Although Traditionalists do not agree on the status of the claimant to the Papacy in Rome, they do agree on their claim to authority.
Bishop Pivarunas on Jurisdiction
(Reprinted with permission from the July, 2009, issue of Adsum)
http://www.christorchaos.com/BishopPivarunasonJurisdiction.htm
In an excellent article on this topic, Traditional Priests, Legitimate Sacraments, 1 Fr. Anthony Cekada makes reference to moral theologians who teach that there is a moral obligation for priests without faculties 2 to administer the Sacraments when the faithful are in serious need:
When priests who have the cura animarum are lacking, other priests are bound out of charity to administer the sacraments.… in serious need for a community, [such priests] are bound to administer the sacraments, even at the risk of their lives, as long as there is reasonable hope of assisting and there is no one else who will help.” This obligation binds under pain of mortal sin. (Merkelbach 3:87. Father Cekada’s emphasis) . . . .
S.S.P.X. Jurisdiction For Confession & Matrimony
(From the works of Rev. Fr. GLOVER, J.C.D. and of the priests of CAMPOS – Brazil – cf. “Cor Unum” #16)
http://www.advancedchristianity.com/pages/MPC/Docs/SSPX_Supplied_Jurisdiction.htm
By his ordination every priest receives the radical power to absolve, but to exercise this power validly he must receive also the power of jurisdiction; for, in the sacrament of penance the priest acts as a judge, and a judge must have authority over those whom he judges or his sentence is not binding.
1.The following have ORDINARY jurisdiction (as attached to their office):
*the Pope over the whole Church
*the Bishop over his diocese’s flock
*the pastor over his parishioners
2. All other priests have DELEGATED jurisdiction (can. 967) 3
and such a jurisdiction is delegated either by someone who has ordinary jurisdiction or by Church law itself. Now, jurisdiction for Confession is delegated by Canon Law itself:
* to all priests if the penitent is in danger of death (see can. 976)
* to the priests who absolve in common error, or in a positive & probable doubt (can. 144) 4
In such cases the Church is said to supply jurisdiction. Thus, even though SSPX priests do not enjoy a jurisdiction which is ordinary or delegated by the local bishop, the Church gives them the necessary jurisdiction in cases of danger of death, common error, and positive doubt. The Code of Canon Law directly gives them the power they need.
True Or False Pope
In 2015 two men, Sisco and Salza, wrote a book claiming to refute Sedevacantism, True or False Pope. Let us consider what they have to say about the authority of Traditionalist bishops on page 73: “We see that the only way Fr. Cekada can defend his Sedevacantist position is to reject the teaching of Pius XII that bishops receive their jurisdiction (mission) “directly from the Supreme Pontiff.” For Fr. Cekada, there is apparently no longer a distinction between the reception of orders and the reception of jurisdiction (a dogmatic distinction that is rooted in divine revelation and taught by the Church since the4 very beginning) because, well he doesn’t think we have a valid Pope.” Then they quote from John Lane, another Sedevacantist: “Private judgment erecting ministers of Christ. No public authority involved. This is worse than Anglicanism, which at least replaced the authority of the Church with secular authority. … Who’s to judge the fitness of this potential bishop? … What’s the authority of of a bishop without a mission from the Church?”
We could ask Sisco and Salza the same question about the Society of Saint Pius X, which has no more authority than the bishops of the sedevacantists? None of the Traditionalist bishops have any mission from the Church, because no Pope has given it to them. John Paul 2, who Archbishop Lefebvre claimed was his Pope, never gave Lefebvre or any of the four he consecrated as bishops any mission in the Church.
1 http://www.traditionalmass.org/images/articles/TradPriestLeg.pdf
2 Emphasis mine
3 A priest can have no jurisdiction at all and therefore could not validly hear confessions, except in danger of death.
4 This is the 1983 code, but he 1917 Code in Canon 209 says the same thing.