I never quite understood the thesis that a man is pope materialiter, but not formaliter.   The form adheres within the matter and causes it to be what it is.  If John XXIII is pope materially, then he is pope formally.  I liken it to a man who receives baptism sacrilegiously. Is he a member of the Church?  Yes. Did he receive the graces of baptism? No.  He did receive the grace of becoming a member of the Church by virtue of his baptism, albeit a sacrilegious baptism, but he did not receive the other graces of the sacrament. A man can receive the papacy and not receive the special graces which the Holy Ghost would have ordinarily bestowed had he been in the state of grace. 

A man can be in the state of mortal sin and become pope, why not?  This whole idea that personal sin exempts one from becoming pope, or, once becoming pope, causes him to lose the office once obtained, is frankly absurd.  The whole problem that I have with Pope Leo is the questionable new rites of both priesthood and episcopacy.  But theoretically one could be elected pope and then receive the priesthood and bishopric after his election. Of course, we have the compounded problem that the majority of the electors were new-rite bishops themselves.  That was not the case with, say, John XXIII.     

I never quite understood the thesis that a man is pope materialiter, but not formaliter.  The form adheres within the matter and causes it to be what it is.  If John XXIII is pope materially, then he is pope formally.  I liken it to a man who receives baptism sacrilegiously. Is he a member of the Church?  Yes. Did he receive the graces of baptism? No.  He did receive the grace of becoming a member of the Church by virtue of his baptism, albeit a sacrilegious baptism, but he did not receive the other graces of the sacrament. A man can receive the papacy and not receive the special graces which the Holy Ghost would have ordinarily bestowed had he been in the state of grace.

A man can be in the state of mortal sin and become pope, why not?  This whole idea that personal sin exempts one from becoming pope, or, once becoming pope, causes him to lose the office once obtained, is frankly absurd.  The whole problem that I have with Pope Leo is the questionable new rites of both priesthood and episcopacy.  But theoretically one could be elected pope and then receive the priesthood and bishopric after his election. Of course, we have the compounded problem that the majority of the electors were new-rite bishops themselves.  That was not the case with, say, John XXIII.   

Well, your first statement is quite accurate, where you rightly declare that you do not understand the thesis, but then it goes downhill from there where you're quite confident that you do understand it in delcaring it to be "absurd".  While nobody's required to accept it, let's show a bit of humility, eh?  Bishop Guerard des Laurier undoubtedly forgot more theology than the combined posters on this forum ever knew.  He's the only legitimate Traditional Catholic theologian, despite a number of pretenders among the SSPX, having been personal confessor to Pius XII for a time, having collaborated in the dogmatic definition of the Assumption, and having ghost-written the Ottaviani Intervention.  If we disagree, we disagree with all due respect, and don't bloviate in ignorance about it being "absurd" ... as if a great theological mind like that were some buffoon who could barely outwit us armchair theologians wielding the Baltimore Catechism #3.

So you make two serious mistakes immediately after your accurate opening sentence.  Firstly, you're considering the formal / material distinction only in the context of an absolute ontological sense, whereas there are multiple senses and applications of that same distinction (logical, epistemological, theological).  Secondly, you completely reverse the concepts by claiming that if something is X materially, then it is so formally.  If anything it's the opposite.  Materialiter refers to a potency, and it's the formaliter that actualizes a potency.  So, because a block of wood is materially picnic table, then it's also formally a picnic table?  Nonsense, "frankly absurd".

Bishop Guerard des Laurier actually expounds upon the different sense of material / formal distinctions (there's a famous interview he gave where he goes into this).

So, it was actually St. Robert Bellarmine who expounded on the difference between the material and formal aspects of papal election and papal authority.  When a conclave elects a Pope, the conclave, the Cardinals, the electors do NOT confer any formal authority upon the Pope, as in the erroneous democratic systems where (formal) authority derives from the people and the consent of the governed.  No, the Cardinals merely express their choice, and then God honors their choice and responds to it by Himself directly bestowing the formal autority.

If a Conclave were to elect a layman, would he have formal papal authority?  Since a Pope is a Bishop of Rome, is he the Bishop of Rome by mere election?  No.  He's merely the Pope-Elect, the Pope-Designate.  In fact, he can do nothing as a layman, since no layman can exercise any kind of oversight in the Church.  Now, if he were made a cleric, at that time he could begin exercising some aspects of papal authority, e.g. making appointments, various administrative tasks.  Does this aspect of papal authority require faith even?  Sedeprivationsts would say no.  But can this cleric now exercise Magisterium?  No, since only bishops are part of the Ecclesia Docens, and, again, he's not the "Bishop of Rome" and therefore not the Pope.

Yet another way to think of the distinction here is to liken a pope-elect who hasn't received the ability to formally exercise papal authority to a marriage that's been ratum sed non consummatum, where the marriage is legally in place, the contract valid ... but which is not been formally completed or fulfilled, and where the material aspect of the contract can be severed.  Once FORMALLY united in marriage, when it has been consummated, then no man can sever (materially) what God had joined (formally).  Similarly, if a heretic were elected, the Church could withdraw their election or designation, at any time before the election were consummated formally by God bestowing the authority upon the papal elect, but authority cannot be conferred on one who's not materially capable of receiving it.

I have that book.  He does not "express his position in great detail".  The focus of that book was on attacking the dogmatic Sedevacantism of Fr. Anthony Cekada.  Furthermore, I don't believe that Fr. Chazal rejected Jorge Bergoglio as pope at the time of publication.

Father Chazal never "rejected" Bergoglio as pope simpliciter, but secundum quid, where he holds that Bergs remained in office, as the visible principle of unity among Catholics, but remained suspended where he could not formally exercise papal authortiy (or exercise is validly, as he termed it).

You have this binary, black-or-white, all-or-nothing, view.

So many errors are being committed by these binaries and false dichotomies, and no greater service has ever been done for theology than when the scholastics applied the Aristotelian notion of the distinction.  Of course, many slippery eels abuse distinctions to pretend they're not denying dogmas when they really are.

When it comes R&R, Dogmatic Sedevacantism, Sededoubtful, or Sedeprivationist, my policy is “Don't ask; don’t tell.”  This question and wrestling over it has lead many away from the Faith into Home Aloneism. The Enemy is laughing all the way to the bank of hell. The issue won’t be settled until the reign of Mary. 

Fr. Chazal held that Jorge Bergoglio was not ontologically pope.  Therefore, he rejected Jorge Bergoglio as pope simpliciter.  That he was "elected" by the cardinals is accidental.  It would be up to the Church to formally declare the See vacant, but Fr. Chazal held the private judgment that Jorge Bergoglio was a public manifest formal heretic and therefore not a member of the Church.  He didn't hold some wacko distinction between formally pope and materially pope.  You, like Pax Vobis, take logical parts that are beings of reason and make them into real beings.  Stop with your nonsense.

But how would that work, given that "the first See is judged by no one"?

An invalid Pope wouldn't be likely to say "okay, I'm busted, I'm not Pope after all".  If he did, I have to think this would constitute a de facto resignation, and thus the whole question of sede vacante would disappear, the see would be manifestly vacant without question.