By what standard are you judging that?  Rejection of Vatican II?  Are you assuming that someone who accepts Vatican II isn't Catholic?  Or are you judging by some other standard?

Let Archbishop Lefebvre answer your question:

“To whatever extent pope, bishops, priests or faithful adhere to this new Church, they separate themselves from the Catholic Church.” (July 29, 1976, Reflections on the Suspension a divinis)

The smoking chimney only proves that an election was THOUGHT to have occurred.  It’s circuмstantial evidence only.  But when added to other circuмstantial eveidence (Siri comments, FBI investigator comments) it paints a picture.  But hardly an airtight case.  

Secondly, even if it could be proved that Siri was elected, it doesn’t matter at all for today.  Because Siri is dead, JPII is dead and the conclave after both of them would’ve been legitimate (ie Benedict XVI).  So now, are we dealing with ANOTHER forced resignation (Benedict), which allows a false pope (Francis) to rule just as Siri’s forced resignation potentially allowed Paul VI and JPII to falsely rule?  I don’t know, but the Siri thesis has no bearing on the Francis problem today.  

As we've seen, Fr. Boulet in 2004 had the best response to the Cardinal Siri thesis, "But, the most important reason why we must discard the "Pope Siri" theory is the fundamental principle that a peaceful acceptance of a pope by the Universal Church is the infallible sign and effect of a valid election. All theologians agree on that point. Cardinal Billot says: "God may allow that a vacancy of the Apostolic See last for a while. He may also permit that some doubt be risen about the legitimacy of such or such election. However, God will never allow the whole Church to recognize as Pontiff someone who is not really and lawfully.  Thus, as long as a pope is accepted by the Church, and united with her like the head is united to the body, one can no longer raise any doubt about a possible defective election… For the universal acceptance of the Church heals in the root any vitiated election."21 (4.3 the Case of Cardinal Siri) http://fsspx.com/Communicantes/Dec2004/Is_That_Chair_Vacant.htm What may have happened is (1) God is showing us just how close we came to getting a good Pope (Cardinal Siri also wanted the dogma of Mediatrix of All Graces to be defined, and I think favored Russia's explicit Consecration to the Immaculate Heart - H.E. was handpicked by Pope Pius XII, and would have been a good Pope), but because of lack of prayer and desire for it, lukewarmness and indifference to sanctity, we did not deserve to obtain it, (2) there may have been interference in the Papal election and so on, and therefore God withdrew His protection just a little, e.g. by the Church no longer using anathemas. But at the end of the day, as Siscoe's article also says, all these conspiracy theories will be endless unless we understand and believe the doctrine of universal acceptance - universal acceptance (1) heals in the root every fault committed (2) proves infallibly the existence of all the required conditions, as Cardinal Billot has explained for us. Siscoe rightly says, "The doctrine of the peaceful and universal acceptance, when properly understood, proves beyond any possible doubt that Francis’s election was valid and refutes each and every objection that has been raised against it. Those who understand this “sound doctrine” (2 Tim. 4:3), and accept it, will know who the true pope is, while those who “turn their hearing away from the truth” by rejecting it will continue to be “tossed to and fro and carried about” by the latest conspiracy theory or fallacious argument. Before continuing, I should note that there was a time when I also had doubts, or at least questions ..."

Note that both our Holy Father Pope St. Pius X and the "last Pope" Pope Pius XII have decreed concerning Papal Elections, "34. No Cardinal, by pretext or reason of any excommunication, suspension, interdict or other ecclesiastical impediment whatsoever can be excluded in any way from the active and passive election of the Supreme Pontiff. Moreover, we suspend such censures for the effect only of this election, even though they shall remain otherwise in force.” (Pope Pius XII, Cons. “Vacantis Apostolicae Sedis,” 8 December 1945)

Rather strange if God knew an alleged 61+ year sede vacante was to come after him, don't you think? In any case, UA settles the issue.

So, it is certain Pope Paul VI became Pope in the 60s (though it is likely he had an impersonator, or others who falsified some docuмents). And it is likely, though again God may well have preferred Pope Benedict XVI remain Pope if we had prayed for it and done the same, He has indeed allowed Pope Francis to become Pope, as the universal acceptance of his election to the pontificate proves in the external forum. Note that as Pope Pius XII said, the OUM's acceptance is a sufficient sign of the Church's declaration.

It is understandable that some want to take the route of "the Man on the Cross is not Jesus; not the same Man we saw working miracles". Or "He Who suffers like this cannot be God", as Archbishop Lefebvre said, regarding the mystery of the Church's Passion.

What God wants imho is souls who will make reparation, offer ceaseless prayers and sacrifices for this crisis to end. We must love the Holy Father, the Vicar of Christ, the Successor of St. Peter, and labor to save his soul. That's what Sr. Catherine Emmerich also asked of us, there's a text in St. Bridget where she says the Pope who tries to lift the discipline of clerical celibacy and dies like that will be lost. https://veritas-vincit-international.org/2014/08/16/blessed-anne-catherine-emmerichs-prophecy-on-the-two-popes/

“I came to the Church of Peter and Paul (Rome) and saw a dark world of distress, confusion, and corruption, through which shone countless graces from thousands of saints who there repose…”  “I saw the fatal consequences of this counterfeit church: I saw it increase; I saw heretics of all kinds flocking to the city. I saw the ever-increasing tepidity of the clergy, the circle of darkness ever widening…”

“Again I saw in the midst of these disasters the twelve new Apostles laboring in different countries, unknown to one another, each receiving streams of living water from on high They all did the same work. They know not whence they received their tasks; but as soon as one was finished, another was ready for them…”  

“The Jews shall return to Palestine, and become Christians toward the end of the world.

October 22, 1822

“Very bad times will come when non-Catholics will lead many people astray. A great confusion will result. I saw the battle also. The enemies were far more numerous, but the small army of the faithful cut down whole rows of enemy soldiers. During the battle, the Blessed Virgin stood on a hill, wearing a suit armor. It was a terrible war. At the end, only a few fighters for the just cause survived, but the victory was theirs…”

Say Siri was elected, and was really pope, and John XXIII to JPII were antipopes.  That could arguably make the cardinals appointed by those 4 popes not real cardinals.

But even aside from that, if the 1968 rites of episcopal consecration were set up by antipopes, they could be invalid, which would mean Benedict could have not truly been a bishop.  In which case he couldn't be a bishop of Rome.

Great points.  Maybe those Cardinals weren't true Cardinals, but there were still other Cardinals who voted.  Would the election still be binding?  Maybe?  I don't know.

The episcopal rites are not 100% invalid, so his episcopal orders could've been ok.  My understanding is that his election would've still been valid, even if he's not a bishop.  He would've been "pope elect" until receiving full orders.  In theory, any catholic male can be elected pope (i'm sure there's an age threshold), they just do not receive full papal powers until they are made a bishop.

I'm not saying that's true, but if it was why wouldn't it matter?

I think it doesn't matter, for a number of reasons:
1) Even if he was made a bishop and validly elected, I believe he's not orthodox, so his spiritual office is impaired through the various ecclesiastical penalties that he would incur for his heresies that he promoted before his papacy, which he never publically abjured.  So, in the theory of sedeprivationism, he holds the material office only and his spiritual office is impaired, until (if) he converts.

2) We don't have concrete proof that Siri was forced to resign, so we cannot draw any conclusions from this situation.  We must presume that Siri was not elected.  I think this is a situation where the Church will re-write the history books, when facts come to light in the future, proving the Siri thesis.

3) Even if +Benedict's election were invalid, due to the false-Cardinals who voted and due to his not being a bishop, we would be in the same boat as situation #1, i.e. a person occupies the material office but he has no spiritual office (or it's impaired).  Same result from no pope or a bad pope - no spiritual guidance and no protection of the Truth.

The fact that a non-pope is in charge of the material office is small problem, because the material office deals with administrative tasks, much like a CEO of a company.  Not much damage one can do here.

4) I truly believe, based on numerous prophecies, that at some point in the near future, the muslims will invade rome, cause havok and kill most all Cardinals in the Vatican.  Then, we may be without a pope for a while, or some prophecies say a new pope will be elected but he will flee rome under duress and this is when the truly bad, false pope will arise, who will try to usher in the one-world religion.  Either way, this will last a short time, until the 3 days of darkness, when most of the world will die, and the Chruch will start fresh, completely fresh, in those days afterwards.  So...all of the scenarios we talk about today and all of the problems and "what ifs" are extremely complex.  And God surely knows our doubts and fears and He certainly knows how to clean up His Church and to make a new start, so that there would be absolutely NO question as to who was the true pope.  He will do this some day and it will be glorious.

2) We don't have concrete proof that Siri was forced to resign, so we cannot draw any conclusions from this situation.  We must presume that Siri was not elected.  I think this is a situation where the Church will re-write the history books, when facts come to light in the future, proving the Siri thesis.

Right, we don't have smoking-gun proof for the Siri Thesis.  Nevertheless, the known facts are sufficient to constitute a positive doubt.  And I'll come back to your objections to sede-doubtism when I have time.

I don't want to get derailed by quibbling over whether a layman can be elected.  Let's just say it's a tonsured cleric who is not a priest and not a bishop.

You're claiming that the second he accepts the election, he receives full papal power.  But that's nonsense.  He cannot suddenly exercise Magisterium if he's not even part of the Ecclesia Docens.  He can receive the material aspects of power, such as making appointments, but he cannot formally exercise the office of teaching and ruling.  How can a mere cleric rule over bishop?

This is an earlier article, https://gloria.tv/article/PydgSw3mJAWF2w9rLGgxGBa4h

"The Conditions for a Valid Election

John of St. Thomas also addresses issues related to the conditions for a valid election – both the conditions required for the electors, and the conditions required for the one elected. For example, the electors must be true Cardinals; they must have the intention of electing the Popes, and they must follow the laws currently in place for a valid election. There are also conditions for a person to be validly elected Pope. He must be a male and baptized (positive conditions) and he must not be insane or a public heretic (negative conditions). John of St. Thomas explains that the infallible certitude we have that the man is Pope (which we know when he is peacefully elected and/or accepted as Pope by the entire Church as soon as it becomes known) provides infallible certitude that all of the pre-requisite conditions for his validity have been met. He wrote:

"The acceptance and definition of the Church [i.e., that this man is Pope], inasmuch as it gives the certitude of faith [concerning his legitimacy], does not touch upon the conditionsof the election, or the intention and genuine identity of the electors, without intermediary, but rather mediately, and as a logical consequence of what it immediately touches upon: namely, that whoever is elected by the persons that the Church designates to choose a Pope in her name, by the very fact that he is accepted by the Church as legitimately elected, is in fact Pope. This latter is what the definition of Martin V[37], related above, as well as the acceptance of the Church, is really about. But from the de fide truth that this man is Pope, it follows as a consequence that all the requisite conditions must have been observed. (…) because it is de fide that this man in particular, accepted by the Church as canonically elected, is the Pope, the theological conclusion is drawn that there were genuine electors, and a real intention of electing, as well as the other requisites [e.g., the proper intention of the resigning Pope, if the election follows a papal resignation], without which the de fide truth could not stand. Therefore, we have the certainty of faith, by a revelation implicitly contained in the Creed and in the promise made to Peter, and made more explicit in the definition of Martin V, and applied and declared in act (in exercitio) by the acceptance of the Church, that this man in particular, canonically elected according to the acceptance of the Church, is Pope."

He then addresses the conditions for the one being elected. What he says refutes the claim of the Sedevacantists that the recent Popes were all “public heretics” prior to their election, which allegedly rendered their elections null. He begins with the following objection: “We do not have the certitude of faith that this subject is susceptible of this dignity [i.e., that he meets the conditions]; neither, then, do we have the certitude of faith that he has, in fact, received this dignity.” He replies as follows:

“The answer here is similar to the preceding. Prior to the election, there is a moral certainty that all these conditions required in the person are actually met. After the fact of the election and its acceptance, the fulfillment of these conditions is known with the certainty of a theological conclusion, since they have, per se, a logical implication with a truth that is certain, and certified by faith. (…) the truth that is defined and accepted by the Church is not that this man is baptized or ordained, etc., but that this man is truly pope. (…) That he is baptized and meets the other requirements [i.e., that he is not a public heretic] … is inferred as a consequence; (…) the truth that this man has been ordained, and has the power of order (that is, of the priesthood or episcopate), is certain in the same way as the truth that he is baptized is certain; namely, not as a truth immediately de fide, but as a theological conclusion necessarily connected with the truth that he is the Pope and the rule of faith in the Church."

He goes on to explain, as Cardinal Billot did above, that God will not permit a man to be elected Pope, and accepted as Pope by the Church, who does not meet the necessary conditions:

"t is not merely a pious belief, but a theological conclusion (as we have stated), that God will not permit one to be elected and peacefully accepted by the Church who in fact does not meet the conditions required; this would be contrary to the special providence that God exercises over the Church and the assistance that she receives from the Holy Ghost."

Next, he addresses the objection of those today who hold to the novel “Material/Formal Pope thesis” – that is, that the Pope has indeed been legitimately and validly elected and truly holds the office, but, due to an alleged impediment (heresy), he did not receive the jurisdiction from God to become a true Pope. John of St. Thomas refutes this novel thesis by stating the obvious:

"Nor is there a real difference between the proposition, 'This man is properly elected,' and, 'This man is Pope,' since to be accepted as the Supreme Pontiff and to be the Supreme Pontiff are the same, just as it is the same for something to be defined, and for the definition to be legitimate.

As we saw earlier, Fr. Catechini taught that those who would reject the legitimacy of a Pope, who had been accepted as such by the Church, would be guilty of a mortal sin against the faith. John of St. Thomas goes one step further by saying such a person would be a heretic.

"Whoever would deny that a particular man is Pope after he has been peacefully and canonically accepted, would not only be a schismatic, but also a heretic; for, not only would he rend the unity of the Church… but he would also add to this a perverse doctrine, by denying that the man accepted by the Church is to be regarded as the Pope and the rule of faith. Pertinent here is the teaching of St. Jerome (Commentary on Titus, chapter 3) and of St. Thomas (IIa IIae Q. 39 A. 1 ad 3), that every schism concocts some heresy for itself, in order to justify its withdrawal from the Church. Thus, although schism is distinct from heresy, in … the case at hand, whoever would deny the proposition just stated would not be a pure schismatic, but also a heretic, as Suarez also reckons."

namely, that whoever is elected by the persons that the Church designates to choose a Pope in her name, by the very fact that he is accepted by the Church as legitimately elected, is in fact Pope.

Notice.  Legitimately Elected + Accepted by the Church

Both are required.

Siri was the man who was legitimately elected in 1958.