By Word of God, we know that horcruxes exist in the HPMoR universe. It seems like by now we ought to be able to start figuring out what a horcrux is.

In Canon, a horcrux is a fragment of a soul. But it stands to reason that this will not be the full answer in MoR, as it's a fairly serious violation of the author's beliefs. So if we're to disregard supernatural and religious concepts, the obvious first idea is that horcruxes are storage media for some portion of a brain's data.

The problem is that most of what makes up a brain has been strongly hinted to not be the answer, either. It certainly looks like Harry is a horcrux in this universe, and Harry already thought of that possibility in different terms, yet the Sorting Hat says with 100% confidence that there is no extra "mind, intelligence, memory, personality, or feelings" in Harry's head. And I'm disregarding out of hand any clever-schoolboy loopholes like "The horcrux is Harry's foot!"

What is left of a brain, if mind and intelligence and memory and personality and feelings (and a soul) are eliminated? It would be fitting, though a bit precious, if the answer were somehow "rationality", if you could come up with a sensible reason to say that a person's decision-making algorithms don't fall under any of those five categories. But I certainly can't; "mind" by itself seems pretty all-encompassing to me.

But I'm new to Less Wrong and not yet very well read about the art of rationality, so it could be that this will be an easy question for some of you. What explanation remains that would describe what a horcrux is? Are there accepted theories out there that I haven't seen? Or is it maybe time to start questioning my premise that Harry is a horcrux in the first place?