Okay, I'll explain here. I didn't come up with this theory; I've seen it but hadn't considered it seriously until as of late. It works like this.

Sept 22: John, Cortana and the other survivors of the Pillar of Autumn hijack the Ascendant Justice. Cortana takes control of the ship, during which she encounters a hostile Covenant AI. Once they have the ship, the humans fly it to Reach to look for survivors.

Sept 7: Due to shenanigans from the Forerunner crystal on Reach, the Ascendant Justice arrives at Reach two weeks earlier than when they left. While John and co rescue Blue Team and their pals, Cortana grapples with the Covenant AI of the Justice. She kills it and discovers its code appears to have been copied over and over for seemingly no reason, and that its code patterns are "maddeningly familiar". She then gets the idea to make a copy of herself using the Covenant AI's code, though it has slight imperfections from the process.

Sept 13: Blue Team and the captured Ascendant Justice initiate Operation: FIRST STRIKE. They board and attack the Unyielding Hierophant, which is accompanied by an enormous Covenant escort fleet. Cortana's copy comes with John and it makes a slightly imperfect copy of itself, which later on makes a copy of itself, which makes a copy of itself, etc, until the Hierophant is swarmed with scrambled Cortana copies.

Speculation: At some point in the battle, one, maybe more, of the Cortana copies makes it onto the Covenant escort ships, one of which is the Ascendant Justice, the original that hasn't gone in time yet. By this point the Cortana copy has been replicated with so many imperfections that it doesn't even act like her anymore. The Covenant don't destroy it but instead repurpose it to work for them. A week later news arrives of the discovery of Halo, and the Ascendant Justice heads to the ring with the AI aboard, completing the loop.