I don't see how this is a "side thing" when there's a lot of hints throughout the main story that basically reference some sort of loop or prescience related to it (i.e. the Syndicate member who said "we have seen it, we have tried," Idriel saying "this time, it'll be the right one" as her parting words to Orion before his first entry to Ion, the game's logo being an infinity symbol, the Founder's chest tattoo being an Ouroboros). Whether it's a straight time loop or a reincarnation loop is still up in the air, however.
Also, I disagree with cycles being an Eastern thing. The Greeks (more specifically the Stoics) have a concept of "eternal recurrence," where the universe dies after a certain point but is reborn as the same universe over and over. And in case you haven't noticed, the entire VN has a lot of Greek references to it (i.e. presence of oracle figures like Idriel and Luna, the default name being Orion and his character development referencing a lot from his mythological counterpart, Penelope and Calypso and Ogygia and Ulysses being references to characters and places in "The Odyssey," Thanatos being a reference to the Greek's version of Death, Psyche (and its latin counterpart Anima) who is a reference to the Greek Goddess of the Soul, the Greek rivers of the dead Styx and Acheron being referenced as a server and a temple respectively). So Cari utilizing another Greek concept in the game wouldn't be out of the ordinary.
Also for the record, he's Spanish.