No one on earth possesses the jurisdiction to "officially" warn or correct the pope. Ecclesiastical warnings are of the nature of an act of a superior over a subject.
Cardinals who rebuke a pope are not doing so jurisdictionally; they are not "pulling rank". They are rebuking per Scripture, as St Paul explains where he rebuked St Peter. Even +Bellarmine says that St Paul "orders" that "2 warnings" be given, to determine 'manifest obstinacy'. You're falsely inserting the idea of jurisdiction.
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A formal correction done as a charitable act can be done by any private person, as Ballerini explains.
Ok, but not in the case of the pope. It's very clear that Cardinals elect the pope and they are allowed to rebuke him (by way of council, committee or letter, etc).
If the obstinacy is manifest, it is perceptible to the senses and is manifested in such a manner that is seen to be obvious, such as when one refuses correction,
Exactly! Correction = St Paul's 2 warnings = rebuke = official warning.
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When the obstinacy is manifest, it is public or will become public, and pertains to the external forum.
Obstinacy can only be determined by a rebuke/correction process. A passionate, open or public assertion of error is not obstinacy.
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Once the manifest material heretic remains obstinate even after being corrected by someone (by anyone who is capable) with an explanation that would suffice to convince a reasonable man (as Fr. Charles Augustine and St. Alphonsus explain), then the formal heresy is obvious and manifest, even before any judgment or declaration is made by the Church.
Agree, except for your insertion of the phrase "by anyone who is capable". In the matter of fraternal correction, we are allowed to fraternally correct those who are our peers, and only correct superiors if they have given us permission, or if we know that such correction can be done in a respectful way. In the case of a pope, since the Cardinals are the "princes of the Church" and they elected him, only they are allowed to rebuke him.
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If the obstinacy of heresy is manifested in such a manner that the dolus of heresy is obvious even without correction, then the form of heresy is already manifest entirely by itself, even without warnings (as Bordoni and de Lugo explain); and therefore is manifest even before being judged and declared by the Church
This is the crux of the debate. The V2 popes claim that their novelties are consistent with Tradition, or at least, they are "pastoral" applications of orthodox doctrine. They continue to deny that they are heretics; they continue to explain that their novelties are catholic. Thus while the heresy is obvious; the obstinacy is not. The heresy may be manifest (in the sense that it is open and public) but it is not "manifest obstinate" heresy, per +Bellarmine's use. Until they are formally rebuked, their obstinacy is not legally established, no matter how open is the error.
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Let's not forget that the PURPOSE of the rebuke/warning/correction is to bring back the heretic from his sin. The purpose is the salvation of his soul. The purpose of all canonical penalties (including excommunication) is to "shock the system" so that the heretic/schismatic will "wake up" from his unorthodox views and come back to Truth. Many of you want to rush to judgement, to see obstinacy where it is not yet been proven. You have no patience for the Church's processes, which as history shows, can take YEARS to develop. Let's not forget that from the day that Martin Luther nailed his 99 heresies on the church door, to the day he was excommunicated, was over a year and a half. Was anyone more of a clear-cut case of heresy than Martin Luther? Yet, he was not declared obstinate for months and months.
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I still say, you need to provide +Bellarmine's definition of "manifest" heresy, before you can apply his quote properly. You are interpreting 'manifest' incorrectly, according to the current use by theologians, over 400 years after +Bellarmine lived. The use of this term has changed.
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By the fact of the manifestly evident obstinacy,
Manifest does not equal obstinate. Obstinate does not equal manifest. You use these terms as if they were connected and they are not. They have 2 COMPLETELY separate meanings, both in law and in theology.