There exist
  1. the true Catholic Church; and
  2. a false, revolutionary conciliar church.
Our salvation depends upon belonging to the first, and staying out of the second:
This conciliar church is not merely a liberal way of thinking. It is a human organization (which deceptively uses the name “Catholic”) and has its own:
This conciliar church is a different (and false) religion. It uses the power structures of the Catholic Church for its own ends, punishing (including excommunicating) those who resist this false conciliar church.8 Again, the conciliar church is a human organization, not just a mindset.
The “old” SSPX knew and taught these truths concerning the conciliar church.9
Because this conciliar church is a new religion, we must entirely shun it, because Catholics must avoid false religions.
Some conciliar leaders are not in the Catholic Church’s hierarchy. A faithful Catholic must avoid them and never follow them.
Other leaders (e.g., Pope Francis) are both conciliar church leaders and in the Catholic Church’s hierarchy. Though they more often act as conciliar church leaders, Catholics must obey them when they command something Catholic because Catholics must obey all good or neutral commands of their lawful superiors.10

Summary of the above section

We always belong to the Catholic Church by living faithful Catholic lives. Bad superiors are unable to exclude us from the Catholic Church for being faithful Catholics.
The false, conciliar church is a human organization and a new religion sharing some members with the Catholic Church. (Of course, it is an objective mortal sin to belong to the conciliar church.)

Pope Pius XI Condemns Joining—and Even Being Willing to Join—a Religious Association which includes Members of a False Religion

In Mortalium Animos, Pope Pius XI condemned joining, and even being willing to join, a religious association with those who do not have our Traditional Catholic Faith. He declared:
Since charity is based on a complete and sincere faith, the disciples of Christ must be united principally by the bond of one faith. Who then can conceive a Christian Federation, the members of which retain each his own opinions and private judgment, even in matters which concern the object of faith, even though they be repugnant to the opinions of the rest? And in what manner, We ask, can men who follow contrary opinions, belong to one and the same Federation of the faithful?
Mortalium Animos, Pope Pius XI, §9.
Pope Pius XI’s words condemn those groups recognized by the conciliar church, because they have joined a religious association (federation) with modernist Rome. This is one aspect of the manifold wickedness of the indult groups (e.g., Society of St. Peter). To join the false conciliar church is evil like joining the false Lutheran church.
Moreover, Pope Pius XI condemns even the idea of joining with (i.e., being “recognized” by) the modernists of the conciliar church. He rhetorically asks what faithful Catholic can even conceive of this idea? This is exactly the “new” SSPX’s evil desire. The N-SSPX both conceives and desires recognition from the conciliar church.
Seeking “recognition” from the modernists can never make one more Catholic, because anyone faithfully professing and practicing the Catholic faith already belongs to the true Catholic Church. He cannot be made “more Catholic” by being recognized by the modernists of the conciliar church. Similarly, if the conciliar church ostracizes a faithful Catholic because of his fidelity, this does him no harm.11
Although conciliar revolutionaries cannot make a Catholic more part of the true Catholic Church, they can make him a member of the false conciliar church. A faithful Catholic must boldly reject even the idea of this “recognition”! He must reject all possibility of joining the conciliar church. As Pope Pius XI teaches, a Catholic betrays Christ by even desiring to join a religious federation with persons who don’t profess the Catholic Faith. Pope Pius XI explains that no faithful Catholic would even conceive of this betrayal!
Recognition (i.e., joining the conciliar church) is evil, just like joining with the Lutherans! The “new” SSPX seeks this evil. Concerning recognition by modernist Rome, Bishop Fellay explained:
We have a sine qua non condition: if this condition is not met, then we will not move. And this condition is for us to be able to remain as we are.12
Unlike Bishop Fellay, a faithful Catholic would no more consider joining with the (false) conciliar church than he would consider joining with the (false) Lutheran church which agreed to his condition to be left “as he is”!
For a faithful Catholic, the issue is not whether modernist Rome will grant the “right condition” for recognition. It is evil, regardless of conditions, to join (or desire to join) a religious federation with modernist Rome, which does not profess the Catholic Faith.

Faithful Catholics must reject even the idea of recognition by modernist Rome.

Following Pope Pius XI, faithful Catholics may not even consider joining the conciliar church. The N-SSPX and its followers betray Christ by even considering union.
The “old” SSPX knew and taught that priests already betray Christ by even considering possible recognition by the conciliar church. Faithful priests have a duty to plainly reject this possibility ahead of time. Here is how all SSPX major superiors stated this truth in 1988:
The Catholic faithful have a strict right to know that the priests to whom they have recourse are not in communion with a counterfeit Church which is evolutionary, pentecostalist, syncretist. ... [W]e ask for nothing better than to be declared outside of this impious communion of the ungodly.13
The “new” SSPX leaves open the possibility of accepting full recognition by the conciliar church and has already accepted (unnecessary) confession and marriage indults from it. The N-SSPX not only betrays Christ but violates the “strict right ... [of] the Catholic Faithful ... to know that the priests to whom they have recourse are not in communion with a counterfeit Church”—to use the words of the (old) SSPX.
Beginning in 1988, having just survived the trap of the conciliar church proposing to recognize him, Archbishop Lefebvre refused to even discuss making a practical deal with Rome until Rome converted. He declared that:
[I]f I live a little while, supposing that Rome calls for a renewed dialogue, then, I will put conditions. I shall not accept being in the position where I was put during the dialogue. No more.
I will place the discussion at the doctrinal level: “Do you agree with the great encyclicals of all the popes who preceded you? Do you agree with Quanta Cura of Pius IX, Immortale Dei and Libertas of Leo XIII, Pascendi Gregis of Pius X, Quas Primas of Pius XI, Humani Generis of Pius XII? Are you in full communion with these Popes and their teachings? Do you still accept the entire Anti-Modernist Oath? Are you in favor of the social reign of Our Lord Jesus Christ? If you do not accept the doctrine of your predecessors, it is useless to talk!14

Conclusion

The “new” SSPX slowly poisons its priests and faithful into gradually accepting union with the modernists of the conciliar church. However, the N-SSPX betrays Christ and the faithful by even considering this possibility.
Let us pray for these priests, who betray Christ and their followers. Let us pray also for their naïve, trusting followers they lead astray.
  1. For more information and quotation of the dogmatic decrees, read this article: Bishop Williamson Promotes Vatican II’s Heresy That People Can be Saved Outside the Catholic Church↩
  2. The Summa explains this truth as follows:
    An excommunication may be unjust ... on the part of the excommunication, through there being no proper cause, or through the sentence being passed without the forms of law being observed. In this case, if the error, on the part of the sentence, be such as to render the sentence void, this has no effect, for there is no excommunication.
    Summa Supp., Q.21, a.4, respondeo (emphasis added).
    Emphasizing the ineffectiveness of a void excommunication on a man’s charity, the Summa adds:
    No man can be justly excommunicated except for a mortal sin, whereby a man is already separated from charity, even without being excommunicated. An unjust excommunication cannot deprive a man of charity, since this is one of the greatest of all goods, of which a man cannot be deprived against his will.
    Summa Supp., Q.21, a.1, ad 2.↩
  3. For an explanation of our duty to avoid conciliar churches and why they are especially bad places to pray, read this article: The evil of going into a conciliar church to pray↩
  4. Archbishop Lefebvre pointed out that the conciliar church’s new priesthood corresponds to its new mass. Declaration of November 21, 1974.
    Doubtful sacraments must be treated as invalid. For a thorough explanation of the doubts about the validity of the new ordinations and consecrations, see these Catholic Candle articles:
    ↩
  5. Archbishop Lefebvre declared this was one of the hallmarks of the new conciliar church, that it had “its own Bible, the ecumenical Bible”. Sermon, July 29, 1976.↩
  6. The Vatican lists the new conciliar decalogue here: http://w2.vatican.va/content/john-paul-ii/en/letters/2002/documents/hf_jp-ii_let_20020304_capi-stato.html↩
  7. The new conciliar “beatitudes” are listed here: https://www.lifesitenews.com/news/pope-proposes-new-beatitudes-for-a-new-age↩
  8. Archbishop Lefebvre was punished by the conciliar church. He wrote on July 29, 1976: “What can be clearer! In the future, one must obey and be faithful to the conciliar Church and no longer to the Catholic Church. This is precisely our problem; we are suspens a divinis by the conciliar Church and for the conciliar Church, of which we do not want to be a part.” (Emphasis added.)↩
  9. Here is Bishop Tissier’s article explaining that the conciliar church is an organized, human association—a false church—not merely a mindset. Is there a conciliar Church?
    To take one of countless earlier examples, the “old” SSPX published the following explanation in the SSPX’s Angelus magazine in April 2003, as part of the Q & A section of this issue:
    [Vatican II] is the introduction of a new religion, all within the exterior structure, hierarchy, language and ceremonies of the Catholic Church. ...
    It consequently cannot be denied that Vatican II attempts to constitute a new religion in radical rupture with all of Catholic Tradition and teaching, a new religion whose principal purpose is to exalt the natural dignity of the human person and to bring about a “religious” unity of mankind. However, the subtle cleverness of this operation must also be noted. It is the traditional hierarchical structure of the Church, its Mass, its devotions and prayers, its catechisms and teachings, and now even its Rosary that have all been infiltrated with the principles of the new religion. This new religion has been swallowed down unwittingly by many Catholics precisely because it hides, as a caricature, behind the outward appearance of Catholicism. The end result is a strange mixture of Catholicism and the new religion.
    This is the reason for which we have every right to condemn the post-Conciliar revolution for the new religion that it is, while at the same time we must respect the offices and functions of those who hold positions in the Church.
    See, Angelus magazine Q & A, April 2003↩
  10. The men in the Catholic Church’s hierarchy speak and act in ways which harm the Catholic Faith. However, a faithful and informed Catholic does not conclude that he knows those men are not “really Catholic” because they have excluded themselves from the Catholic Church. That conclusion would be the mortal sin of rash judgment. For a full explanation of this fact, read this article: Rash Judgment: Concluding the Pope is a Formal Heretic↩
  11. This truth is explained in the above quotes from the Summa.↩
  12. http://sspx.org/en/news-events/news/bp-fellay-gives-rome-clear-condition (emphasis added).↩
  13. Open Letter to Cardinal Gantin, Écône, July 6, 1988.↩
  14. Interview of Archbishop Lefebvre published in the November-December 1988 Fideliter Magazine. (emphasis added) http://www.sspxasia.com/Documents/Archbishop-Lefebvre/Archbishop_Lefebvre_and_the_Vatican/Part_II/1988-11.htm (emphasis added).↩