Are you sure that there's no doctrine on how to deal with those who are in schism? I'm pretty sure that there is.
How are we jumping from the question of how the Church is to deal with a pope who becomes a heretic to the question of how to deal with schismatics?
The situation with the anti-popes is a different situation.
It nevertheless presents a solution, unless you are an ecclesia-vacantist. The heretic pope is an antipope and the cardinals, discernign this, elect a true pope, just as happened with each valid election during the Western schism. The bishops in union with the true pope then have the authority to declare to deliver definitive judgment. "Simple", in that sense. The problem arises if one rejects the "Novus Ordo" bishops and their ordinations in their entirety, leaving no authority to even designate a pope.
This is something separate from how the individual Catholic is to behave according to his best knowledge.
There HAS to be a method (Church teaching), if you are going to adopt the Ipso Facto methodology on how to deal with a heretic Pope, after he is proclaimed by laypersons to no longer be the Pope.
No, there doesn't. That's a non sequitur.
if you are going to adopt the Ipso Facto methodology on how to deal with a heretic Pope, after he is proclaimed by laypersons to no longer be the Pope.
It's not a methodoloy. It's a doctrinal fact. JURISDICTION comes from God, and heretics ipso facto LOSE jurisdiction. This is not a theory. It's not a methodology. It's not positive law. It's not open for discussion. And it's truth does not depend on how this fact works itself out in the recognition of thsi loss of office by the Church.
Though, of course, there is no doctrine which says that laymen are to proclaim and judge that the Pope is not a Pope. If you could find such doctrine, I would be interested in seeing it.
Again: nobody can judge a pope. You're confusing issues here. It's always, in the final anaylysis, going to be the individual who decides to recognise or not to recognise someone as the pope: usually that comes from simply accepting, as one is bound to, what the whole Church and his superiors appear to accept - the man elected as pope - but if there's obvious manifest heresy then that acceptance begins to conflict with ones conscience and reason.
Again the Western schism is important: there were saints who recognised antipopes and saints who rejected valid popes, even though following the Church to the best of their knowledge and ability, precisely because of that necessary element of fallible personal discernment. They were is material schism - saints no less - but they were not formal schismatics. Sedes don't deny the Papacy as such; they (good-willed) simply cannot in good conscience accept Francis as a pope when they know without doubt that manifest heretics lose office. Might they be wrong? Yes. Does this endanger their souls eo ipso? If it's in good conscience and not rashly decided, then I don't think so.
As has been said before, the well-intentioned Sedevacantist is at worst in material schism; the R&R'er is, even if he is right, in formal schism; the FSSP type is probably guilty of nothing much at all; the typical Novus Ordo type endangers his soul by submitting to teachers of heresy.
You are venturing into unknown territory when dealing with a heretical Pope, and yet you sedes act as if the there's a specific doctrine that says that laypersons are to judge and proclaim as to whether the seat is empty or not.
Stop calling me a "sede". Are
to judge? No. Can
discern, at least to the point of grave doubt? Absolutely. And yes, Cum ex apostolatus officio
supplies the basis upon which I am permitted to withdraw my submission to and avoid as an heresiarch such a manifest heretic.
And that's exactly what I do: I have grave doubt, therefore I do not formally recognise, therefore I can materially withdraw and avoid. Is the seat empty? I don't know, and though I wish to discover, I don't need to know.