Sedevacantism denies the 4th mark, Apostolicity

Sedevacantist theology attributes the liturgical abuse and decline in authentic catholicity to the string of supposed anti-popes and heretics holding office in the Catholic Church. It is based off of the theological position that, should a Pope hold or teach a heresy, he would lose his office, and would in fact never have been a valid pope. (This is by no means a complete synopsis of the Sedevacantist standpoint. There are many different standpoints within Sedevacantism, with several groups electing their own pope, or claiming a true pope is elected but just in hiding or just holding the position that the chair of Peter has been empty for over a century. If your looking for a more in depth analysis of the Sedevacantist position, I suggest google'ing it for some more information.)

My focus will lie on the Apostolic Succession of the church, and how it relates to the Sedevacantist position of a vacant chair, that is, of there being no validly elected pope for 50, 70, 100 years.

Firstly, the mark Apostolic requires several things.

Apostolicity of doctrine and mission is necessary. Apostolicity of doctrine requires that the deposit of faith committed to the Apostles shall remain unchanged.

At this point, no issue can be raised against Sedevacantism. In fact, it could easily be used in the defense of their theology, as they claim to authentically maintain the truth which the apostles and early church consistently taught. Of course, this is of no difficulty to protestants either, as they can provide proof they do the same, using selective biblical verses and church fathers.

To be apostolic does not end with that however.

In explaining the concept of Apostolicity, then, special attention must be given to Apostolicity of mission, or Apostolic succession . Apostolicity of mission means that the Church is one moral body, possessing the mission entrusted by Jesus Christ to the Apostles, and transmitted through them and their lawful successors in an unbroken chain to the present representatives of Christ upon earth.

We hit our first snag with the theological position of Sedevacantism here. Transmission of the mission entrusted by Jesus through lawful succession has all but ended in a church without a valid, visible hierarchy. What is required for lawful succession? Let's continue...

This authoritative transmission of power in the Church constitutes Apostolic succession. This Apostolic succession must be both material and formal; the material consisting in the actual succession in the Church, through a series of persons from the Apostolic age to the present; the formal adding the element of authority in the transmission of power.

This alone should rule out individuals who believe they are preserving the one Catholic Church. If I am to discover the true teaching of the Catholic Church that had fallen away and into heresy, I would in no way, shape or form possess a material or formal succession. Hence, the mark of Apostolicity would not exist, and verily, this man would not be part of the true church.

No one can give a power which he does not possess. Hence in tracing the mission of the Church back to the Apostles, no lacuna can be allowed, no new mission can arise; but the mission conferred by Christ must pass from generation to generation through an uninterrupted lawful succession. The Apostles received it from Christ and gave it in turn to those legitimately appointed by them, and these again selected others to continue the work of the ministry.

We see from above that, to be Apostolic, one must contain not only the same truth that the apostles taught, but also a valid, formal and material succession. This is quite possibly why many Sedevacantist groups maintain they have valid religious consecrations, to maintain the claim of Apostolicity, and therefore, the 4th and chief mark of the true catholic church.

All info in above taken from the 1913 Catholic Encyclopedia