Well, in this hypothetical cosmology I'm operating under the assumption that timelines can't just fork and continue on indefinitely as that requires energy (in

Well, in this hypothetical cosmology I'm operating under the assumption that timelines can't just fork and continue on indefinitely as that requires energy (in the form of the time traveler) to come from an outside source with no connection to this timeline. From the perspective of the new timeline that event has no origin. By forming a meta-stable loop the two timelines are actually just different parts of the same meta-timeline. Of course, there's no real reason to believe this is a requirement, it's all just an intellectual exercise.