specific facts. Which shows how far modern sedevacantists are from the mind of the Church.
It appears to me that Cardinal Billot's point is that we can't second guess the Church from a historical perspective. We can't convict Pope Alexander VI of public heresy a few centuries later after the fact. Even if we found Savonarola's evidence of heresy and we considered it credible, as long as the Church at the time did not find him guilty of heresy we can confidently assume that he remained the Pope.
I don't see how that argument applies to the present situation. Cardinal Billot was certainly not arguing (as you seem to be) that because a reigning Pope is accepted by the Church, he cannot be found to be a heretic. That would be a patently absurd argument.
If you are arguing only that Pope John XXIII cannot now be found to be a heretic, you may have a point. In that case, because no one at the time accused him of being a heretic (that I am aware of), then we cannot now revise our acceptance of his claim. That sounds like a good argument to me. But I am open to counter-arguments.
However, I definitely would not agree with you if you were claiming that because Cardinal Mueller, Cardinal Kasper, Cardinal O'Malley, Cardinal Mahoney, et al accept Pope Francis as the true Pope, we must all accept him. That's a ridiculous idea. If you believe that you already have one foot in the door of the Conciliar Sect.
But does the whole Church accept Francis as Pontiff?
You're finally asking the right questions. According to all theologians, this is the only question that matters. Because so long as a Pope has universal acceptance, it is impossible absolutely that he be a true heretic.
So whose recognition counts as a true recognition? And if a Pope becomes a heretic in fact, who will immediately cease to recognize him? First and foremost, the bishops/ordinaries, secondly the Roman clergy.
I believe that last paragraph is contrived. Please provide a source. Even if you can docuмent such a claim, there were at least two ordinaries who thought Paul VI was a public heretic - namely, Archbishop Thuc and Bishop Castro de Mayer.
This is well known and standard doctrine, (frequently mentioned by the Society) although almost no modern sedevacantist (with few exceptions) takes it into account.
What is "well known and standard doctrine"? Are you talking about the idea that a reigning Pope cannot possibly be found to be a heretic? Or are you talking about the idea that only ordinaries and Roman clergy are permitted to judge a claim to the papacy?
To see or to know something is not to judge. -St. Francis de Sales.
Through tacit resignation, accepted by the law itself, all offices become vacant ipso facto and without any declaration if a cleric: ...n.4. Has publicly forsaken the Catholic Faith.
[8] But though we, or an angel from heaven, preach a gospel to you besides that which we have preached to you, let him be anathema. [9] As we said before, so now I say again: If any one preach to you a gospel, besides that which you have received, let him be anathema.
There is no Catholic God.