Assuming for the sake of argument that Francis is indeed the Holy Father, how are someone like the Dimonds (or whoever) not schismatic?

Because of the widespread doubt regarding his legitimacy.

Again, material vs. formal isn't related to good faith or sincerity.

SV Catholics have every intention of submitting to the Holy Father, but (assuming Bergoglio is legit) are in material error in considering him not to be Pope.

Protestants have no such intention of submitting to the Holy Father.  They are not wrong merely with regard to the fact of identity, but do not submit IN PRINCIPLE to the Pope.

During the Great Western schism, neither side were formal schismatics, because all were Catholics ... but some of them were mistaken about the identity of the Pope.  Protestants on the other hand do not submit out of principle to the Pope, and not due to a mere error of fact.

That is the difference between formal vs. material ... and it has nothing to do with sincerity.  You could have sincere and insincere formal heretics, and sincere and insincere material heretics.

That's a true point regarding the Great Western Schism, but that still raises questions.

One: What counts as "widespread doubt"?  I'm guessing 99% of Catholics have zero doubt regarding the identity of the Pope, or 99% of the world.  That wouldn't have been the case during the Great Schism.

If during the reign of Pope Pius XII someone said that on principle they want to submit to the Holy Father, they just don't think that man is the Holy Father, would that be only material schism?  What if they thought Pius IX was an antipope and rejected Vatican I on that ground?

That's a true point regarding the Great Western Schism, but that still raises questions.

One: What counts as "widespread doubt"?  I'm guessing 99% of Catholics have zero doubt regarding the identity of the Pope, or 99% of the world.  That wouldn't have been the case during the Great Schism.

If during the reign of Pope Pius XII someone said that on principle they want to submit to the Holy Father, they just don't think that man is the Holy Father, would that be only material schism?  What if they thought Pius IX was an antipope and rejected Vatican I on that ground?

Well, I'm guessing (and this is backed by their own polls) that 95% of nominal "Catholics" aren't actually Catholic and don't even have the faith.  So of those remain, the vast majority of the TRADITIONAL CATHOLIC movement, those who still hold the Traditional Catholic faith, have doubts about the legitimacy of the Vatican II popes.

No matter how it's sliced, it's widespread acceptance, not widespread doubt. Neither really matters though, since the Church is not a democracy.

No matter how it's sliced, it's widespread acceptance, not widespread doubt. Neither really matters though, since the Church is not a democracy.

We've already discussed this.  Widespread simply means that it's not localized, i.e. not just a single crackpot losing his mind.  There are a great many Traditional Catholics who harbor doubts about their legitimacy (including +Lefebvre, +Williamson, and +Tissier).  Bishop Castro de Mayer became a sedevacantist in his later days.  And these doubts are based on very specific and grave reasons, thereby constituting positive doubt.

Then, give me one or two texts from pre-Vatican II manuals that (1) speak of "material offices" (2) disagree with the CE's explanation of Apostolicity of mission (you won't find any) as I asked (or with that of Fr. Gueranger, St. Anthony's Catechism etc). You have not done this. When something is true doctrine, you will easily find it in multiple sources, widely attested.

If you think St. Robert has said something about "material offices", cite the text or give the reference. The Doctor never did.

Perhaps nothing is found about "material offices". (Des Laurier's  Cassisiacuм Thesis is just that, an emerging hypothesis in response to a current problem). However, the distinction in the Papacy between a material aspect and a formal aspect is definitely there, and it existed before. Des Lauriers did not invent it. (Bellarmine, St Antoninus, and even Cajetan touched this point. I will try to provide quotes as time permits).

There are only a few ways the Church will be saved, either (1) Pope Francis is miraculously enlightened by God, or (2) a good Tradition-leaning Catholic Cardinal becomes Pope, or (3) a good Traditional Catholic Bishop (like Bp. Fellay) becomes Cardinal and then Pope etc.

All of these solutions are consistent with Sedeprivationism.

We've already discussed this.  Widespread simply means that it's not localized, i.e. not just a single crackpot losing his mind.  There are a great many Traditional Catholics who harbor doubts about their legitimacy (including +Lefebvre, +Williamson, and +Tissier).  Bishop Castro de Mayer became a sedevacantist in his later days.  And these doubts are based on very specific and grave reasons, thereby constituting positive doubt.

Ladislaus, I know you ascribe to the sedeprivationist theory, which one of its great advantages is that doubt is irrelevant, since we only judge on the external matters, that the pope is not orthodox.  If you truly belive in this theory, then arguing for a doubt-based theory (i.e. straight sedevacantism) is hypocritical.

Just because one has a doubt, doesn't mean it's a legitimate doubt.  Just because there's "widespread" doubt doesn't mean it's the kind that warrants positive doubt.  Many people have doubts about the papacy because they have the wrong foundational understanding of the limits of infallibility and indefectibility.  They read 1 part of +Bellarmine, where he argues that the pope's faith could never fail, and assume that our times indicates that the seat is vacant.  Obviously, if this was the ONLY thing that +Bellarmine wrote on the topic, they would have a point.  But he wrote much more.  (And there are other non-Bellarmine texts that are taken out of context, like "cuм Ex").

Many other theologians, including +Bellarmine, say the pope's faith can fail and he can be a heretic.  Thus, the question of having a doubt about a papacy solely based on the orthodoxy of the pope, is not a legitimate doubt.  So, the high % of people today, who base their doubt on faulty theology and incomplete and partially outdated laws (i.e. cuм Ex was revised by both St Pius X and Pius XII, but that is purposefully ignored).  All doubts are not equal.  Only legitimate doubts, based on sound principles, count towards positive doubt.  

Just because one has a doubt, doesn't mean it's a legitimate doubt.  Just because there's "widespread" doubt doesn't mean it's the kind that warrants positive doubt.

While what you wrote is certainly correct, hypothetically speaking, it's clear that the doubts are both legitimate and positive in the case of the V2 papal claimants.