End Of Empire.
I finally thought to check out the list of Emperors of Rome. It is confusing. Constantine and Constantius turn out not to be the same name. And Constantius III is a long way from the end of the list. However, the line of Western Emperors does end in that same century - and the nominal last Emperor has the same name as the Founder of the City.
So Poul and Karen Anderson's The King Of Ys really is set very near the end of the Western Roman Empire. In the closing chapters of Volume IV, Imperial rule has effectively ended in Western Europe but it takes time for some of the participants to realize and acknowledge this.
As I said here:
"...there is a Roman King of Ys both at the beginning and at the end of the Age of Brennilis."
The Age of Brennilis began when Ys became allied to Rome, immediately before the Empire, and ends when the Empire is in terminal decline and soon to end. Thus, the fates of Ys and Rome became inextricably interlinked even though Ys was veiled from Rome for much of the intermediate period. Volume I is appropriately entitled Roma Mater.