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Tom, to present the subject correctly, anyone presenting themselves as candidate for pope in the Vatican, would have to claim to be the Successor of Saint Peter.
The person who the Vatican says is St. Peter's first successor is Linus, followed by every pope thereafter, with the exception of a few anti-popes.
My studies show me that St. Mark the Evangelist is St. Peter's Successor, as he wrote the first version of the Gospel, that which Peter the Apostle taught him.
Peter did not write a version of the Gospel, even though he was one of the twelve disciples of Jesus, whereas Mark was not a direct disciple of Jesus. Hence Mark wrote what Peter told him.
Peter also called Mark his "son". This in New Testament language has the spiritual meaning of disciple and successor.
In fact Linus did not write a version of the Gospel nor was he mentioned by Peter.
So Peter's Successor must be St. Mark the Evangelist, and on his version of the Gospel Matthew and Luke later structured each their own respective version of the Gospel, thus "succeeding" St. Mark and continuing his Apostolic mission.
Hence the term "synoptic Gospels".
St. John wrote his own version of the Gospel, following his own scheme, for which reason it is not "synoptic", meaning it is not structured on St. Mark's Petrine version. Being one of the twelve disciples of Jesus, St. John did not need to "succeed" Peter, having followed Jesus directly as one of the Twelve.
Without the four versions of the Gospel, the Christian Church would have failed.
And as none of the popes, starting with Linus, ever wrote a version of the Gospel, or indeed any piece of Scripture, none of them can be considered Peter's Successor.
The Vatican does not recognise any author of the New Testament as the successor of Saint Peter.
Incidentally, Linus, who is mentioned casually by St. Paul in one of his Epistles, never claimed to be either "pope", "successor of St. Peter", "holy father" or any other fanciful title that later was given to the popes.
It is reckoned that the Vatican first used the title "pope" around the fifth or sixth century, and that the Liber Pontificalis, possibly initiated around the sixth century (historians are not exactly sure when), ascribed to Roman bishops the title of "pope" and "successor of St. Peter", long after they had died, going all the way back to Linus, but leaving out all the Evangelists.
Linus, who lived in the first century, possibly would have turned in his grave had he known what would happen about 500 years later when the Liber Pontificalis (the book of the popes) was initiated. He certainly never gave his consent to being placed above St. Mark.