Ok if anyone has already said this, sorry in advance.
I think that because cortana was part of chiefs armor which was connected straight to his nueral system, she could still be part of HIM. When chief set the nuke off, he was holding the bomb. How could he have suvived? Cortana, still part of his armor, created a shield around chiefs armor that protected him from the blast. The only way that this is possible is if cortana never actually left cheif, and she is embeded into his neural system.
This also explains how you can still hear her talking to chief after she was "destroyed".
*****FUN FACT
The symbol of the Didact, which is shown all through halo 4, is shown at the very end of the last terminal of halo annaversary. 343 guilty spark also speaks of "a horde of prometheans", and the librarian. He also hints to the didact. Kinda a coincidence that this was the halo before halo 4 and was made by 343 industries. maybe they were givin us all hints for halo 4
