The single most important argument for the current position of sedevacantism is not what’s required for a pope to lose his office, because sedevacantists don’t argue that a pope has lost his office. The primary issue involves whether a pope can be head of a false religion.
To distinguish itself apart from all false churches, the Catholic Church teaches that it can be identified by four marks: One, Holy, Catholic, and Apostolic.
If the church under Francis I and the Second Vatican Council (Vatican 2) lack any one of the four marks, then it necessarily follows that Francis I and his church are counterfeit.
First Mark “One”
The Roman Catechism explains the first mark as unity:
“The faith which all are bound to believe and to profess is one.”
That faith which all are bound to believe and to profess consists in the written word of God and in tradition, and those which are proposed by the Church, either in a solemn pronouncement or in her ordinary and universal teaching power, to be believed as divinely revealed, all the judgments and decrees of the Apostolic See, and all points of doctrine proposed by the Church’s authoritative magisterium. 
Pope Pius XI declared in Mortalium Animos #10: “For since the mystical body of Christ, in the same manner as His physical body, is one, compacted and fitly joined together, it were foolish and out of place to say that the mystical body is made up of members which are disunited and scattered abroad: whosoever therefore is not united with the body is no member of it, neither is he in communion with Christ its head.”
To the contrary, The Catechism of the Catholic Church promulgated by John Paul II echoes Vatican 2 on the mystery of the Church:
This is the one Church of Christ which in the Creed is professed as one, holy, catholic and apostolic, (12*) which our Saviour, after His Resurrection, commissioned Peter to shepherd, (74) and him and the other apostles to extend and direct with authority, (75) which He erected for all ages as “the pillar and mainstay of the truth”. (76) This Church constituted and organized in the world as a society, subsists in the Catholic Church, which is governed by the successor of Peter and by the Bishops in communion with him, (13*) although many elements of sanctification and of truth are found outside of its visible structure. These elements, as gifts belonging to the Church of Christ, are forces impelling toward catholic unity. (LG ch 1)
If the one Church, constituted and organized in the world as a society, subsists in (subsistit in) the Catholic Church, then the one Church must necessarily be more than the Catholic Church. 
However, Pope Pius XII repeatedly taught that “the Mystical Body of Christ and the Roman Catholic Church are one and the same thing.” (Mystici Corporis Christi, 1943, Humani Generis, #27, 1950.)
Vatican 2’s explanation is in line with the Protestant understanding of the profession of the Creed insofar as its teaching consists of a visibly divided Church unified in spirit only. It denies the mark of oneness as taught by the Roman Catechism and the popes and councils before the 1960’s.
Second Mark: “Holy”
The Roman Catechism explains the second mark:
“The Church is called holy because she is consecrated and dedicated to God; for so other things when set apart and dedicated to the worship of God were wont to be called holy, even though they were material. Examples of this in the Old Law were vessels, vestments and altars.”
A prime principle in being dedicated to God is seen in the First Commandment. Thou shalt not have strange gods before me. Thou shalt not make to thyself a graven thing, nor the likeness of anything that is in heaven above, or in the earth beneath, nor of those things that are in the waters under the earth. Thou shalt not adore them, nor serve them. (Ex. 20:3)
The conciliar popes, John Paul II and Benedict XVI, instituted and praised as good three Assisi Events where strange gods were worshiped before God. The rejection of the First Commandment is institutionalized by the church of Vatican 2.
Woe to you that call evil good, and good evil: that put darkness for light, and light for darkness: that put bitter for sweet, and sweet for bitter. (Isaias 5:20)
The Roman Catechism continues:
“The Church is also to be called holy because she is united to her holy Head, as His body; that is, to Christ the Lord,’ the fountain of all holiness, from whom flow the graces of the Holy Spirit and the riches of the divine bounty.”
St. Paul teaches in Galatians, “But though we, or an Angel from heaven, preach a gospel to you besides that which we have preached to you, let him be anathema. As we said before, so I say now again: If any one preach to you a gospel, besides that which you have received, let him be anathema.”
Therefore, anyone who brings another gospel is cut off from Christ, thus, is not holy!
Vatican 2 and the conciliar popes officially teach another gospel on the nature of the Church, and encourage prayer with non-Catholics, which has been solemnly condemned as contrary to the Divine law from Holy Scripture to Pope Pius XII. 
Therefore, the church of Vatican 2 lacks the mark of holiness.
Third Mark: “Catholic”
The Roman Catechism:
“She is also called universal, because all who desire eternal salvation must cling to and embrace her, like those who entered the ark to escape perishing in the flood. This (note of catholicity), therefore, is to be taught as a most reliable criterion, by which to distinguish the true from a false Church.”
Pope Eugene IV, Council of Florence, Cantate Domino: “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, cannot share in eternal life and will go into the everlasting fire which was prepared for the devil and his angels, unless they are joined to the Church before the end of their lives.”
Pope Leo XIII, Satis Cognitum # 9: “The practice of the Church has always been the same, as is shown by the unanimous teaching of the Fathers, who were wont to hold as outside Catholic communion, and alien to the Church, whoever would recede in the least degree from any point of doctrine proposed by her authoritative magisterium…But he who dissents even in one point from divinely revealed truth absolutely rejects all faith, since he thereby refuses to honor God as the supreme truth and the formal motive of faith… The Church alone offers to the human race that religion – that state of absolute perfection – which He wished, as it were, to be incorporated in it. And it alone supplies those means of salvation which accord with the ordinary counsels of Providence.”
To the contrary, The Catechism of the Catholic Church of John Paul II states:
“In her subsists the fullness of Christ’s body united with its head; this implies that she receives from him ‘the fullness of the means of salvation.’”
The catechism makes special reference to UR3 of Vatican 2 supporting the above statement:
“It follows that these separated churches and communities as such, though we believe them to be deficient in some respects, have by no means been deprived of significance and importance in the mystery of salvation. For the Spirit of Christ has not refrained from using them as means of salvation whose efficacy comes from that fullness of grace and truth which has been entrusted to the Catholic Church.”
Of course, the Vatican 2 religion advocates that clinging and embracing its religion is best because it has the fullness of means, but it doesn’t teach that its membership is a necessary precept because non-Catholic churches have a means to salvation, just not the fullness thereof.
Therefore, the church of Vatican 2 lacks the mark of catholicity.
Fourth Mark: “Apostolic”
It’s been demonstrated that the Vatican 2 church is not one, holy, or catholic, and it most certainly is not apostolic. It’s a sect founded in the 1960’s claiming to be Catholic, which necessarily means the church and its pope are counterfeit.
The Roman Catechism:
“The true Church is also to be recognised from her origin, which can be traced back under the law of grace to the Apostles; for her doctrine is the truth not recently given, nor now first heard of, but delivered of old by the Apostles, and disseminated throughout the entire world. Hence no one can doubt that the impious opinions which heresy invents, opposed as they are to the doctrines taught by the Church from the days of the Apostles to the present time, are very different from the faith of the true Church.
That all, therefore, might know which was the Catholic Church, the Fathers, guided by the Spirit of God, added to the Creed the word Apostolic. For the Holy Ghost, who presides over the Church, governs her by no other ministers than those of Apostolic succession. This Spirit, first imparted to the Apostles, has by the infinite goodness of God always continued in the Church. And just as this one Church cannot err in faith or morals, since it is guided by the Holy Ghost; so, on the contrary, all other societies arrogating to themselves the name of church, must necessarily, because guided by the spirit of the devil, be sunk in the most pernicious errors, both doctrinal and moral.”
The church of Vatican 2 has erred in both faith and morals. Therefore, it can’t be the one true Church founded by Christ and guided by the Holy Ghost.
The church of Vatican 2 is guided by the spirit of the devil. Therefore, it must be sunk in its most pernicious doctrine and moral errors along with its head.
A true pope can’t be head of a false religion.
The church of Vatican 2 is a false religion.
The Vatican 2 popes are not true popes.
 The First Vatican Council (Vatican 1) stated:
[The object of faith]. Further, by divine and Catholic faith, all those things must be believed which are contained in the written word of God and in tradition, and those which are proposed by the Church, either in a solemn pronouncement or in her ordinary and universal teaching power, to be believed as divinely revealed. (Dogmatic Constitution concerning the Catholic Faith, Ch. 3, FIRST VATICAN COUNCIL, Pope Pius IX) (Denz. 1792)
Pope Pius IX stated: “And, we cannot pass over in silence the boldness of those who “not enduring sound doctrine” [II Tim. 4:3], contend that “without sin and with no loss of Catholic profession, one can withhold assent and obedience to those judgments and decrees of the Apostolic See, whose object is declared to relate to the general good of the Church and its right and discipline, provided it does not touch dogmas of faith or morals.” There is no one who does not see and understand clearly and openly how opposed this is to the Catholic dogma of the plenary power divinely bestowed on the Roman Pontiff by Christ the Lord Himself of feeding, ruling, and governing the universal Church.” (Pope Pius IX Quanta Cura Dec 8, 1864)
“You will firmly abide by the true decision of the Holy Roman Church and to this Holy See, which does not permit errors.” (Lateran Council V, Bull ‘Cum postquam’ by Pope Leo X)
Pope Leo XIII, Satis Cognitum (# 9), June 29, 1896: “The practice of the Church has always been the same, as is shown by the unanimous teaching of the Fathers, who were wont to hold as outside Catholic communion, and alien to the Church, whoever would recede in the least degree from any point of doctrine proposed by her authoritative Magisterium.”
 In an interview with the German newspaper, Frankfurter Allgemeine, Ratzinger, aka Benedict XVI, stated: “When the Council Fathers replace the word ‘is,’ used by Pius XII, with the word ‘subsistit,’ they did so for a very precise reason. The concept expressed by ‘is’ (to be) is far broader than that expressed by ‘to subsist.’ ‘To subsist’ is a very precise way of being, that is, to be as a subject, which exists in itself. Thus the Council Fathers meant to say: the being of the Church as such extends much further than the Roman Catholic Church, but within the latter it acquires, in an incomparable way, the character of a true and proper subject.”
Vatican 2 theologians confirm this meaning taught by Ratzinger.
Avery Cardinal Dulles, a member of the International Theological Commission: “The Church of Christ is not exclusively identical to the Roman Catholic Church. It does indeed subsist in Roman Catholicism but it is also present in varying modes and degrees in other Christian communities.” (Toward the Church of the Third Millennium: Verso la Chiesa del Terzo Millennio, Brescia: Queriniana, 1979)
Fr. Edward Schillebeeckx, one of the main drafters of Vatican II documents, stated: “It is difficult to say that the Catholic Church is still one, Catholic, apostolic, when one says that the others (other Christian communities) are equally one, Catholic and apostolic, albeit to a lesser degree. —- at Vatican Council II, the Roman Catholic Church officially abandoned its monopoly over the Christian religion.”
On May 25, 1995, John Paul II, in Ut Unum Sint, n. 59, approved the Balamand declaration.
1993 Balamand Statement:
On each side it is recognized that what Christ has entrusted to his Church – profession of apostolic faith, participation in the same sacraments, above all the one priesthood celebrating the one sacrifice of Christ, the apostolic succession of bishops – cannot be considered the exclusive property of one of our Churches.
Comment: Apostolic faith consists on recognizing the authority of Rome as Head of the Church. Only the Catholic Church professes the apostolic faith. Yet the document denies this truth.
14. It is in this perspective that the Catholic Churches and the Orthodox Churches recognize each other as Sister Churches, responsible together for maintaining the Church of God in fidelity to the divine purpose, most especially in what concerns unity.
In a Jan. 22, 2013 L’Osservatore Romano article titled: The divisions among Christians disfigure the face of the Church, it was written that Benedict XVI said, “One of the gravest sins ‘that disfigure the Church’s face’ is the sin ‘against her visible unity’, and, in particular, ‘the historical divisions which separated Christians and which have not yet been surmounted’”.
The two references of the “Church’s face” and “Against her visible unity” is a point blank denial of the dogma that the visible Church is one in faith.
 Vatican 2: UNITATIS REDINTEGRATIO – Decree on Ecumenism
CHAPTER II – THE PRACTICE OF ECUMENISM
8. This change of heart and holiness of life, along with public and private prayer for the unity of Christians, should be regarded as the soul of the whole ecumenical movement, and merits the name, “spiritual ecumenism.”….
In certain special circumstances, such as the prescribed prayers “for unity,” and during ecumenical gatherings, it is allowable, indeed desirable that Catholics should join in prayer with their separated brethren. Such prayers in common are certainly an effective means of obtaining the grace of unity, and they are a true expression of the ties which still bind Catholics to their separated brethren. “For where two or three are gathered together in my name, there am I in the midst of them”.