Let's make this real simple:
St Alphonsus:
...In this case, the Church would not depose him, because no one his authority above the Pope. It (the Church) would simply declare that he had fallen from his pontificate.
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St. Robert Bellarmine:
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“A Pope who is a manifest heretic automatically ceases to be a Pope and head, just as he ceases automatically to be a Christian and a member of the Church. Wherefore, he can be judged and punished by the Church. This is the teaching of all the ancient Fathers who teach that manifest heretics immediately lose all jurisdiction.”
--Comment: You are not the Church. You have no authority to declare a pope has fallen from his pontificate. You have no authority in any ecclesiastical way, to do ANYTHING related to canon law, or Divine Law.
St Alphonsus
....must be a question of manifest and external heresy, not of an occult or mental heresy.
--Comment: Who decides if such heresy is manifest or mental? Answer: The Church.
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St. Alphonsus Liguori:
“If ever a Pope, as a private person, should fall into heresy, he should at once fall from the Pontificate. If, however, God were to permit a pope to become a notorious and contumacious heretic, he would by such fact cease to be pope, and the apostolic chair would be vacant.”
--Comment: St Alphonsus uses the word "contumacious" which means "obstinate" or stubborn. Who determines if such heresy is obstinate? THE CHURCH.
All these other opinions speak of heresy, of which we can presume they mean obstinate/pernicious. As St Paul teaches:
A man that is a heretic, after the first and second admonition, avoid: Knowing that he, that is such an one, is subverted, and sinneth, being condemned by his own judgment. (Titus 3, 10-11).-- St Paul was writing to Titus and other CLERICS on how to deal with heretics. He was not writing to tell laymen to form a heretic militia and practice wild west justice.
..Xavier da Silveira explains,“The Apostolic See being vacant, an ecclesiastical organ such as the College of Cardinals or the imperfect Council can legally declare the loss of office by the heretic who was Pope, to render the fact official and make it unequivocally known by all, proceeding to elect a new Pontiff.”.-- Again, some "ecclesiastical organ" must make a decision or a declaratory act. Laymen have no authority to do anything.
..Xavier da Silveira explains,All other opinions on how a heretic loses the pontificate presuppose at least one jurisdictional act by the imperfect Council (that is, the Council without the pope), the College of Cardinals, or some other ecclesiastical organ. The only opinion of the classical doctors that does not resort to a jurisdictional pronouncement against the still reigning pope is the fifth opinion of St. Robert Bellarmine, also adopted, complemented and enriched on some points by Ballerini, Wernz-Vidal, Billot and others..--If we compare St Bellarmine's views against ALL OTHER CLASSICAL DOCTORS, then St Bellarmine is the ONLY ONE with such an opinion. Meaning, the CONSENSUS is that there MUST be an act/decision by the ecclesiastical authorities on the matter.
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St Bellarmine was correct, in theory, that an obstinate heretic loses office. What he was incorrect on, and WAY outnumbered in, is his lack of a practical way to make this happen. Life is not a theoretical vacuum. We must have rules, and practical signs from the Church to tell us all what to do. This is part of the Church's UNITY, so that we all act, agree and believe the same. Laymen running around screaming that the pope is a heretic is ANYTHING BUT unifying. It is the definition of chaos and disorder. Christ created the papacy and the Bishops for order. If the pope falls, then the Bishops/Cardinals step in and decide on matters. Not the laity. There's no Scriptural, canon law or theological basis for a laymen to judge the pope's status or ANY cleric's status.