You're called to a table where AP controls Azor’s Gateway, which is currently tapped. AP explains that he activated Azor’s Gateway, drew a card, then instead of exiling a card from his hand, he put a card from his hand on the bottom of his library. He explains that for a moment, he forgot what he was doing and thought he was scrying to the bottom of his library. Both players agree that the card on the bottom of the library should be exiled under Azor’s Gateway.

Our study group discussed at length what the proper penalty for this call ought to be, specifically whether this fits into the definition of HCE or GRV. The relevant lines under HCE Definition and Philosophy are as follows (emphasis mine):
Definition: A player commits an error in the game that cannot be corrected by only publicly available information and does so without their opponent's permission.
Philosophy: "If cards are placed into a public zone, then their order is known and the infraction can be handled as a Game Rule Violation. Order cannot be determined from card faces only visible to one player unless the card is in a uniquely identifiable position (such as on top of the library, or as the only card in hand.)"

The question: Is the bottom card of the library considered ‘uniquely identifiable?’ Further, does that mean it's a known location and therefore publicly available information (its position, not identity)? Our conclusions were: If so, this becomes a GRV since it can be corrected with publicly available information. We perform a simple backup, returning the erroneous card to hand and having the player properly resolve the effect. If not, meaning we can't identify the card based on its location, it sounds like this fix becomes way more messy and dire, or we say the set no longer exists and we can't apply the fix.

Thanks in advance for helping offer some clarity on this issue!

Originally posted by Kade Goforth:

The question: Is the bottom card of the library considered ‘uniquely identifiable?’

Yes.

And yes, this is a GRV, for exactly that reason; the physical location of the card makes it uniquely identifiable.

Consider Enigma Sphinx, which - when it dies - goes into your library third from the top. You resolve this trigger, but mistakenly put three cards on top of it - just a simple misunderstanding of that process. Your opponent says “no, it's the third card, not under three cards”. The card is uniquely identifiable (it helps that we know it's Enigma Sphinx, of course) because we know its exact position, even though it's now in a hidden zone.

d:^D