Why is the Last Place on Earth St. Peter Founded the Seat of his Authority? The (Roman) Catholic Church agrees that Peter founded different churches, like at Antioch or Rome, but it claims that it is only at the last place Peter lived or died is to where his supposed authority from Christ as the head of the apostles remains.

What is the proof of this claim?  Why would it matter where Peter lived last or died at as to where his supposed authority remained?
The question is about the logical basis for the argument that the last place St Peter resided at should retain his authority as Pope. 

When Peter left Antioch he did not confer his papal authority on anyone in that city, or on anyone anywhere else.  He retained it personally wherever he went. It was only when he came to die that it became necessary to choose a successor. We do not know whether St Peter nominated his successor, or arranged a process for his election. But the need for a successor arose only on his death and the person who succeeded him succeeded him as Bishop of Rome and Pope,  because these were the roles that his death left vacant. 

It would not have been possible to summon a conclave from all over the world.  Even today, notionally at least, the Bishop of Rome is elected only  by the clergy of Rome,  and as Bishop of Rome is ex officio Pope.   The way this is done is that every cardinal, on appointment, becomes a deacon or priest or suburban bishop of a parish in the diocese of Rome itself or one of the suburban dioceses of Rome. In this way it is still the case that the Pope is chosen by the clergy of Rome.