The material/formal distinction is not one that sedevacantism has ever depended on; theologians say that manifest (public) heretics are not members of the Church, regardless of whether or not they are formal or material.  Bergoglio et al. are manifest heretics, and that's all that really matters.

This seems to be equating "public" with "manifest" heretic. I don't agree that they are the same thing. Something public is not necessarily manifest.

II. MISTAKEN ASSUMPTIONS ABOUT
“MANIFEST,” “PUBLIC,” “NOTORIOUS”
——————————————————————––
Most anti-sedevacantist controversialists over the years have, like Fr. Boulet, made exactly the same error. Why? The answer lies in their false assumptions about the meaning of technical terms.
The long line of theologians and canonists over the centuries who examined the question of a heretical pope distinguished between two general types of papal heresy according to the “notice” or “publicity” it received.

Quote

(1) “Occult” (i.e., secret or hidden) heresy. (E.g., written in a diary, uttered in private to a few discreet people, etc.)
(2) A second type of heresy that is not occult. (E.g., published in an official docuмent, proclaimed in a public discourse, etc.)

For the latter, the various theological and canonical treatises did not always use an identical term, but instead employed a variety of expressions to describe the papal heretic or his heresy: “public,” “notorious,” “manifest,” “openly divulged,” etc.
These were generic terms that did not have a uniform meaning in sources and authors before the 1917 Code, and were simply used in contradistinction to “occult.” (See F. Roberti, “De Delictis et Poenis,” schemata praelectionum [Rome: Lateran 1955] 80–1) Authors writing after the 1917 Code about the question of a heretical pope continued to use the same generic language to distinguish between occult and non-occult heresy.
Because of this, Fr. Boulet and many others like him have fallen into anachronism about the terminology. They mistake this generic language used by authors writing about papal heresy before the Code, and subsequently taken up even by authors after the Code, as an indication that all the minute criteria of the Code’s criminal legislation must be satisfied before a loss of papal office can kick in.
This, alas, is a fatal error, so none of their arguments on this point can be used against the sedevacantist case.

http://www.fathercekada.com/2007/10/10/a-pope-as-a-manifest-or-public-heretic/

The significant distinction is between "occult" and "not occult," which has been described by various terms, including both "public" and "manifest." A non-occult heretic is outside the Church according to the majority view. 

Van Nort has something on this which I can get for you. And Father Cekada speaks of the distinction being between "occult" and "not occult":
For the latter, the various theological and canonical treatises did not always use an identical term, but instead employed a variety of expressions to describe the papal heretic or his heresy: “public,” “notorious,” “manifest,” “openly divulged,” etc.
These were generic terms that did not have a uniform meaning in sources and authors before the 1917 Code, and were simply used in contradistinction to “occult.” (See F. Roberti, “De Delictis et Poenis,” schemata praelectionum [Rome: Lateran 1955] 80–1) Authors writing after the 1917 Code about the question of a heretical pope continued to use the same generic language to distinguish between occult and non-occult heresy.
Because of this, Fr. Boulet and many others like him have fallen into anachronism about the terminology. They mistake this generic language used by authors writing about papal heresy before the Code, and subsequently taken up even by authors after the Code, as an indication that all the minute criteria of the Code’s criminal legislation must be satisfied before a loss of papal office can kick in.
This, alas, is a fatal error, so none of their arguments on this point can be used against the sedevacantist case.

http://www.fathercekada.com/2007/10/10/a-pope-as-a-manifest-or-public-heretic/

If you want to argue about a distinction of theological significance between "public" and "manifest" to ahead; I"ll obviously read it.

I don't understand where people get this from! Of course we are to obey our superiors, but not when it comes to sin.
 
Bishop Sanborn recently gave an analogy on this subject. It went something like this. As passengers on a jetliner we trust the pilots and do as they say because they are trained to fly the airplane. But if one of the co-pilots suddenly tries to steer the plane into the ground, everyone knows he was NOT trained to do this, and that he is putting everyone's lives in jeopardy. Should the passengers sit back and say, "We can't judge the pilot because he is trained and knows what he is doing"? Of course not. Do we have to get the pilot's boss on our cell phone to ask for approval to remove the nutty pilot from the cockpit? Of course not. Everyone reading this knows that the passengers would do their absolute best to remove the crazy co-pilot immediately, and bind him and put him in the back of the plane.
 
It's similar with Francis; he is wearing a pilot's uniform but he is clearly not doing what a pilot should, and he is clearly trying to steer the Church into the ground, taking all the passengers with it. The Church teaches a man doing this is not a pope, nor is he a Catholic. We don't need to wait for someone to tell us this -the Church has already taught us this over and over, and why people ignore it is just insane.