According to theologians post Bellarmine and pre Vatican II, manifest/public heretics depends on the large number of people their heresy is made manifest to.  It does not matter if they are ignorant (material) nor willful (formal).  See Van Noort Dogmatic Theology, Members of the Church.

2Vermont, I think in the passage you may be thinking of, Van Noort is speaking of Protestants. Van Noort says some hold even Protestant material heretics to be members of the Church, but most others teach that they are not externally united as members.

Similarly, when it is said by Pope Pius XII, that the sin of heresy is such as of its own nature to cause a person to be separated from the Church, since only formal heretics have committed the sin of heresy, it seems clear that the reference is to formal heretics.

As for St. Robert Bellarmine, he said the heretic must at least show himself to "be manifestly obstinate" as was mentioned earlier. Maybe a Council could make a presumption of heresy even for material heresy, but I don't think that laity can. 

Here's a study by Xavier Da Silveira: "As is obvious, we are not discussing the possibility of the Pope being in material heresy. No one denies, that mistakenly or by inadvertence, the Supreme Pontiff can fall into material heresy, as a private person"

Now, as regards what was being discussed earlier, I'm open to the idea as a hypothesis that a Council of Bishops, after warnings, could determine the Pope "to be manifestly obstinate". But what happens if the Council rebukes him, and then the Pope retracts?

So personally - and ironically I'm agreeing with Fr. Kramer here, though I commend Siscoe and Salza for the publication of their book, which has been endorsed by the Society of Saint Pius X, and already proved helpful for many - I don't believe a Pope will ever lose the Faith. The reason for this, although St. Robert and Bp. Gasser at Vatican I, called this only a pious opinion and not absolutely certain, is that Our Lord Jesus indeed prayed that the Faith of St. Peter and his Successors may not fail. Now faith does not fail in someone who falls into heresy but without pertinacity, as was perhaps the case with Pope John XXII, nor does it fail if a man commits ordinary mortal sins, because then only grace is lost, but not faith; but it does fail when he becomes a formal heretic, by committing the mortal sin of heresy, and then not only sanctifying grace, but even the internal virtue of supernatural faith itself is lost. 

On the occult heretic thing, it is very unlikely occult heretics lose membership imo. Pope Pius XII said, to be members of the Church, it is necessary to be validly baptized, and to profess the true faith, among other things. So validly baptized professing Catholics are all included in that imo. If a person is an occult heretic but still claims to be Catholic, he remains a member. As for one unjustly excommunicated, I believe St. Robert himself says somewhere, that such a person remains within the Church but by internal bonds. Similarly, Pope Pius XII seems to endorse that opinion when he excludes from Church membership only those excluded by Church authorities for real "grave faults committed", not for mistaken penalties or injustices borne. Anyway, the Pope-heretic thing is broader.

But the prayer of Our Lord Jesus Christ in Luk 22:32 appears to be meant as an efficacious one, just like Our Lord's Prayer for the Unity of His Church in Jn 17. Thus, while the faithful are invited to pray for the same, e.g. that the Church's unity may be strengthened, that the Holy Father may not fail in the faith etc, nevertheless, it doesn't seem that the prayer of Jesus Christ can fail. And therefore, it seems more likely than not, that when corrected and rebuked by the Bishops in Council, the Pope will retract. Therefore, he will remain a Catholic, although one who erred graved, and once his error is corrected in Council, the Church goes on.

Cardinal Billot: "I said: ‘admitted the hypothesis”. But it appears by far more probable that this hypothesis is a mere hypothesis, never reducible to act, in virtue of what St. Luke says (22: 32): “I have prayed for you that your faith not fail, and you, once being converted, confirm your brethren”. That this ought to be understood of Saint Peter and of all his successors, is what the voice of Tradition attests ... "

Relatio of Bp. Gasser at Vatican I: "the opinion of Albert Pighius, which Bellarmine indeed calls pious and probable, was that the Pope, as an individual person or a private teacher, was able to err from a type of ignorance but was never able to fall into heresy or teach heresy.  To say nothing of the other points, let me say that this is clear from the very words of Bellarmine, both in the citation made by the reverend speaker and also from Bellarmine himself who, in book 4, chapter VI, pronounces on the opinion of Pighius in the following words: "It can be believed probably and piously that the supreme Pontiff is not only not able to err as Pontiff but that even as a particular person he is not able to be heretical, by pertinaciously believing something contrary to the faith."  From this, it appears that the doctrine in the proposed chapter is not that of Albert Pighius or the extreme opinion of any school, but rather that it is one and the same which Bellarmine teaches in the place cited by the reverend speaker and which Bellarmine adduces in the fourth place and calls most certain and assured, or rather, correcting himself, the most common and certain opinion". Anyway, it's only a Council of Bishops, after warning the Pope, that can determine pertinacity or the lack thereof for sure imo. We can determine an error.

I look forward to your thoughts. I found it very powerful and reasonable. I will have to read it again myself.

I just finished reading it, and it's a great article.  Because I haven't had the time to really study it, some of his train of thought lost me about halfway through.  I need to have the time to read it more closely.  It's certainly worth studying.  His citations from Pius XII do in fact suggest that there can be no real membership in the Church without supernatural faith.  Msgr. Fenton does not thoroughly address all of these and I feel that his refutation was inadequate.  More than anything, it was based on the assertion that, well, all these theologians who follow Bellarmine can't all be wrong.  But Father Lawlor isn't saying that they are completely wrong, or wrong outright, but that the Bellarmine position must be refined.  Bellarmine was so intent on defending the visibility of the Church, that he went to an extreme, leading to a purely material conception of the Church ... as suggested by the fake Catholic theory.  This was of course due to the Protestant heresy of portraying the Church as entirely invisible.  He was arguing for a subtle refinement of the thinking.

He states that a fair number of "modern" (in his day) authors, including Franzelin, hold that occult heretics are not members of the Church.

I think that there can be many lessons regarding the truth of the Mystical Body by considering the human body.  God clearly designed the human body in such a way as to help teach us about the reality of the Church, foreknowing before He designed and created it that He Himself would liken the Church to the body.

So, for instance, we can say that the Baptismal character is very much like the DNA that identifies the various cells of the body.

If some non-baptized person attaches himself to the Body, it does not share the DNA and is not properly part of the body but has merely attached itself to the body materially or per accidens like some parasite.

Just as we say that the parasite is not actually part of the body, even though it has managed to attach itself to or even burrow into it.

I'm going to keep meditating on these metaphors.

So with this metaphor we can rule out the speculative theory that non-baptized Catholics are members of the Body of Christ.

SOME of it was, 2Vermont, not all.  Just because some of his thinking as adopted doesn't mean that all of it was.  I think there's a quote from one of the Vatican I fathers to the effect that the heretical pope issue was not being definitively settled.  So, for instance, just because some pieces of St. Robert were adopted by the Church, this does not mean the Church endorsed his theory that even a fake Catholic was a member of the Church.

No, but it still seems inappropriate at best to place him on the same level as all other theologians given he is a canonized Saint and Doctor of the Church. Were there any other theologians who spoke to these issues who the Church honored in the same way?

No, but it still seems inappropriate at best to place him on the same level as all other theologians given he is a canonized Saint and Doctor of the Church. Were there any other theologians who spoke to these issues who the Church honored in the same way?

Nobody's placing him "on the same level," but that does not mean he's infallible, nor does it mean that every single position of his has been adopted by the Church.  Not a few canonized Church Fathers held opinions that were later rejected as heretical.  Does this mean that he got it "more right" than many others on a lot of issues?  I think that's a fair conclusion.  But to say that his opinion on any given is definitive merely because he's been declared a Doctor is an exaggeration.  We've had canonized Doctors disagreeing with one another on various issues.

2Vermont, in the areas where the Church has adopted +Bellarmine’s thinking, he has been properly elevated for his “Doctor” status.  In all other areas, his opinion is on the “same level” as all other theologians.  This is the proper distinction. 

St. Robert Bellarmine's Refutation of the Second Opinion

The argument: "Thus, the second opinion is that the Pope, in the very instant in which he falls into heresy, even if it is only interior, is outside the Church and deposed by God, for which reason he can be judged by the Church. That is, he is declared deposed by divine law, and deposed de facto, if he still refused to yield." The argument sets forth that even if the heresy is only interior, the heretic is already deposed by God, for which reason, he can be judged by the Church. "That is, he is declared deposed by divine law, and deposed de facto, if he still refused to yield." Therefore, having already been deposed by God and no longer pope, the Church can judge him by declaring him deposed by divine law; and if he refuses to yield, the Church can actually depose him de facto. Thus, it is not even a question of judging or deposing an actual pope, but of judging anddeclaring deposed the man who was pope before he fell from office when he fell into heresy. Bellarmine counters by pointing out that "Jurisdiction is certainly given to the Pontiff by God, but with the agreement of men, as is obvious; because this man, who beforehand was not Pope, has from men that he would begin to be Pope, therefore, it is not removed by God unless it is through men. But a secret heretic cannot be judged by men to have been deposed by divine law, nor would such wish to relinquish that power by his own will." Bellarmine explains that God will not depose a heretic pope without the cooperation of men. The reason why is that a secret heretic invisibly judged and deposed by God cannot be judged by men: The man who would be invisibly deposed by God, and no longer pope, cannot be judged by men to have been deposed by God because his heresy cannot be detected. But unless that judgment be made by men, God will not invisibly withdraw his jurisdiction, because it was given with the visible cooperation of men. Thus, in Bellarmine's refutation of the second opinion, there is absolutely not even the question an antecedent judgment against a still validly reigning pontiff; but only that God will not invisibly depose the pope because the ipso facto fall from office cannot be judged by men to have taken place. The Salza/Siscoe argument which holds that the Bellarmine's refutation of the second opinion demonstrates that in the fifth opinion an antecedent judgment is required for a manifest heretic to fall from office, is entirely without foundation.  

St. Robert Bellarmine's Refutation of the Second Opinion

The argument: "Thus, the second opinion is that the Pope, in the very instant in which he falls into heresy, even if it is only interior, is outside the Church and deposed by God, for which reason he can be judged by the Church. That is, he is declared deposed by divine law, and deposed de facto, if he still refused to yield." The argument sets forth that even if the heresy is only interior, the heretic is already deposed by God, for which reason, he can be judged by the Church. "That is, he is declared deposed by divine law, and deposed de facto, if he still refused to yield." Therefore, having already been deposed by God and no longer pope, the Church can judge him by declaring him deposed by divine law; and if he refuses to yield, the Church can actually depose him de facto. Thus, it is not even a question of judging or deposing an actual pope, but of judging anddeclaring deposed the man who was pope before he fell from office when he fell into heresy. Bellarmine counters by pointing out that "Jurisdiction is certainly given to the Pontiff by God, but with the agreement of men, as is obvious; because this man, who beforehand was not Pope, has from men that he would begin to be Pope, therefore, it is not removed by God unless it is through men. But a secret heretic cannot be judged by men to have been deposed by divine law, nor would such wish to relinquish that power by his own will." Bellarmine explains that God will not depose a heretic pope without the cooperation of men. The reason why is that a secret heretic invisibly judged and deposed by God cannot be judged by men: The man who would be invisibly deposed by God, and no longer pope, cannot be judged by men to have been deposed by God because his heresy cannot be detected. But unless that judgment be made by men, God will not invisibly withdraw his jurisdiction, because it was given with the visible cooperation of men. Thus, in Bellarmine's refutation of the second opinion, there is absolutely not even the question an antecedent judgment against a still validly reigning pontiff; but only that God will not invisibly depose the pope because the ipso facto fall from office cannot be judged by men to have taken place. The Salza/Siscoe argument which holds that the Bellarmine's refutation of the second opinion demonstrates that in the fifth opinion an antecedent judgment is required for a manifest heretic to fall from office, is entirely without foundation.  

Your reasoning here is solid, and I can find no fault with it.  This is how I think all theologians have read Bellarmine when he's talking about removing a heretical Pope -- (Fr. Kramer:  "it is not even a question of judging or deposing an actual pope, but of judging and declaring deposed the man who was pope before he fell from office when he fell into heresy")