I understand that formal heresy (moral and canonical) requires pertinacity. I do not agree, however, that it requires admonitions. St. Robert Bellamarine did not teach this.
"It is manifestly evident that in the clearly stated context where he speaks of warnings, that Bellarmine is not stating or implying that there is any need for canonical admonitions for a pope to lose office. It was the exponents of Opinion No. 4 who maintained that canonical admonitions are necessary for a pope to be deposed for heresy. Canonical admonitions are proper only to a superior, and are administered by a superior as the initial phase of a penal process. Bellarmine qualifies the sense of the term as he means it to be understood with the words, 'that after two censures, that is, after he appears manifestly pertinacious', thereby making it clear that the warnings are not part of a juridical procedure, but only serve the purpose of fraternal correction in order to determine whether or not the pope in question 'ceases by himself to be pope' by his heresy, ipso facto, and not by the ministerial instrumentality of any judgment at the conclusion of an official procedure, whereby the Church authorities would render a judgment on the supreme judge and ruler, and as a consequence of which the heretic Pontiff would only then fall from the Pontificate upon such judgment passed on him by his subjects and inferiors. Ballerini, a follower of Bellarmine’s opinion, understood perfectly that Bellarmine was not speaking of canonical admonitions being necessary in this context when he commented (in the above cited passage), 'In such a crisis for the faith, cannot even inferiors warn their superior by fraternal correction[?]'; and, 'For any person, even a private person, the words of Saint Paul to Titus hold' – and he demonstrates that he understood well that Bellarmine’s reason for the Pontiff’s subjects to and he demonstrates that he understood well that Bellarmine’s reason for the Pontiff’s subjects to administer the warnings would not be an official act of ecclesiastical authority, but 'a duty of charity and not of jurisdiction', for the purpose only of discerning whether or not there is pertinacity, in order to determine whether or not the individual in question is properly a heretic who has fallen from the Pontificate: 'having been once or twice corrected, does not repent, but remains obstinate in a belief contrary to a manifest or defined dogma; by this his public pertinacity which for no reason can be excused, since pertinacity properly pertains to heresy, he declares himself to be a heretic'."
Kramer, Paul. To deceive the elect: The catholic doctrine on the question of a heretical Pope (p. 500-501). Kindle Edition.
In Synopsis Theologiae Dogmaticae (1897), Tanquerey explains that in the extraordinary case of a heretical Pope or multiple doubtful Popes, begins by noting that during the First Vatican Council, Bishop Gasser took the occasion to discussed the hypothesis of a Pope falling into heresy, which was considered unlikely, but not impossible. He then explains that if such were to happen, “he would either be ipso facto deprived of the Pontificate, or the body of bishops could (indirectly) depose him, as in the case of doubtful pope: for in these extraordinary cases, the authority devolves to the episcopal body.”[42]
In Tractatus De Romano Pontifice (1891), Palmieri explains that it is God, not man, who deprives a heretical Pope of his jurisdiction, but says it doesn’t happen until the Church declares him a heretic. He then references Suarez as the authority for his position:
"If a Pope is obstinate in his heresy—obstinate, I say: for, if he heeds the Church’s admonitions, nothing further is necessary—such a Pope is deposed, not by man, but by God himself, who takes away the jurisdiction that He had given him; the Church, for her part, only declares the man to be a heretic, and then (ideoque) God deprives him of his jurisdiction." (cf. Suarez, Defensio Fidei Catholicae, lib. iv c. 7 n. 5).
In the quotation Palmieri referenced, Suarez says, “although in the case of heresy [the Pope] could be deposed, he is in truth not deposed by man but by God himself, after the declaration of a legitimate Council has preceded.” (Defensio Fidei Catholicae, lib. v c 7 n. 5).
In Sacrae Theologiae Summa (1955), Joachim Salaverri teaches the same as Palamerri, and he too references Suarez:
"Theologians concede that a general Council can licitly declare a Pope heretical, if this case [of a Pope falling into heresy] is possible, but it cannot depose him authoritatively since he is superior to the Council … see Suarez, De fide d.10 s.6."[43] The fact that Salaverri means a sitting Pope can be licitly declared a heretic, is evident from the following quotation from Suarez that he referenced to support his position:
I say thirdly, if a Pope were a heretic and incorrigible, when first, through the legitimate jurisdiction of the Church, a declaratory sentence of the crime is pronounced against him, he ceases to be Pope. This is the common opinion among the doctors. (Suarez, De Fide, disp x, sect. 6.)
In Elements of Ecclesiastical Law (1887), Fr. Smith observes that there are two main opinions concerning how a heretical Pope is deprived of his jurisdiction, and then explains that, “Both opinions agree that he must at least be declared guilty of heresy by the church, i.e., by an ecuмenical council or the College of Cardinals.”[44] In Summa Apologetica de Ecclesia Catholica (1890), Vincent Groot also teaches that a Pope who falls into heresy is not deprived of his jurisdiction until his heresy is juridically established and declared:
"In the case of a Pope who is a public, legal, notorious, and contumacious heretic… he would have to be deposed by a council of bishops. But the deposition would not be an act of jurisdiction, since there is no power greater than the Pope, but a declaratory sentence, by which the fact of heresy is juridically established; and once established, the Pope is believed to be deprived of his dignity by divine law."[45]
http://www.trueorfalsepope.com/p/the-true-meaning-of-bellarmines-ipso.html