We aren’t judging the internal forum, we are making a judgement solely from external words and deeds.

OK, but that's the debate.  

Among trads and even those who are close to being trad, there's really no dispute that Francis says things that are, at best, grossly erroneous and confusing, almost certainly with no valid excuse for doing so, probably outright heretical.

The difference is the Sedevacantists believe its appropriate to judge that he must be a *formal* heretic and therefore must have fallen from office, whereas other trads don't think they can make that leap.  

That's the issue.

QVD: And that is exactly what I and many others have done.  We've made just such a judgment.

I'm not interested in judging *you* for making such a judgment, but it seems like you guys are judging that he's *definitely* a formal heretic, and thus can't be a valid pope.

Whereas some trads will judge the actions as objectively wrong, but believe they can't judge the internal forum, and thus they believe they have to take his claim to be Catholic at face value (despite material heresies) and thus accept him as their hierarchical leader (despite resisting unjust commands.)

Lefebvre saying his reason for not tolerating sedevacantism is dogmatic (not practical):

“Does not the exclusion of the cardinals of over eighty years of age, and the secret meetings which preceded and prepared the last two Conclaves render them invalid? Invalid: no, that is saying too much. Doubtful at the time: perhaps. But in any case the subsequent unanimous acceptance of the election by the Cardinals and the Roman clergy suffices to validate it. That is the teaching of the theologians.
The visibility of the Church is too necessary to its existence for it to be possible that God would allow that visibility to disappear for decades. The reasoning of those who deny that we have a Pope puts the Church in an extricable situation. Who will tell us who the future Pope is to be? How, as there are no cardinals, is he to be chosen? This spirit is a schismatical one for at least the majority of those who attach themselves to certainly schismatical sects like Palmar de Troya, the Eglise Latine de Toulouse, and others.
[…]
Thus, I have never refused to go to Rome at his request or that of his representatives. The Truth must be affirmed at Rome above all other places. It is of God, and He will assure its ultimate triumph.
Consequently, the Society of St. Pius X, its priests, brothers, sisters and oblates, cannot tolerate among its members those who refuse-to pray for the Pope or affirm that the Novus Ordo Missae is per se invalid. Certainly we suffer from this continual incoherence which consists in praising all the Liberal orientations of Vatican II and at the same time straining to mitigate its effects. But all of this must incite us to prayer and to the firm maintenance of Tradition rather than to the affirmation that the Pope is not the Pope.”
https://www.google.com/amp/s/mundabor.wordpress.com/2014/05/01/archbishop-lefebvre-on-sedevacantism/amp/

I think what Matthew means is not that the reasoning was pragmatic in a purely utilitarian sense, but rather that he saw it as his opinion, one that was backed by reasons, yes, but still as his opinion not as de fide.

For instance if you denied the immaculate conception or transubstantiation, that would make you objectively *not Catholic* and comparable to a Protestant or some other kind of heretic.

Whereas it seems like Lefebvre didn't see it at that level.  He disagreed with Sedevacantism (while sometimes considering it) and did so for theological reasons, but he thought it was his opinion.

Whether he was right or not, I don't know.  But that seems to be what the Archbishop thought.

I started this topic, but can honestly report that I had never heard the terms “internal forum” and “external forum.” If this is some kind of ecclesial legalese, then I confess a total ignorance of it. Yes, the terms “formal heretic” and “material heretic” were somewhat familiar. But in finally coming to closure about Francis, I swept all that stuff aside. Francis, in my mind, became an intolerable papal fraud, a supposed pontiff whom no sane or fair minded person, of whatever Catholic stripe, can possibly recognize. I don’t care what ABL, on balance, said or didn’t say in the past. I don’t care what Bp. Williamson and the R&R say in the present. Francis is an anti-pope.

That's not quite right.  Pertinacity can in fact be determined in the external forum.  If it cannot, then the Church can never make a judgment about it.  Only God can judge something in the internal forum.

The way I look at it, the Church uses the 2 rebuke process in the external forum, to determine the internal forum.  To put it canonically, She would twice-rebuke a material heretic, and if they were obstinate in their material heresy (externally), then She would presume they were a manifest heretic (internally).  Much like the Church says that a sacrament which is (externally) performed according to the proper prayers, is assumed to be valid (internally).
.
What all sedes ignore, is St Paul's teaching on the 2 rebuke process (which is now part of canon law), which is carried out by the Church.  If this process is not used, then you cannot judge someone to be manifest.  No layperson, priest or non-jurisdictional bishop has any canonical authority to rebuke any other catholic, so they can never call anyone manifest, and so they cannot consider anyone to have "ipso facto" lost their office.  The fact that most all of them presume to do so is against canon law and against catholic thinking.  

Quo Vadis,
The fact that you are missing (or probably ignoring, since you dodged this fact on the other thread), is that one cannot be considered a manifest/pernicious heretic based on the external forum alone.  Pertinacity is stubbornness of the will (internal forum), which must be determined by St Paul’s twice-rebuke process.  Unless you can prove that a person has been formally rebuked and yet still clings to error, then you cannot label them a manifest heretic.  Ergo, they have not ipso-facto lost office.

I didn’t dodge you in the other thread, I was just finished arguing with someone who spreads error and thinks they know more than any preconciliar theologian.
Your erroneous idea of pertinacity being determined solely in the internal forum is certainly quite bizarre. Care to give us any reputable authority that concurs with you on this?

The way I look at it, the Church uses the 2 rebuke process in the external forum, to determine the internal forum.  To put it canonically, She would twice-rebuke a material heretic, and if they were obstinate in their material heresy (externally), then She would presume they were a manifest heretic (internally).  Much like the Church says that a sacrament which is (externally) performed according to the proper prayers, is assumed to be valid (internally).
.
What all sedes ignore, is St Paul's teaching on the 2 rebuke process (which is now part of canon law), which is carried out by the Church.  If this process is not used, then you cannot judge someone to be manifest.  No layperson, priest or non-jurisdictional bishop has any canonical authority to rebuke any other catholic, so they can never call anyone manifest, and so they cannot consider anyone to have "ipso facto" lost their office.  The fact that most all of them presume to do so is against canon law and against catholic thinking.  

“The way I look at it” ......yes I know. Sorry, but no reputable theologian sees things the way you look at them. 
This is precious: “ No layperson, priest or non-jurisdictional bishop has any canonical authority to rebuke any other catholic.....” Really? Have you ever heard of the spiritual work of mercy in which we are to admonish the sinner?