There is no such thing as a "material office"

That's not what Catholics are supposed to do, read it again: "recognizes him as the head of the Church, the vicar of Christ, and the successor of blessed Peter,"

And these bishops have a lawful mission to guard always, in their teaching and management of the Church, the unity of Faith and of communion with their head and center, the Roman Pontiff

Yes, that's where it becomes tricky.  Heretics can be slippery.  They will SAY that it matters to them that their position be consistent with Church dogma.  But the question is whether they really do.  At the end of the day, no one but God can judge that.

We have to deal with things as manifest in the external forum.  That's why St. Robert Bellarmine speaks about MANIFEST heresy deposing from the Church rather than FORMAL heresy.  So this is the wrong argument here.  You could have someone who on the outside appears to be perfectly orthodox, but in his soul doesn't have the faith ... and vice versa, someone who's a heretic outwardly but inside has the sincere intention to accept Church teaching.  But de internis Ecclesia non judicat, the Church does not judge regarding matters of the internal forum.

We can only judge the PERTINACIOUS adherence in the external forum to heretical doctrine.  And there's no doubt that the V2 papal claimants adhere pertinaciously to their errors.  I'm certain of it that if Bergoglio were to resign, and some orthodox Pope came along and ordered Bergoglio to submit to traditional Church teaching, he would refuse.

And I do believe that these men are active infiltrators and conscious destroyers.

But this is the wrong argument.  Personal heresy doesn't even matter.  If the V2 Magisterium had taught perfectly orthodox doctrine and we were still using the Tridentine Mass, etc. ... then I would not waste 10 seconds of my time attempting to resolve the question of whether Bergoglio's insane ramblings constitute pertinacious heresy.

So the problem we have here is whether the V2 Magisterium is in fact the Catholic Magisterium, and not the personal state of Bergoglio's soul.  Do I even recognize this Conciliar establishment as the Catholic Church?  I most certainly do not.  As to how or why this has happened, God only knows.  Archbishop Lefebvre famously speculated about the possible explanations.  Was Paul VI drugged?  Was Paul VI insane?  Was Paul VI being blackmailed?  Was there a double put in his place?  Was Paul VI a heretic?  At the end of the day, along with the Archbishop, we don't know for sure.  All we know is that their Magisterium is not the Magisterium of the Catholic Church, and they are not (at least freely) exercising the papal Magisterium in a formal way.

Here's how faith starts.  We look at the Church and recognize its marks.  Then, based on these motives of credibility, we submit to this Church.  So there's a lead up of natural reason towards supernatural faith.  All this is taught at Vatican I.  But we do not see these marks in the Conciliar Church, so we withdraw from submission to it ... categorically.  R&R however says that we can recognize it as the Church and at the same time submit to what we like and not submit to what we don't like.  That, in a nutshell, is the debate between Sedevacantism and R&R.

What does "rejecting the conciliar establishment as the Catholic Church" mean exactly?  My question goes deeper than someone like Bergoglio here.

Say you meet a Catholic layperson who according to all appearances seems to love the Church and the faith. But he believes that Vatican II is in fact part of the magsiterium and interprets it through some sort of hermeneutic of continuity.  Would you view him as outside the Church or presume he was a formal heretic?

Sedeprivationism addresses many of the valid points made by Siscoe/Salsa but avoids their mistakes and errors; and then it also navigates away from the extremes of radical sedevacantism ... which would allow lay people even to usurp the prerogatives that belong only to the Church.  Sedeprivationism avoids the pitfalls of both extremes.

I don't see how this is necessarily the case. Some of the most hardened tenants of sectarian sedevacantism, like Bishop Sanborn, are also sedeprivationists.

It's a recognition of his office, around which Catholics are unified.
Pax Vobis, you are not reading Ex Quo, but only reiterating what the sedeprivationists have confused you with. There is no such thing as a "material office", nor a mere "recognition of his office". Even the Orthodox or Old Catholics could claim they recognize the office. That's not what Catholics are supposed to do, read it again: "recognizes him as the head of the Church, the vicar of Christ, and the successor of blessed Peter," so, the Bishops recognize the Pope as the Vicar of Christ. And all who pray for the Pope as the Pope in the Canon by doing the same manifest for that reason "the profession of a mind and will which firmly espouses Catholic unity." Sedevacantists/privationists on the other hand who of their own will drop the name of the Pope from the Canon or choose to attend the Masses of schismatic clergy who do so fall under the latter category here "the commemoration implies a profession of due subjection to the Roman pontiff as head of the Church, and of a willingness to remain in the unity of the Church. On the other hand the omission of this commemoration signifies the intention of steadfastly espousing schism." - Dropping the name of the Pope is a sin of schism.

Instead you say: "This is why, assuming you believe that +Francis is pope, you are obligated to continue to pray for him" - is that really what Pope Benedict XIV says? he says you are separated from the communion of the entire world if you don't.

Just read the CE on what it says about Apostolicity and it word for word refutes the sedeprivationists. The SPs believe Apostolicity of mission and the Apostolic Church is one thing and yet we have to separate and continue elsewhere apart from Rome. Wrong.

Is that what any Saint has ever taught? Please show me even one in such a case. I can show you a hundred, but I'll just cite one.

St. Anthony Mary Claret in his Catechism on Apostolicity: "The fourth note or mark of the Church is to be Apostolic. That is to say, it was founded by the Apostles and is governed by their successors, the bishops, who. since the Apostles, have succeeded without interruption. And these bishops have a lawful mission to guard always, in their teaching and management of the Church, the unity of Faith and of communion with their head and center, the Roman Pontiff ... All of us know that the Apostles fulfilled the mission that Jesus Christ gave them. And it is sufficient to read the list of the Catholic bishops, especially of the Supreme Pontiffs of Rome as the continuing Head or principal leader of Christianity - better said, of Catholicism - in order to see that ... You will notice above that with the word mission I added the word lawful, that is, coming from that one who has the keys of the kingdom of heaven or of the Church, who is the Pope ...

For this reason you cannot doubt that the only true Church is our Catholic, Apostolic, and Roman Church, in which you must persevere, inwardly and outwardly." http://catholicvox.blogspot.com/2009/03/eens-saint-anthony-mary-claret-from.html