My theory (as recently seen in the discord) is that Cortana may or may not be alive depending on your own philosophical beliefs. I believe that many versions of Cortana exist and have survived one way or another via the various personality splinters (or alt flash clones as stated by
Ashuraa), however this doesn't necessarily mean Cortana has survived, only some version of her has. If you believe that the brain is the source of thought/personality (which is relatively common) then to understand whether the Cortana we know is alive or not I propose this argument I've borrowed from metaphysics.
Suppose there's a brainless clone of your body, and half of your brain is transplanted in it (doesn't matter which half). Additionally, let's call yourself before the operation Person A, the body with the left half of the brain after the procedure Person B, and the body with the right half of the brain after the procedure Person C, so that we have 1 original Person A and two new people Person B and C. The question we now have, is if Person A survives through the existance of Person(s) B/C. There are 4 possible scenarios
1. Person A is not identical to person B or C
2. Person A is identical to B but not C
3. Person A is identical to C but not B
4. Person A is identical to both B and C
If we assume that this thought experiment is similar to Cortana's personality splinters, then we can either conclude that Cortana as we knew her is either alive or dead depending on our answer to the above question.
This is an overview on my thoughts about Cortana's survival going into Infinite.
If we are to say that 1 is true, then she is already dead.
If we are to say that either 2 or 3 are true, then she may be alive via the personality splinter that ran the story in Halo 5 (as she would be identical to that splinter) or she is identical to some other splinter not shown in Halo 5 (to my knowledge) and is either alive or dead depending on the status of that other splinter.
If 4 is true, then she is alive and there may now be more than 1 equally valid Cortana's in the Universe.