That faith they claim to have has the indefectability of the papacy as its element. I understand that an Orthodox or a Protestant might interpret Mt. 16:18 the way that suits their worldviews, but a Catholic has no such liberty. Sedevacantism maintains its connection to Catholicity as long as, while claiming that a particular Pope is not really a Pope, it adheres to these two truths: (1) Valid Mass is offered by those priests whose ordination traces to true Popes and (2) Masses invalid due to improper line of ordination remain an exception. Denial of (1) means denial of papacy altogether, and denial of (2) means that the gates of Hell have prevailed.
For example, if a sedevacantist states that Novus Ordo masses are invalid because they are Novus Ordo, he crosses over from sedevacantism, which question apostolicity, into a general schism with Rome on ecclesiological grounds.
"For example, if a sedevacantist states that Novus Ordo masses are invalid because they are Novus Ordo, he crosses over from sedevacantism, which question apostolicity, into a general schism with Rome on ecclesiological grounds."
As with Protestant ecclesiology, there are inherent problems with the SV position, given the papist position of the SVers. I personally think this can only be resolved by SVers going to either an Old Catholic position or to Orthodoxy. As Old Catholicism is imbued with a heavy dose of Jansenism, however, I don't see SVers going there. And if they go to Orthodoxy, they'll have to dump a lot of 19th century RC piety. In other words, the SV position, as it is now, is not tenable. They have to go somewhere and when they do, they'll have to dump something. My guess is they'll go wherever they don't have to dump as much. Off hand, I can think of two Orthodox synods they might fit into.
Or maybe they'll become the Western equivalent of the Russian Old Believers?