The thing with Alexander is that while, from one viewpoint, things go around and around infinitely.... there is only one timeline, and there was only ever one timeline. It's a stable time loop, where everything plays out the same way every time (except there's only really one time).
From the perspective of each person involved in the story, they only experience their sequence of events once. That sequence may move strangely through time, and they may experience the same time period twice at different times in their personal timeline, but nobody is stuck "going around" the same actual sequence of events and carrying out the same actions over and over. They all have an entrance and exit point from the story.
There are loops and oddities, but no alternate outcomes. There are no "potential yous" that failed - there is one sequence of events where you did save yourself, and it's only an infinite loop if you stop tracing our single (slightly non-linear) path through the events, and start thinking of "past you" and "future you" as different people who might act differently the next time around.
Quickthinx isn't any more a product of the time loop than we are. He found Backrix's lost journal (three years 'before' it was written) that chronicled future events, giving him apparent omniscience, but his only actual time travel was the brief "detour" to three years ago that everyone experienced, before returning to our normal place in time.
And Mide and Dayan? Yes, their existence infinitely depends on time travel having happened in the first place - but there was never a timeline where it didn't happen, and they individually are not reliving the same time over and over.
Mide's timeline is fragmented from an external view of time moving from past to future, but she individually is still born, lives, and dies. Her personal timeline begins 26 years ago, plays out normally up to the events of Alexander, which ultimately lead to her being teleported back into the past - where she lives the rest of her life, has children and eventually dies at what is probably an earlier date than her birth.
Weird genetics ensue, and somehow one of her descendents ends up as a perfect genetic match. Perhaps it's chance she also got the same name, or perhaps her great-grandmother insisted.
We're all set for it to play out again - but young Mide has never experienced it before.



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