POPE PAUL IV (1559-1566)
         
        "If ever it should appear that any bishop, even one acting as an
Archbishop, Patriarch, or Primate, or a Cardinal of the Roman Church,
or a legate, or even the Roman Pontiff, whether prior to his promotion to
cardinal, or prior to his election as Roman Pontiff, has beforehand deviated
from the Catholic faith or fallen into any heresy, We enact, we decree, we
determine, we define:  Such promotion or election in and of itself, even
with
the agreement and unanimous consent of all the Cardinals, shall be null,
legally invalid, and void.

        "It shall not be possible for such a promotion or election to be
deemed valid or to be valid, neither through reception of office,
consecration, subsequent administration, or possession, not even through the
putative enthronement of a Roman Pontiff himself, together with the
veneration and obedience accorded him by all.  Such promotion or election
shall not through any lapse of time in the foregoing situation be considered
even partially legitimate in any way....  Each and all of the words, as
acts,
laws, appointments of those so promoted or elected -- and indeed, whatsoever
flows therefrom -- shall be lacking in force, and shall grant no stability
and legal power to anyone whatsoever.

        "Those so promoted or elected, by that very fact and without the
need
to make any further declaration, shall be deprived of any dignity, position,
honor, title, authority, office, and power....  Therefore, it is permitted
to
no one to impair this page of Our approval, renewal, sanction, statute,
wills
of repeal, of degrees, or to go contrary to it by a rash daring deed.  If
anyone, moreover, will have presumed to attempt this, he will incur the
wrath
of Almighty God an of the blessed Apostles Peter and Paul."  (cuм ex
apostolatus officio, February 16, 1559, sec. 9, apud "Fontes Iuris
Canonici,"
1971)
   
      Two popes, Innocent III and Paul IV, both vigorous defenders of
papal authority, together with many theologians, admitted the principle
that a pope, in his personal capacity, can defect from the Faith or
become a heretic; that when the fact of defection becomes manifest, such
a pope automatically (ipso facto) loses his office and authority; and
that such a pope should be resisted.

        What is even more significant is that in his cuм ex apostolatus
officio, Paul IV even perceives the possibility of a Protestant being
elected to the throne of Peter.  He says that in such a case, the pope's
acts would be automatically void, and he would not be the pope, even if
he had been accepted and obeyed as true Pope by the whole Church.  This
papal docuмent shows the mind of the Church on this matter.  In such a
case, Paul IV is calling upon Catholics to resist such a "pope" with all
their might.

I don't have sources for any of the above, so take with a grain of salt.

But I'm pretty sure Pius X and Pius XII changed the conclave rules.  I've also heard that this was brought up as a challenge to Vatican I on papal infallibility.

TBH I don't see how a bull like this is practically enforceable.  Like even if a heretic actually does (like in the eyes of God) automatically lose office, who gets to decide that he's lost his office?

Yeah, I think for that to be practically applicable, I think some future pope would have to declare that that pope had in fact been an antipope and that that bull had applied to him.  That would still be questionable, but I could see a case where, say, if some future Pius X type pope made such a declaration about Francis, we could be more sure about said future Pius X's pontificate than that of Francis.

But I don't see how the way that sedes practically use this isn't quasi Protestant.  "Well I think he's a heretic so I won't obey him."  If that was the correct conclusion under the Roman paradigm, wouldn't it be more reasonable to just conclude that Rome was wrong about what the Church is and go EO?
I mean my problem here is this is basically unfalsifiable.  The Church's claims can't ever be tested if, in any event where they would be falsified, you can just be like "nah, antipope"

"We have been concerned also lest it may befall Us to see the abomination of desolation, which was spoken of by the prophet Daniel, in the holy place."

https://www.mostholyfamilymonastery.com/catholicchurch/cuм-ex-apostolatus-officio-pope-paul-iv/#.XL_gK5NKi1Y

obviously we don't have the authority to give any formal declaration...nevertheless Jєωgorlio is  a heretic..and not a Catholic...and not a true pope

we as catholics can judge him a heretic

which can only be judged by Church authorities.

If there had been no Papal Authority for 61 years, there are now no Church Authorities either, for Episcopal Authority comes from Papal Authority. If Pope Pius XII was the last "formal" Pope, there would be no "formal" episcopal authority in the Church today, and consequently the Catholic Church has died, disappeared and defected, which is false and heretical to maintain. Consequently, not only sedevacantism but also the thesis called sedeprivationism is now known to be false.