Quote from Plenus Venter - "He [The Pope] is only guaranteed such infallible fidelity when he teaches on faith and morals under very strict conditions as laid down by Vatican I." Bergoglio is a "formal heretic" that has taught "formal heresy" [e.g. Amoris Laetitia #297 (2016)] and it's been a string of formal heresy ever since his election 10 years ago. Nobody can honestly say he's only making mistakes or just speaking off-the-cuff "material" heresy or believe he is the Vicar of Christ simply because he hasn't made these statements with the words "Ex Cathedra" alongside them.
Apparently, according to the R&R crowd, Bergoglio could be like The Exorcist Girl spewing vomit, twisting his head around, walking upside down on all fours, speaking in the growling tones of a demon and he would still be the Vicar of Christ on earth because he hasn't spoken "Ex Cathedra."
Not impossible, Hank. As St Robert Bellarmine says, even if we could not depose a Pope WHO WANTS TO DESTROY THE CHURCH (note: a pope can have the desire to destroy the Church), we must pray God Who will either CONVERT HIM (note: a Pope may need converting) or abolish him from the midst before he achieves his evil designs.
This is a new situation in the Church. Whether or not he can be deposed even for heresy is not settled. The Church has tolerated all theological opinions. Much caution and prudence are required. It is not impossible, in spite of appearances, that Pope Francis is nothing but a modernist who could indeed have the Faith. But even if he does not... He needs to be held to account. If an imperfect Council is called to do that I will certainly watch with interest.
It is so important for you to make a definitive judgement that he is not Pope. I don't think there is any certitude according to our Holy Catholic Faith that you can do that. It doesn't change what you do, does it? It might lead to some of your friends never accepting a new and certainly valid pope from what you consider a defected church, though. It may lead to many Catholics in the Conciliar Church (the great number of Catholics in the world today, even if it is a small percentage of that Church) being left for dead... decisions have consequences. Such a grand decision, requires certainty, and it is not for me, so my Catholic sense tells me.
You mention formal heresy. But formal heresy requires admonitions from a superior. Who would give such monitions to a Pope? So many ifs and buts and unanswerable questions. Let us not try to be the Church, we end up making ourselves popes and settling disputed matters. Not Catholic for me, not at all.