Mr. Conclavist, I don't know if you've read the original text of Cardinal Billot I cited earlier - your opinion is very common among modern sedevacantists, but it is mistaken and exactly backward, universal acceptance proves infallibly that a certain claimant fulfils all the conditions necessary for the validity of the election, in other words it proves that the Pope is not a public and formal heretic, and at most in material error without public pertinacity.
“Finally, whatever you still think about the possibility or impossibility of the aforementioned hypothesis (of a Pope heretic), at least one point must be considered absolutely incontrovertible and placed firmly above any doubt whatever: the adhesion of the universal Church will be always, in itself, an infallible sign of the legitimacy of a determined Pontiff, and therefore also of the existence of all the conditions required for legitimacy itself. God can permit that at times a vacancy in the Apostolic See be prolonged for a long time. He can also permit that doubt arise about the legitimacy of this or that election. He cannot however permit that the whole Church accept as Pontiff him who is not so truly and legitimately.
Therefore, from the moment in which the Pope is accepted by the Church and united to her as the head to the body, it is no longer permitted to raise doubts about a possible vice of election or a possible lack of any condition whatsoever necessary for legitimacy. For the aforementioned adhesion of the Church heals in the root all fault in the election and proves infallibly the existence of all the required conditions ... For this very reason, Alexander VI was not a false Pope, but a legitimate one. Therefore he was not a heretic ...
Pope Alexander VI was accused both of heresy and simony. Convalidation would only apply to the simony, but the fact of universal acceptance proves all the required conditions even in the internal forum are satisfied, in other words, it proves the Pope is the Pope, and so not a heretic. This is the unanimous teaching of all theologians. Wernz-Vidal say that universal acceptance is " a sign and infallible effect of a valid election."
After a Pope has been validly elected, you have to prove pertinacity in heresy which is required for loss of office - the bishops and Cardinals can declare sede vacante, after they are satisfied the Pope has become a public and formal heretic. For the sin and crime of heresy alone, the Pope can be declared to have been deposed, and lost his office. Notice that when it happens that a Pope has fallen into error, and then is contested by Cardinals and Bishops, the Pope is no longer universally accepted by the hierarchy, and therefore he can lose his office. After this, a new Pope can be elected by the Cardinals or Roman clergy.
For the person who, admonished once or twice, does not repent, but continues pertinacious in an opinion contrary to a manifest or public dogma - not being able, on account of this public pertinacity to be excused, by any means, of heresy properly so called, which requires pertinacity - this person declares himself openly a heretic. He reveals that by his own will he has turned away from the Catholic Faith and the Church, in such form that now no declaration or sentence of any one whatsoever is necessary to cut him from the body of the Church ... Therefore the Pontiff who after such a solemn and public warning by the Cardinals, by the Roman Clergy or even by the Synod, maintained himself hardened in heresy and openly turned himself away from the Church, would have to be avoided, according to the precept of Saint Paul. So that he might not cause damage to the rest, he would have to have his heresy and contumacy publicly proclaimed, so that all might be able to be equally on guard in relation to him. Thus, the sentence which he had pronounced against himself would be made known to all the Church
Pope John XXII never lost his office even though he publicly fell into material heresy, which is not heresy properly so called, but only error and does not make a man a true heretic nor cause the loss of the Papal office, because pertinacity is what gives heresy its form and is essential to the effect.
Every Catholic who examines this issue has to ask himself two questions - where is the hierarchy having succession from the Apostles today, which includes both orders and jurisdiction? Second, do the remaining valid bishops with jurisdiction who are not heretics, at this moment of time, recognize the Pope? If the answer is yes, that is an infallible proof that the Pope, at this moment, is still truly the Pope, and consequently is not a heretic in the true sense. He can, of course, become a true heretic with manifest pertinacity in the future, and after that time declared by the Church to be outside Her. Personally, I think it is important for traditional Catholics to say frankly and openly that the Pope can be deposed for heresy by the Church - the Cardinals and Bishops - if he becomes a notorious and contumacious heretic, as St. Alphonsus puts it.
Clemens Maria, the opinion that the majority of the episcopate cannot fall into error is only a pious opinion, not an absolute certitude. It is, however, certain that the entire episcopate cannot cease to exist, as you agree, and this suffices for the point that it is necessary to (1) recognize the Pope, which is all that is treated on this thread. On another thread, like the running one on episcopal consecrations, we can discuss the (2) extent to which a Pope can be resisted, and also the manner in which the new rites are harmful, namely that they do not profess the Faith in its integrity, and consequently cause a loss of numerous graces compared to the traditional form, even though they are not per se invalid or explicitly formulate any heresy, which the teaching of theologians does not in any way preclude. We will see the other reasons to be skeptical of the new rites there. Suffice here to say, that in practice, Quo Primum grants all priests the right to celebrate exclusively the true Mass, that this right was re-affirmed by a commission of nine Cardinals in 1986, is explicitly stated in Summorum Pontificuм, and that last docuмent also states all other traditional Sacramental rites may also freely be used.