(Dr. Scott) Hahn Family spends its first Holy Week in Rome

That's Catholic's version of Catholic history...That's not church history...
Polycarp was among the first of the church fathers and he never mentioned a word about your religion.

There WERE other church fathers besides St. Polycarp.
St. Ignatius of Antioch was consecrated Bishop of Antioch there in the year 69 A.D. by St. Peter himself and was bishop there for some 30+ years. He was dragged away to Rome in 107 A.D. On his way to martyrdom he wrote letters to the Christians in the known world.
In his letter to the people of Symrna he wrote: Where Jesus is, there also is the CATHOLIC CHURCH.
THAT is just history. If you refuse to believe history, and that is your choice, then do so.
Google it. It's not esoteric.

From Catholic Answers:
Full Question:
After the death of Christ, Christians were the ones who evangelized and spread the gospel. Ignatius of Antioch was the first to refer to them as Catholics. What were they called before?

Answer
Originally Christians weren’t even called Christians. They were called "disciples" (i.e., "students") of Jesus of Nazareth. Later, in the city of Antioch, they received the name "Christians" (Acts 11:26). This probably happened in the A.D. 30s. This term spread very quickly—probably to the chagrin of those Jewish individuals who did not wish to acknowledge Jesus as the Messiah (Christ).

Ultimately, however, different groups began to break off from the Christian community, falling into either heresy or schism. These groups still wished to represent themselves as Christian—and many of them were, retaining valid baptism and a profession of faith in Christ. Consequently, a new word was needed to distinguish the Christians belonging to the Church that Christ founded from those belonging to the churches that had split off from it.

The term that was picked was kataholos, which means according to the whole or universal in Greek. The thought was apparently that these were Christians who believed and practiced according to what body of Christians as a whole did, in contrast to what some particular group thought or did. Over the course of time, kataholoscame to be represented by the parallel English word "Catholic."

Ignatius of Antioch did not introduce kataholos. However, his letters contain the earliest known uses of it. It may well have been used in other Christian writings prior to this, but we have simply lost them. It certainly was in general use in speech before this point, because Ignatius writes in such a way that he already expects his readers to know this term and what it means. He also uses the term in more than one of his letters, meaning that he expects people in more than one place to know the term.

This indicates that in his day—at the beginning of the second century (circa A.D. 107)—the term was already in widespread use. For it to be used in such a broad manner, it would have required some time to pass into currency in the Christian community, meaning that the term probably was coined sometime in the second half of the first century. We don’t know who first used it, but it was a suitable description of the Church Christ founded and so was already in general use by the time Ignatius wrote.

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