So the evidence for Clara being related to CAL seems pretty strong at this point. She is being re-born—she could be being beamed out by CAL at different points, then returned to and stored in the library. The different time things could be hand-waved by the future’s easily available time-travel technology.
The foreshadowing is there: she is being repeatedly downloaded, associated with computers, and a few others that more observant fans have doubtless written up somewhere else.
Here are my problems.
1) Clara isn’t just being beamed out. She wasn’t just inserted into a timeline. Her parents meant, she became a glint in their eye, she was a baby, then a toddler, then a little girl who gave advice to a sad man, then an adult.
CAL could have beamed Clara in somewhere in the timeline. But how could she have beamed Clara into her mother’s womb? The Doctor was scrutinizing her timeline pretty closely: how could he miss that?
2) If Clara were CAL, or were River Song reincarnated through CAL, then a) why would CAL not just remake River’s body and b) why would CAL bother to go through all this trouble just to save the Doctor at Trenzalore and c) how would CAL get anyone out of the library with the Vashtnarada everywhere?
3) Everything about Clara points to her being a perfectly ordinary girl, except that she keeps popping up again. Why would CAL make sure Clara couldn’t remember the entire point of her? And if Clara were some sort of computer program downloaded into a person, why couldn’t people like Emma Grayling tell?
Remember the Ladies 