It shall appear to WHOM?  To Grandma Jones?  I knew a guy who considered Pius IX a non-pope because it "appeared" to HIM that Pius IX had embraced heresy.

That's between him and God. But there is such a thing as manifest heresy which clearly and explicitly contrary de fide teachings. Why is the ability to know this constantly questioned in general because many people don't possess the necessary faculties to discern it?
If my priest stands before the congregation and says:" I've converted to Islam", is his apostasy, in good conscience, not to be taken as an objective fact? If a bishop states publically that Jesus Christ did not come in the flesh and die on the cross, is there some reaosonable doubt left somewhere as to my obligation to shun him as a heretic? And if an apparent pope teaches moral relativism and gradualism as part of his authentic magisterium and, after being corrected, persists in this, what am I to think?
I do not know with certaintiy, but my doubt would be sufficient that in good conscience I cannot accept him, but neither will I publically denounce him: rather I will shun and let God judge my intentions, as he has not given me sufficient knowledge to make a final judgment.
And I am apparently permitted to act according to my conscience in discernment of this:
7. Finally, [by this Our Constitution, which is to remain valid in perpetuity, We] also [enact, determine, define and decree]:- that any and all persons who would have been subject to those thus promoted or elevated if they had not previously deviated from the Faith, become heretics, incurred schism or provoked or committed any or all of these, be they members of anysoever of the following categories:
(i) the clergy, secular and religious;
...
shall be permitted at any time to withdraw with impunity from obedience and devotion to those thus promoted or elevated and to avoid them as warlocks, heathens, publicans, and heresiarchs (the same subject persons, nevertheless, remaining bound by the duty of fidelity and obedience to any future Bishops, Archbishops, Patriarchs, Primates, Cardinals and Roman Pontiff canonically entering).” - Cum ex

Withdraw and avoid.
If there's division in the hierarchy, which an "imperfect council" implies, it all boils down to having to make an act of personal discernment until such time as a new pope is elected (and even then ...). Such a council may help many, but it is not a solution.
But that's precisely why I have been sede-agnostic ... totally confused ... what I am NOT going to do is become a formal schsimatic who akcnowledges a pope but refuses to submit to him.

No ONE PERSON can judge a Pope, that's true. And there isn't a set doctrine in regards to how to deal with a heretical Pope. Surely you know that. Various theologians have different formulas for dealing with the possibility for a Pope who is in heresy.

No, not "no one person". Read it again:
“… Neither by Augustus, nor by all the clergy, nor by religious, not by the people will the judge be judged… ‘The first seat will not be judged by anyone.’"

NOR BY ALL THE CLERGY. Nobody can judge a pope. No person, No group. No isntitution. No body.

Sedevacantists tend to believe that the Pope is equal to God,

Hyperbolic lies.

but he isn't. He can be judged by a body of men. How it is that they separate a heretic Pope from the papacy is not clear.

He CANNOT.

Can 1556. Prim Sedes a nemine iudicatur.

You can quote canon law till the cows come home in defense of your position, but quotes can be provided to the contrary.

The quotations still stand, they have legal force, you have not addressed them, and nothing you can quote can contradict that fact.

Also, you cannnot provide a single authoritative quotation that a pope may be judged by a body. Stop lying.

We do not live in the wild west, where law doesn't matter, and where the populace then takes matters into their own hands and administers frontier justice. That's not how the Church works.

Irrelevant.

Submit yourself to your pope.

And yet, Cum Ex assumes a trial.
Cum Ex...

"(vii) if perchance they shall have been Judges, their judgements shall have no force, nor shall any cases be brought to their hearing.;"

So, an HEARING is presumed in Cum Ex. That means, this declaration is only a guideline for determining within an hearing that someone proven guilty of heresy, Pope included, must be treated as written. But an hearing is presumed.

Bellarmine’s thinking regarding this matter is perfectly consistent with the mind of the Church, as we see expressed in Canon 10 of the Fourth Council of Constantinople. In response to the schism of Photius, the Council attached the grave penalty of excommunication to any layman or monk who, in the future, separated himself from his patriarch (the Pope is Patriarch of the West) before a careful inquiry and judgment by a synod.

“As divine scripture clearly proclaims, ‘Do not find fault before you investigate, and understand first and then find fault’. And does our law judge a person without first giving him a hearing and learning what he does? Consequently this holy and UNIVERSAL SYNOD justly and fittingly declares and lays down that no lay person or monk or cleric should separate himself from communion with his own patriarch before a careful inquiry and judgment in synod, even if he alleges that he knows of some crime perpetrated by his patriarch, and he must not refuse to include his patriarch's name during the divine mysteries or offices. (…) If anyone shall be found defying this holy synod, he is to be debarred from all priestly functions and status if he is a bishop or cleric; if a monk or lay person, he must be excluded from all communion and meetings of the church [i.e. excommunicated] until he is converted by repentance and reconciled”.

This proves nothing at all. Any trials would be for the heretic to prove his innocence (not the other way around). A Catholic always presumes dolus when encountering heresy in the external forum. Also the above quote from Constantinople IV says "OUR law". Ecclesiastical Law does not equal divine law. Here are some more quotes that show a heretic falls "ipso facto" out of the Church without a declaration by divine law and that dolus must be presumed...

St. Francis De Sales (17 th century), Doctor of the Church,  The Catholic Controversy , pp.  305-306: " Now when he [the Pope] is explicitly a heretic, he falls ipso facto from his dignity and out of the Church ..."  

St. Robert Bellarmine,  De Romano Pontifice, II, 30 :  “... for men are not bound, or able to read hearts; but when they see that someone is a heretic by his external works, they judge him to be a heretic pure and simple, and  condemn him as a heretic.


St. Robert Bellarmine,  De Romano Pontifice , II, 30:  “For, in the first place,  it is proven with arguments from authority and from reason that the manifest heretic is ‘ipso facto’ deposed . The argument from authority is based on St. Paul (Titus 3:10), who orders that the heretic be avoided after two warnings, that is, after showing himself to be manifestly obstinate –  which means before any excommunication or judicial sentence . And this is what St. Jerome writes, adding that the other sinners are excluded from the Church by sentence of excommunication, but the heretics exile themselves and separate themselves by their own act from the body of  Christ.”  


What have the Popes said?...

Pope Pius VI,  Auctorem fidei , Aug. 28, 1794:  “47. Likewise, the proposition which teaches that it is necessary, according to the natural and divine laws, for either excommunication or for suspension, that a personal  examination should precede , and that, therefore , sentences called ‘ipso facto’ have no other force than that of a serious threat without any actual effect” –  false, rash,  pernicious, injurious to the power of the Church, erroneous.  

Pope Pius XII,  Mystici Corporis Christi (# 22):  “As therefore  in the true Christian community there is only one Body, one Spirit, one  Lord, and one Baptism, so  there can be only one faith . And therefore if a man refuse to  hear the Church let him be considered –  so the Lord commands – as a heathen and a  publican. It follows that those who are divided in faith or government cannot be living in the unity of such a Body , nor can they be living the life of its one Divine Spirit  


Pope Innocent III,  Eius exemplo , Dec. 18, 1208:  “By the heart we believe and by the mouth we confess  the one Church, not of heretics ,  but the Holy Roman, Catholic, and Apostolic Church outside of which we believe that no  one is saved.”  

Pope Eugene IV,  Council of Florence , “Cantate Domino,” 1441:  “The Holy Roman Church firmly believes, professes and preaches that  all those who are  outside the Catholic Church , not only pagans  but also Jews or heretics  and  schismatics...  

Pope Leo XIII,  Satis Cognitum (# 13), June 29, 1896:  “You are not to be looked upon as holding the true Catholic faith if you do not teach  that the faith of Rome is to be held.


Pope Innocent IV,  First Council of Lyons , 1245:  “The civil law declares that  those are to be regarded as heretics, and ought to be subject  to the sentences issued against them, who even on slight evidence are found to have  strayed from the judgment and path of the Catholic religion.    

Pope St. Celestine:  “ The authority of Our Apostolic See has determined that the bishop, cleric, or simple  Christian who had been deposed or excommunicated by Nestorius or his followers,  after  the latter began to preach heresy shall not be considered  deposed or excommunicated .  For he who had defected from the faith with such preachings, cannot depose or remove anyone whatsoever .  

Pope Pius IX,  Quartus Supra (# 12), Jan. 6, 1873:  “Since the faction of Armenia is like this,  they are schismatics even if they had not yet  been condemned as such by Apostolic authority.”  


Pope Paul IV, Bull Cum ex Apostolatus Officio , Feb. 15, 1559: “6.  In addition, [by this Our  Constitution, which is to remain valid in perpetuity We enact, determine, decree and  define:] that if ever at any time it shall appear that... the Roman Pontiff, prior to his  promotion or his elevation as Cardinal or Roman Pontiff, has deviated from the Catholic  Faith or fallen into some heresy... (ii)  it shall not be possible for it to acquire validity  (nor for it to be said that it has thus acquired validity) through the acceptance of the  office, of consecration, of subsequent authority, nor through possession of  administration, nor through the putative  enthronement of a Roman Pontiff, or  Veneration, or obedience accorded to such by all, nor through the lapse of any period  of time in the foregoing situation;...”  


What does Canon Law say and how is it interpreted? We shall now see...

Canon 188 No. 4:
“All offices shall be vacant ipso facto (without a declaration required) by tacit resignation... #4 by public defection from the Catholic Faith.”

Canon 2200.2, 1917 Code of Canon Law: “ When an external violation of the law has  been committed, malice is presumed in the external forum until the contrary is proven.

A commentary on this canon by Rev. Eric F. Mackenzie, A.M., S.T.L., J.C.L, states: “The very commission of any act which signifies heresy, e.g., the statement of some doctrine contrary or contradictory to a revealed and defined dogma, gives sufficient ground for juridical presumption of heretical depravity...  Excusing circumstances have to be proved in the external forum, and  the burden of proof is on the person  whose action has given rise to the imputation of heresy. In the absence of such proof, all such excuses are presumed not to exist.”

That's why theologians such as John of St. Thomas felt that one would, practically speaking, need an Imperfect Council to issue a declaration of heresy.  Otherwise, as he said, there would be complete chaos in the Church.

And there is not chaos amongst those who claim Bergoglio is a true Pope? A publically heretical Pope will always produce chaos, regardless. Just look at all of the divisions currently in those who claim Bergoglio the clown is a true Pope. You have the Resistance (which is divided), the SSPX, St. Benedict Center, FSSP, Conservative NOers and Flaming NOers.  That's just off the top of my head. No declaration is needed for public defection from the faith and dolus is always presumed in the external forum until the contrary is proven...

Canon 188 No. 4:
“All offices shall be vacant ipso facto (without a declaration required) by tacit resignation... #4 by public defection from the Catholic Faith.”

Canon 2200.2, 1917 Code of Canon Law: “ When an external violation of the law has  been committed, malice is presumed in the external forum until the contrary is proven.

A commentary on this canon by Rev. Eric F. Mackenzie, A.M., S.T.L., J.C.L, states: “The very commission of any act which signifies heresy, e.g., the statement of some doctrine contrary or contradictory to a revealed and defined dogma, gives sufficient ground for juridical presumption of heretical depravity...  Excusing circumstances have to be proved in the external forum, and  the burden of proof is on the person  whose action has given rise to the imputation of heresy. In the absence of such proof, all such excuses are presumed not to exist.”

Alright, so please tell what doctrine tells us EXACTLY how we are to deal with a heretical pope. Please be specific. Notice that I'm asking about how to deal in a practical manner with a Pope who is in heresy. What does the Church teach on how to exactly go about it?

There is no such doctrine. That it doesn't exist doesn't change any facts. It's like asking if doctrine existed specifically to deal with the Western schism and antipopes - that's probably also the best case to look at to guess how things might resolve themselves. If one denies valid orders to the Novus Ordo bishops, it becomes more difficult to imagine.
I'll tell you what wasn't anyone's position during those times, though: we recognise you as the true pope but refuse to submit to you and your teaching.

Where does it say that laypersons are to proclaim that the seat is vacant

By the way, if you can discover any public statement by me saying that the seat is empty, you get a cookie.

There is no such doctrine. That it doesn't exist doesn't change any facts. It's like asking if doctrine existed specifically to deal with the Western schism and antipopes - that's probably also the best case to look at to guess how things might resolve themselves. If one denies valid orders to the Novus Ordo bishops, it becomes more difficult to imagine.
I'll tell you what wasn't anyone's position during those times, though: we recognise you as the true pope but refuse to submit to you and your teaching.

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.

The situation with the anti-popes is a different situation.

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. Though, of course, there is no doctrine which says that laymen are to proclaim and judge that the Pope is not a Pope, much less on how to treat the non-Pope after laypersons proclaim he is not the pope. Sedes try to force others to adopt their view, through intimidation. What Church teaching says that this behavior is acceptable and needed? 

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.

There is no doctrine, but there is a precedent that was established in the early Church, and Saint Robert Bellarmine uses this precedent as an example to back up his teaching on the Roman Pontiff.

Let's go back to your original statement where you said: "He can be judged by a body of men."

The POPE cannot be judged by any power on earth - period.  Now, if a pope were to become a manifest (public) heretic, he ceases to be pope, thus he can be judged and punished by the Church.  This is the teaching of St. Robert Bellarmine.  

I understand that there are theologians with differing opinions, but St. Robert's teaching stems from a precedent established during the Arian crisis.  

St. Roberts says in De Romano Pontifice:
"Therefore, the true opinion is the fifth, according to which the Pope who is manifestly a heretic ceases by himself to be Pope and head, in the same way as he ceases to be a Christian and a member of the body of the Church; and for this reason he can be judged and punished by the Church. This is the opinion of all the ancient Fathers, who teach that manifest heretics immediately lose all jurisdiction..."

He continues:
"The foundation of this argument is that the manifest heretic is not in any way a member of the Church, that is, neither spiritually nor corporally, which signifies that he is not such by internal union nor by external union."

He goes on to further explain what happened to Pope Liberius during the Arian crisis:
"In addition, unless we are to admit that Liberius defected for a time from constancy in defending the Faith, we are compelled to exclude Felix II, who held the pontificate while Liberius was alive, from the number of the Popes: but the Catholic Church venerates this very Felix as Pope and martyr. However this may be, Liberius neither taught heresy, nor was a heretic, but only sinned by external act, as did St. Marcellinus, and unless I am mistaken, sinned less than St. Marcellinus.

Then two years later came the lapse of Liberius, of which we have spoken above. Then indeed the Roman clergy, stripping Liberius of his pontifical dignity, went over to Felix, whom they knew [then] to be a Catholic. From that time, Felix began to be the true Pontiff. For although Liberius was not a heretic, nevertheless he was considered one, on account of the peace he made with the Arians, and by that presumption the pontificate could rightly [merito] be taken from him: for men are not bound, or able to read hearts; but when they see that someone is a heretic by his external works, they judge him to be a heretic pure and simple, and condemn him as a heretic."



Seems pretty Catholic to me.  

However, does anyone really believe that the heresiarchs in Rome are going to actually do anything about their fellow heretic?  

Since you admit there's no doctrine, then we do not have to accept the quotes you provide, and therefore we are allowed to form a view from the opposing theologians such as John of St. Thomas and Suarez, correct?