In what year did the Byzantine Empire begin to exist?

Initially, the Byzantine Empire was part of a single Roman Empire, which was divided into 2 parts for ease of administration and protection from barbarians.

This division took shape in 395. This year the Roman emperor Theodosius, nicknamed the Great, divided the empire into western and eastern parts, leaving two of his sons to rule: Arcadius and Honorius.

However, in my opinion, the Byzantine Empire began an independent existence with the fall of the Western Roman Empire in 476. It was then that she became the successor to the Roman Empire.