Since His Excellency is promoting the opinions of John of St. Thomas and Cajetan regarding a manifestly heretical pope, I will post an analysis of their theological opinions made some time ago, while also comparing them with the position of St. Robert Bellarmine (the opinion of Cajetan that His Excellency presents in number III is refuted by St. Robert in the part marked in red). I would ask those in regular contact with His Excellency to please show him this text.
The source for most of the quotes is the famous book by Arnaldo Xavier da Silveira, pp. 161-171.

There are two main positions regarding this matter. One states that a manifestly heretical pope would immediately lose all jurisdiction, i.e. the pontificate. The other states that a manifestly heretical pope does not lose the pontificate, unless the cardinals or the bishops issue a declaration informing the Church of his heresy.

The opinions of those theologians who take the latter position differ in some important points. The two most eminent proponents of this opinion, Cajetan (who originated the opinion) and Suarez (who followed it) both founded their opinions on an error regarding jurisdiction in heretics. I will demonstrate by quoting Suarez himself (who also exposes Cajetan's other crucial error) from Xavier da Silveira's book:

Suarez:

Later, on treating the penalties of the heretics, we will indicate still other authors, and in a general manner show that by divine law no one is deprived of dignity and ecclesiastical jurisdiction because of the crime of heresy.

This assertion is an error contrary to the unanimous teaching of the Fathers and to Divine Law, as St. Robert Bellarmine successfully proves futher below. It is also contrary to Canon 188 of the 1917 Code of Canon Law, which interprets Divine Law by stating that all who publicly defect from the Faith automatically lose all jurisdiction.

Suarez:

Now we will give an a prior argument: since such a destitution is a most grave penalty, one would only incur it ipso facto if it were expressed in the divine law; however we do not find any law which establishes this, either in general as far as the heretics are concerned, or in particular as to the Bishops, nor in a very particular way as far as the Pope is concerned.
Neither is there a certain Tradition over this matter. Nor can the Pope lose his dignity ipso facto by virtue of a human law, for this law would have to be established by an inferior, that is, by a Council, or by an equal, that is, by a previous Pope; but neither a Council nor a previous Pope posses such a coercive power as to be able to punish their equal or superior. Therefore, etc.
(…) You will say that there could be a law interpreting divine law. But this would be without foundation for you do not quote any such divine law; furthermore, up to now there has not been laid down by the Councils or by the Popes any law which had interpreted such a divine law.

This is where Suarez actually unknowingly disproves his own theory, since he says that if it were Divine Law that heretics are automatically deprived of jurisdiction then the penalty would indeed occur ipso facto, and he cites as evidence the lack of any Tradition which would support that automatic deprival of jurisdiction.
However, St. Robert Bellarmine proves that this is precisely a matter of Divine Law (it is based "on the very nature of heresy") and he proves that this is the opinion of all the ancient Fathers. Furthermore, with the promulgation of the 1917 Code of Canon Law, even the third paragraph of Suarez is disproved, since there exists at least since that time a law interpreting Divine Law regarding this matter (Canon 188).

Suarez:

Finally, the faith is not absolutely necessary in order that a man be capable of spiritual and ecclesiastical jurisdiction and be able to exercise true acts which demand this jurisdiction; then, etc. The foregoing is obvious, granted that, as is taught in the treatises on penance and censures, in case of extreme necessity a priest heretic may absolve, which is not possible without jurisdiction.

This is his second error, since he mistakes supplied jurisdiction (which is what happens in the example he gives), which the Church supplies for certain acts, with the jurisdiction of a heretical minister, which is always non-existent (see http://www.newadvent.org/summa/3039.htm#article3). Because of this error and the previously mentioned error which says that heretics are not deprived of jurisdiction Suarez is able to support Cajetan's opinion that a Pope who became a manifest heretic would continue to hold jurisdiction, something which St. Robert showed to be impossible and contrary to the unanimous teaching of the Fathers (whose unanimous teachings no Catholic is free to disagree with).

Suarez:

From this arises a third doubt: by what right could the pope be judged by the assembly, being superior to it? (1). In this matter Cajetan makes extraordinary efforts to avoid seeing himself forced to admit that the Church or a Council are above the Pope in case of heresy; he concludes in the end that the Church and the Council are superior to the Pope, not as Pope, but as a private person. This distinction however does not satisfy, for with the same argument one would be able to say it belongs to the Church to judge or to punish the Pope, not as Pope, but as a private person (…)

Here he exposes the error behind Cajetan's opinion, and it is the error that the Church has the right "to judge or to punish the Pope, not as Pope, but as a private person".

St. Robert Bellarmine's refutation of those opinions (I will quote it in full for the sake of those who are not able to get ahold of Xavier da Silveira's book):

“The fourth opinion is that of Cajetan, for whom (de auctor, papae et conc., cap. 20 et 21) the manifestly heretical Pope is not ipso facto deposed (6), but can and must be deposed by the Church. To my judgement, this opinion cannot be defended. For, in the first place, it is proven with arguments from authority and from reason that the manifest heretic is ipso facto deposed. The argument for authority is based on Saint Paul (Epist. ad Titum, 3), who orders that the heretic be avoided after two warnings, that is, after showing himself to be manifestly obstinate – which means before any excommunication or judicial sentence. And this is what Saint Jerome writes, adding that the other sinners are excluded from the Church by sentence of excommunication, but the heretics exile themselves and separate themselves on their own from the body of Christ. Now, a Pope who remains Pope cannot be avoided, for how could we be required to avoid our own head? How can we separate ourselves from a member united to us?

This principle is most certain. The non-Christian cannot by any means be Pope, as Cajetan himself admits (ibidem, cap. 26). The reason for this is that he cannot be head of what he is not a member; now he who is not a Christian is not a member of the Church, and a manifest heretic is not a Christian, as is clearly taught by Saint Cyprian (lib. 4, epist. 2), Saint Athanasius (Ser. 2 cont. Arian.), Saint Augustine (lib. de grat. Christ. cap. 20), Saint Jerome (cont. Lucifer.) and others; therefore the manifest heretic cannot be Pope.

To this Cajetan responds (in Apol. pro tract. Praedicto cap. 25 et in ipso tract. cap. 22) that the heretic is not a Christian simpliciter, but is one secundum quid. For, granted that two things constitute the Christian – the faith and the character – the heretic, having lost the faith, is still in some way united to the Church and is capable of jurisdiction [this is the principal error of both Cajetan and Suarez]; therefore, is also Pope, but ought to be removed, since he is disposed, with ultimate disposition to cease to be Pope; as the man who is still not dead but is in extremis.

Against this; in the first place, if the heretic, by virtue of the character, would remain, in actu, united to the Church, he would never be able to be cut or separated from her in actu, for the character is indelible. But there is no one who denies that some people may be separated in actu from the Church. Therefore, the character does not make the heretic be in actu in the Church, but is only a sign that he was in the Church and that he must return to her. Analogously, when the sheep wander lost in the mountains, the mark impressed on it does not make it be in the fold, but indicates from which fold it had fled and to which fold it ought to be led back. This truth has a confirmation in Saint Thomas who says (Summa Th. III, 8, 3) that those who do not have faith are not united in actu to Christ, but only potentially – and Saint Thomas here refers to the internal union, and not to the external which is produced by the confession of faith and visible signs. Therefore, as the character is something internal, and not external, according to Saint Thomas the mere character does not unite a man, in actu, to Christ.

Further against the argument of Cajetan; either faith is a disposition necessary simpliciter for someone to be Pope, or it is only necessary for someone to be Pope more perfectly (ad bene esse). In the first hypothesis, in case this disposition be eliminated by the contrary disposition, which is heresy, the Pope immediately ceases to be Pope; for the form cannot maintain itself without the necessary dispositions. In the second hypothesis, the Pope cannot be deposed by reason of heresy, for otherwise he would also have to be deposed for ignorance, improbity, and other similar causes, which impede the science, the probity and the other dispositions necessary for him to be Pope in a more perfect way (ad bene esse papae). In addition to this, Cajetan recognizes (tract. praed., ca. 26) that the Pope cannot be deposed for the lack of the dispositions necessary, not simpliciter, but only for greater perfection (ad bene esse).

To this, Cajetan responds that faith is a disposition necessary simpliciter, but partial, and not total; and that, therefore, the faith disappearing the Pope can still continue being Pope, by reason of the other part of the disposition, which is the character, which still endures.
Against this argument: either the total disposition, constituted by the character and by faith, is necessary simpliciter, or it is not, the partial disposition then being sufficient. In the first hypothesis, the faith disappearing there no longer remains the disposition simpliciter necessary, for the disposition simpliciter necessary was the total, and the total no longer exists. In the second hypothesis, the faith is only necessary for a more perfect manner of being (ad bene esse), and therefore its absence does not justify the deposition of the Pope. In addition to this, what finds itself in the ultimate disposition to death, immediately thereafter ceases to exist, without the intervention of any other external force, as is obvious; therefore, also the Pope heretic ceases to be Pope by himself, without any deposition.

Finally, the Holy Fathers teach unanimously not only that heretics are outside of the Church, but also that they are ipso facto deprived of all ecclesiastical jurisdiction and dignity. Saint Cyprian (lib. 2, epist. 6) says: “we affirm that absolutely no heretic or schismatic has any power or right”: and he also teaches (lib. 2, epist. 1) that the heretics who return to the Church must be received as layman, even though they have been formerly priests or bishops in the Church. Saint Optatus (lib. 1 cont. Parmen.) teaches that heretics and schismatics cannot have the keys of the kingdom of heaven, nor bind nor loose. Saint Ambrose (lib. 1 de poenit., ca. 2), Saint Augustine (in Enchir., cap. 65), Saint Jerome (lib. cont. Lucifer.) teach the same (…).

Pope Saint Celestine I (epist. ad Jo. Antioch., which appears in Conc. Ephis., tom. I, cap. 19) wrote: “It is evident that he has remained and remains in communion with us, and that we do not consider destitute, anyone who has been excommunicated or deprived of his charge, either episcopal or clerical, by the Bishop Nestorius or by the others who followed him, after these latter commenced preaching heresy. For the sentence of him who has already revealed himself as one who must be deposed, can depose no one”.
And in a letter to the Clergy of Constantinople, Pope Saint Celestine I says: “the authority of our Apostolic See has determined that the Bishop, cleric or simple Christian who had been deposed or excommunicated by Nestorius or his followers, after the latter began to preach heresy shall not be considered deposed or excommunicated. For he who had defected from the faith with such preachings, cannot depose or remove anyone whatsoever”.
Saint Nicolas I (Epist. ad Michael) repeats and confirms the same. Finally, Saint Thomas also teaches (S. Theol., II-II, 39, 3) that schismatics immediately lose all jurisdiction, and that anything they try to do on the basis of any jurisdiction will be null.

There is no basis for that which some respond to this: that these Fathers based themselves on ancient law, while nowadays, by decree of the Council of Constance, they alone lose their jurisdiction who are excommunicated by name or who assault clerics. This argument, I say has no value at all, for those Fathers, in affirming that heretics lose jurisdiction, did not cite any human law, which furthermore perhaps did not exist in relation to the matter, but argued on the basis of the very nature of heresy. The Council of Constance only deals with the excommunicated, that is, those who have lost jurisdiction by sentence of the Church, while heretics already before being excommunicated are outside the Church and deprived of all jurisdiction. For they have already been condemned by their own sentence, as the Apostle teaches (Tit. 3, 10-11), that is they have been out from the body of the Church without excommunication, as Saint Jerome explains (1).

Besides that, the second affirmation of Cajetan [the second error of Cajetan], that the Pope heretic can be truly and authoritatively deposed by the Church, is no less false than the first. For if the Church deposes the Pope against his will it is certainly above the Pope; however, Cajetan himself defends, in the same treatise, the contrary of this. Cajetan responds that the Church, in deposing the Pope, does not have authority over the Pope, but only over the link that unites the person to the Pontificate. In the same way that the Church in uniting the Pontificate to such a person, is not because of this above the Pontiff, so also the Church can separate the Pontificate from such a person in case of heresy, without saying it be above the Pope.

But contrary to this it must be observed in the first place that, from the fact that the Pope deposes Bishops, it is deduced that the Pope is above all the Bishops, though the Pope on deposing a Bishop does not destroy the episcopal jurisdiction, but only separates it from that person. In the second place, to depose anyone from the Pontificate against the will of the deposed, is without doubt a penalty; then, the Church, on deposing a Pope against his will, is without doubt punishing him; however, to punish is proper to a superior or to a judge. In the third place, given that according to Cajetan and the other Thomists, in reality the whole and the parts taken as a whole are the same thing, he who has authority over the parts taken as a whole, being able to separate them one from another, has also authority over the whole itself which is constituted by those parts.

The example of the electors, who have the power to designate a certain person for the Pontificate, without having however power over the Pope, given by Cajetan, is also destitute of value. For when something is being made, the action is exercised over the matter of the future thing, and not over the composite, which still does not exist, but when a thing is being destroyed, the action is exercised over the composite, as becomes patent on consideration of the things of nature. Therefore, on creating the Pontiff, the Cardinals do not exercise their authority over the Pontiff for this still does not exist, but over the matter, that is, over the person who by the election becomes disposed to receive the Pontificate from God. But if they deposed the Pontiff, they would necessarily exercise authority over the composite, that is, over the person endowed with the pontifical power, that is over the Pontiff.”

No wonder he is a Doctor of the Church! He successfully defeated both the arguments of Cajetan and Suarez and exposed the errors behind them. It is necessary to say that both Cajetan and Suarez are very eminent theologians, but they are not infallible, and they made at least two crucial errors when formulating and presenting their theory. Those errors are understandable since the theology behind them was not as developed at that time, but they are errors nonetheless.

St. Robert Bellarmine:

"Therefore, the true opinion is the fifth, according to which the Pope manifestly a heretic ceases by himself to be Pope and head, in the same way as he ceases to be a Christian and a member of the body of the Church; and for this reason he can be judged and punished by the Church. This is the opinion of all the ancient Fathers, who teach that manifest heretics lose immediately all jurisdiction...

This is the opinion of great recent doctors, as John Driedo (lib. 4 de Script. et dogmat. Eccles. Cap. 2, par. 2, sent. 2), who teaches that they only separate themselves from the Church who are expelled, like the excommunicated, and those who by themselves depart from her or oppose her, as heretics and schismatics. And in his seventh affirmation, he sustains that in those who turn away from the Church, there remains absolutely no spiritual power over those who are in the Church. Melchior Cano says the same (lib. 4 de loc., cap. 2), teaching that heretics are neither parts nor members of the Church, and that it cannot even be conceived that anyone be head and Pope, without being member and part (cap. ult. ad argument. 12). And he teaches in the same place, with plain words, that occult heretics are still of the Church, they are parts and members, and that therefore the Pope who is an occult heretic is still Pope. This is also the opinion of the other authors whom we cite in book I De Eccles."

Now, to address the arguments of others who are said to have adopted Cajetan's position:

Billuart:

“According to the more common opinion, Christ, by a particular providence, for the common good and the tranquility of the Church, continues to give jurisdiction to an even manifestly heretical pontiff until such time as he should be declared a manifest heretic by the Church” (Billuart, De Fide, diss. V, a. III, § 3, obj. 2).

Billuart adopts the theory of Cajetan and Suarez, takes away the errors which were the basis for the theory in the first place (jurisdiction remaining in heretics and the Church being able to judge the person of the Pope), and replaces them with the reason: "for the common good and the tranquility of the Church".

Thus the justification for the truthfulness of this whole theory changed from the Church being able to judge the Pope as a private person (error, probably proximate to heresy) and jurisdiction being able to remain in manifest heretics (error) to God leaving heretics as popes "for the common good and the tranquility of the Church", until a purely informative declaration is made.

As is probably apparent, this reason is highly problematic because it remains to be demonstrated if and how the Church would profit from a heretic being sustained in the Pontificate - how would such a thing be "for the common good and the tranquility of the Church"?. If he were automatically deprived, his heretical acts and teachings could no longer hurt the Church - the Church would be protected from them, but if he were to remain Pope, then he could still continue to push his heretical agenda and actively attack the Catholic Faith, using his papal authority to try to impose his errors (as Francis and his predecessors have been doing).

Also, to quote Xavier da Silviera:

"It is unquestionable that the concrete application of this opinion [St. Robert Bellarmine's] in the eventual case of a Pope heretic could occasion the gravest confusions and afflictions for the Church. It seems to us nevertheless, that supposing the hypothesis of a Pope heretic, these confusions and afflictions would follow ineluctably, whatever be the sentence of the theologians which one had adopted. Considering things only from the point of view of the schisms, the confusions and the rivalries which could arise, we do not see how to prefer one of the opinions to the rest. We shall take as an example only the position of Suarez: What divisions could not arise if some cardinals and bishops declared the Pope a heretic, while others supported him!

We believe however that the point of view from which the question ought to be focused is not this. It is not a question fundamentally, of asking which is the solution which would conserve “peace” better, but rather asking which would conserve the faith better, and which would be more in accord with the divine institution of the Church. And from this point of view, as we will say further on (pp. 172 ff.), we judge that there are solid reasons to embrace, with Saint Robert Bellarmine, Wernz-Vidal and others, the fifth opinion."

John of St. Thomas also bases his reasoning on an error - that manifest heretics remain members of the Church until the Church condemns them:

"So long as he has not become declared to us juridically as an infidel or heretic, be he ever so manifestly heretical according to private judgment, he remains as far as we are concerned a member of the Church and consequently its head. Judgment is required by the Church"

This would mean that a person who, for example, openly denied Christ's Resurrection, even when made aware of the teaching of the Church by other Catholics, would continue to remain a Catholic until and unless the authority of the Church declared him a heretic. This, apart from being an error, is also ridiculous to hold today (although at the time some believed so).

And Garrigou-Lagrange, who some put among the proponents of Cajetan's opinion, only says that the Pope who is a secret heretic remains the Pope (which is the teaching of St. Robert as well) - he does not discuss the situation of a Pope who is a manifest heretic:

“This condition is quite abnormal, hence no wonder that something abnormal results from it, namely, that the pope becoming secretly a heretic would no longer be an actual member of the Church [the soul], according to the teaching as explained in the body of the article, but would still retain his jurisdiction by which he would influence the Church [the body] in ruling it. Thus he would still be nominally the head of the Church, which he would still rule as head, though he would no longer be a member of Christ, because he would not receive that vital influx of faith from Christ, the invisible and primary head. Thus in quite an abnormal manner he would be in point of jurisdiction the head of the Church, though he would not be a member of it.

“This condition could not apply to the natural head in its relation to the body, but such a condition is not repugnant in the case of the moral and secondary head. The reason is that, whereas the natural head must receive a vital influx from the soul before it can influence the members of its body, the moral head, such as the pope is, can exercise his jurisdiction over the Church, although he receives no influx of interior faith and charity from the soul of the Church. More briefly, as Billuart says, the pope is constituted a member of the Church by his personal faith, which he can lose, and his headship of the visible Church by jurisdiction and power is compatible with private heresy. The Church will always consist in the visible union of its members with its visible head, namely, the pope of Rome, although some, who externally seem to be members of the Church, may be private heretics”

In conclusion, three of the four eminent theologians who held the theory in question based their opinion on errors (reg. doctrines obviously insufficiently understood at the time), and the fourth based it, as it appears, merely on the belief that that would best safeguard "the common good and the tranquility of the Church".

BOTH of the extreme sides on this issue have problems, but the distinctions made by sedeprivationism (formal vs. material) solve the problem.

Where some of the confusion comes from also is from the failure of all these groups to make the following distinction (expressed via the following scenarios):

1) Pope Francis says, "I have become a Buddhist."  IPSO FACTO deposition by ceasing to be a member of the Church for failure to profess the Catholic faith.  Conclave could immediately elect a successor without any further declaration.  If Francis, however, claimed that he is still pope (so that he could use the Vatican funds to aid poor people for instance), the Church would declare that he was no longer the pope (this would only be declaratory and would not effect deposition).

2) Pope Francis says, "I know the Church teaches transubstantiation but I don't believe it anyway."  IPSO FACTO deposition by way of obvious pertinacious heresy.

3) Pope Francis says, "I believe in religious liberty.  I think that it's consistent with Tradition."  There's no ceasing to profess Catholicism, just a debate over whether a certain teaching could be reconciled with Tradition given the appropriate distinctions.  In fact, arguing that a certain doctrine is in conformity with Tradition is prima facie evidence that the person has not committed any formal heresy, since it seems to be a matter of importance to him that the doctrine be reconcilable with Tradition.

St. Robert Bellarmine's position applies to scenarios #1 and #2.

#3 however requires some form or mode of intervention from the Church's authority ... i.e. some form of the non-Bellarmine position.

#3 would be a gradual process whereby first Pope Francis would be admonished.  If he remains pertinacious, then an Imperfect Council could decide the matter of whether in fact his propositions were heretical.  If judged heretical, he would be asked to recant.  If he recanted, then no deposition ever took place.  If he persisted, then the Church would declare him deposed.  This declaration would not effect the deposition but would only declare it to be the case.

But even #3 involves a conundrum.  It would be a pitting of the Ordinary Universal Magisterium against the Papal Magisterium if the pope were to come out and declare the contrary of what the Imperfect Council concluded, and OUM only works when in submission to the Holy Father.  So you'd have a stalemate.

No need for all of these complications. Francis has said and done much worse than #3, much more basic denial of infallible doctrine, and much more unambiguously.

Also, ignorance is not an excuse for someone who has already been taught what the Church infallibly teaches regarding a doctrine (which a priest, and especially a bishop, and especially a cardinal certainly has been). Otherwise absolutely any heretic (a layman, a priest, a bishop etc.) could claim that he is not a heretic because he believes his heresies are compatible with the Christian Faith, and we would have no way to disprove it except by waiting for an official condemnation.

An example might help - if what you said is true then if Francis publicly said "I do not believe in the Incarnation" and then said that he believes his view is consistent with Tradition, we would not know that he is a heretic, which, of course, we would.

No need for all of these complications. Francis has said and done much worse than #3, much more basic denial of infallible doctrine, and much more unambiguously.

Bull.  You are, like most sedevacantists, blinded by bitter zeal; you are so consumed with and defined by contempt that you refuse to look objectively at truth.