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Not necessarily.  Even if Francis isn't a valid bishop, assuming his election was valid, he'd be a papal-elect.  So, in theory, the office would be filled (from a human/govt aspect) but it would be vacant (spiritually) until he were made a bishop and accepted properly.

Papal Elect is still not pope.

Yes, that is interesting. I have to admit when I come across valuable quotes like this, I have to reread it a couple times and then follow with the source and then look for commentary.  I have not come to any hard conclusions at this time. There are usually distinctions that have to made.

What does "character of order" mean? Does it mean these true bishops and priests can be lacking the sacramental character of Holy Orders? The mystery continues...

Hi Comrade,

There are other ways to look at the situation besides focusing on the NREC. Most importantly, Bergoglio is not even a "sacerdotal priest" (in the sense of a priest endowed with the power to "offer the Body and Blood of Our Lord for the living and the dead"). The New Rite of Priestly Ordination removed that power. In doing so, the NRPO has tacitly redefined the priesthood.

The NRPO "priests" are empowered, by the specific prayers of the Rite, to offer the "the oblation of the holy people, the gifts to be offered to God." This new power precisely corresponds to the NO Offertory prayers, which offer the "work of human hands," rather than offering the "immaculate victim." This type of offering is alluded to figuratively in Genesis as the "offering of Cain," an offering that was displeasing to God. This is the "grain offering," the Sacrifice of "the Eucharist," i.e., the sacrifice of "praise and thanksgiving." The Lutherans and Anglicans pulled the same trick hundreds of years before, and Leo XIII invalidated their orders.

So, these NRPO "priests" are "ordained" by a rite approved by the NuChurch. They are called "priests." But they are not true sacerdotal priests as understood by traditional Catholic sacramental theology. The remission of sins only comes from the offering of the Body and Blood of Jesus Christ in the true Holy Sacrifice of the Mass at the hands of a sacerdotal priest. The NRPO does not include prayers that ask God to bestow that power on those Novus Ordo "priests." God does not give what is not asked for. So, in essence, the NRPO "priest" is something in between a deacon and a true sacerdotal priest. He can offer "the sacrifice of praise," but he is a handicapped "priest" from the traditional perspective.

Traditionally, a Catholic bishop must first be a sacerdotal priest. It is the sacerdotal priesthood that is the most important element. 

Now, what does all of this mean for Bergoglio? Since he is not a true sacerdotal priest, it is ontologically impossible for him to be the Vicar of Christ.

I don’t think Matthew thinks there zero doubt that Francis is the pope, just that it’s not his job to solve that problem 

He wouldn’t need to be certain about that one way or another.  Whereas he WOULD need certainty that his own priest is valid.  He’s not saying (I don’t think) that he’s sure about the new bishops one way or another .