Surely this is a Ship of Theseus situation, isn't it? There may have been little change from year to year, but a red-blooded Jupiter-Optimus-Maximus-fearing classical Roman citizen transplanted into the future would have been horrified at various moments, first when the Republic became an Empire, again when Christianity took over, when the Empire was split in the Tetrarchy, and when it became apparent that the Greek-speaking half was all that was left. Sure, there's a continuous line of succession much later, but for Gaius Quinctilius Budweiser, I think the traditional break when the West finally gave up is as good as any.