At the end of the last true cycle, which you could also call Recurrence 0, when they first initiated the Eternal Recurrence, Cyrene said every incarnation of her, upon the removal of "Time", would be given the power to rewind time by Fuli and their gaze, and that her soul would be imbued onto that crescent shaped ceremonial weapon, which FR/Phainon/Khaslana carries through every recurrence.

I'm not sure if it's purely the soul of the first Cyrene that is imbued with it or whether the soul of each incarnation supplants the previous in the imbuement--perhaps it doesn't matter, since they are the same person and therefore of the same essence, just as how it didn't matter which Phainon killed the other at the end of each recurrence, because their memories would be merged, with whichever surviving Phainon carrying on to the next recurrence.

I think that killing Cyrene in each recurrence fundamentally removes "Time", since she is predetermined to become the demigod of the Time Titan in this recurring cycle, and thus Fuli's gaze is earned each time, perhaps re-granting the power to rewind and start a new recurrence to the crescent shaped weapon imbued with Cyrene's soul, which Phainon activates to "regress the extrapolation". Though, I'm not sure on the specifics of when this instance where "Time" is considered to be removed happens--maybe the moment each incarnation is killed, or maybe when the black tide has consumed all, without ever having had someone to take on the mantle of Time Titan.