http://strobertbellarmine.net/books/Concerning_A_SSPX_Dossier_on_Sedevacantism.pdf

7. Fr. Boulet then provides this summary of the so-called Fifth Opinion.

5th Opinion: “If he was to fall into a manifest heresy, the Pope would ipso facto lose his pontificate”. Some authors say that the Pope would lose his pontificate ipso facto at the very moment when his heresy becomes external; some others maintain that the heretical Pope would lose his pontificate only when his heresy becomes notorious and publicly spread. Among the 5 opinions studied by St. Robert Bellarmine, this 5th opinion appears to be the most probable. a) Let us be completely clear. Bellarmine does not say that it is the most probable. He says that it is “true,” and that it is the doctrine of all of the Fathers.

"Therefore, the true opinion is the fifth, according to which the Pope who is 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 immediately lose all jurisdiction, and outstandingly that of St. Cyprian…”6 [6 St Robert Bellarmine, De Romano Pontifice, lib. II, cap. 30.]

b) Bellarmine then cites several fathers, other doctors and Popes, to prove that this opinion is the constant tradition of Holy Church.
 
8. Fr. Boulet proceeds at this point to open up the question of whether a Pope may teach heresy when acting as Pope – that is, officially.

3. The heretical Pope:

3.1. Can a Pope be heretic? It has been taught by various Popes that a Pope can teach heresy against the Faith. Pope Adrian VI († 1523) stated that: “If by the Roman Church you mean its head or pontiff, it is beyond question that he can err even in matters touching the faith. He does this when he teaches heresy by his own judgement or decretal. In truth, many Roman pontiffs were heretics. The last of them was Pope John XXII († 1334).” Venerable Pope Pius IX († 1878) recognised the danger that a future Pope would be a heretic and “teach […] contrary to the Catholic Faith", and he instructed, "do not follow him". He said: "If a future Pope teaches anything contrary to the Catholic Faith, do not follow him." (Letter to Bishop Brizen).

Now this is frankly scandalous doctrine.  It seems that Fr. Boulet has been deceived by fraudulent quotes which he has carelessly lifted from some place unknown, but since he does not provide references we cannot check.

A little research has revealed, however, two modern sources for the first of those two quotes. Apparently it may be found in a book by the theologian Viollet, Papal Infallibility and the Syllabus, (1908).  This book was placed on the Index of Forbidden Books by St. Pius X.  The same scandalous and erroneous, if not heretical, quote may also be found in the anti-Catholic work entitled Vicars of Christ, the Dark Side of the Papacy, by Peter De Rosa (p. 204).

Here are two reviews of this latter work, found on Amazon.com:

From Publishers Weekly

De Rosa (Prayers for Pagans and Hypocrites) is an angry Catholic. In the worst proselytizing tradition, this devil's advocate overstates familiar arguments, bludgeoning the reader with his dossier against the Church. Among De Rosa's tamer charges: Jesus renounced possessions, but his vicars celebrate high mass garbed in cloth of gold; the Church has never lifted strictures against usury, yet the Vatican operates a bank. De Rosa sweeps through Church history to parade Popes who begat children, Popes who fornicated on a grand scale, Popes who married. Then in the second half of this polemic, he addresses Church teaching, conjoining the "immaculate conception" doctrine to decrees governing birth control, abortion, celibacy. The doctrine of papal infallibility is dealt with, as is Church anti-Semitism through the ages leading to the h0Ɩ0cαųst silence of Pius XII, the "one man in the world whose witness Hitler feared." And in wrapping up his catalog of "the sins of the papacy," De Rosa virtually dismisses internal reform: "It is not Catholics but other Christians who chiefly can make the papacy what it ought to be."

From Library Journal

In his history of the papacy, former Jesuit De Rosa aims to undermine belief in papal infallibility. Although he claims to be a friend of the Catholic Church, and does at times express admiration for the holiness of many of the Popes, his book is so heavily weighted with information on the corruption of the Papacy that it would be hard for any reader to see any good in the office. The book cannot be faulted historically or stylistically, though most of the information including the most sordid can be found in the standard Roman Catholic sources. Patrick Grainfeld's The Limits of the Papacy (Crossroad, 1987) offers a more balanced view of the expansion of papal power. Augustine J. Curley, Newark Abbey, N.J.

So much for the first of the two quotes which aim to prove the scandalous doctrine that Popes may officially  teach  heresy.    The  other  quote  refers  to  a  letter  to  "Bishop  Brizen."  Once  again,  no reference is given by Fr. Boulet. Now, my best guess as to the identity of this person is Bishop Vincent Ferrer Gasser, who was Bishop of Brixen from 1856-79. He was also Relator of the Faith at the Vatican Council. Pope Pius IX would probably have written him letters, but no particular letter or published source is provided. The content of the quote is less scandalous than that of the first, but still clearly erroneous.

A man who claims to be Pope can teach heresy even if is not trying to define something.  If he does so publically the wolf in sheep's clothing dons the cloths of the wolf.

Define Pope. Heresy and Pope don't go together. Like water and oil. Heresy, MANIFEST is automatic excommunication. Vatican II had no definitions, because to do so would show the heretics off, one being Ratzinger. Infallibility has it's definition as we know. So does Pope. Matters relating to infallibility include the Holy Sacrifice of the Mass, the 7 sacraments and the teachings of the Catholic Church because they were established by Christ. Pope can not change them. Any True Pope in their right mind would see to it that the sacraments would be unchanged, and so far, that has not happened since Vat. II. A True Pope would put all the sacraments back to the way Christ instituted them. If a Pope does not believe this, he is not catholic and was not when nominated (Manifest) and therefore if elected would be invalid. And yes, we may judge, "you will know them by their fruits!".As for indefectibility , is meant that the Church that CHRIST founded will last until the end of time and that gives us no clue as to the number of people or any at all. Indefectibility also implies UNCHANGEABLE. Therefore, we have a lot destroyed thanks to New Order. Be sure to read Matthew 24 and Chapter 12 of Daniel. The Mass will come to an end, the continual sacrifice and the Blood that Christ shed is eternal. I do envison that our faith will be taken to crucifixion and we will resurrect. There are many people who do not know the Continual sacrifice coming to an end. It is reasonable to believe for this to take place would be anti-popes.

Well put.

Define Pope.

"A Catholic who . . ."

We have to stop right their because Ratzinger is not a Catholic.

I saw mention of valid elections.

I believe Catholics have to elect their head as well.

Not sure how many "Cardinals" were Catholic in 1978.

I would venture to say that none of the "Cardinals" were Catholic in 2005.

Remember the false-popes tried to make sure no valid Cardinals would elect as they put an age limit on who could "elect".