It seemed clear to me in the first few paragraphs...
I implore Euodia and I implore Syntyche to be of the same mind in the Lord. And I urge you also, true companion, help these women who labored with me in the gospel, with Clement also, and the rest of my fellow workers, whose names are in the Book of Life (Philippians 4:2-3).
They are claiming Clement was a disciple of Peter...Peter was long gone preaching to the circumcision...Clement would have been hanging with that group...
And 2, if there was going to be a pope out of Paul's group preaching to the Gentiles, it most certainly would have been Timothy, not Clement...
This next one really got me...
4. CLEMENT I, ST. (88-97)...He was among the first baptized by St. Peter...Clement was the one to introduce the liturgical vestments into the sacred functions and the use of the word Amen. He appointed seven notaries, one for each ecclesiastical area of Rome, to edit and file all information regarding martyred Christians...He can be considered the first pope to have abdicated (Lopes A. The Popes: The lives of the pontiffs through 2000 years of history. Futura Edizoni, Roma, 1997, p. 2).
Oh ya brother, that's it...
Luk_20:46 Beware of the scribes, which desire to walk in long robes, and love greetings in the markets, and the highest seats in the synagogues, and the chief rooms at feasts;
Paul's helper Clement came up with the idea and design for fancy long robes and the highest seats in the synagogues...Ya, Paul tells us all about those guys like 'this' Clement...
>> “They are claiming Clement was a disciple of Peter” <<
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Where do we find any evidence of Peter having disciples?
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I believe that these are the Nethinims. The OT speaks of them taking on priesthood duties when the Levitts got lazy. Most of them are Kenittes.