Martyrdom of Saint Clement
Is the Papacy found in the era of the early church Fathers (ante 500) or is it a medieval invention? Technically, this question is misguided from a Catholic point of view since the Papacy was founded by Christ in about AD 30. Its institution is did not begin sometime later, either earlier or much later.

That being said, we find that in A.D. 96, Clement (the fourth pope and also disciple of Saint Paul mentioned in Phil 4:3) believed that he had special authority over the entire Church. He writes from his place in Rome to the Christians in Corinth:

“If anyone disobeys the things which have been said by Him through us, let them know that they will involve themselves in transgression and in no small danger.” Clement of Rome, Pope, First Epistle to the Corinthians, 1, 59:1 (c. A.D. 96).

In the year A.D. 256, Pope Stephen was called: “Stephen, who announces that he holds by succession the throne of Peter.” Firmilian to Cyprian, Epistle 74/75:17 (A.D. 256).

The writings of Jerome (who lived in Rome and Palestine) testify to the universal authority of the Pope in Rome in the 300s. Writing in A.D. 375, Jerome said:

My words are spoken to the Successor of the Fisherman, to the disciple of the cross. As I follow no leader save Christ, so I communicate with none but your blessedness, that is, with the Chair of Peter.

For this, I know, is the rock on which the Church is built! This is the house where alone the Paschal Lamb can be rightly eaten. This is the ark of Noah, and he who is not found in it shall perish when the flood prevails.” (Jerome, To Pope Damasus, Epistle 15:1-2)

Jerome believes that Pope Damasus is the successor of Peter and that the Pope’s chair (apostolic see) is the locus of unity and communion for the universal Church. Keep in mind that all of these quotes were written before the canon of the books of the Bible had been selected and confirmed by Pope Damasus in A.D. 382.

In other words, the Papacy is older than the canon of Scripture. There was only one church in A.D. 382 and its chief pastor on earth was the Pope in Rome.