Uh, Omega has no business deciding what problem I'm solving.

Could you point me to your preferred writeup of the Unexpected Hanging Paradox?

The solution I consider definitively correct is outlined on the Wikipedia page, but simple enough to be expressed here. The judge actually says "you can't deduce the day you'll be hanged, even if you use this statement as an axiom too". This phrase is self-referential, like the phrase "this statement is false". Although not all self-referential statements are self-contradictory, this one turns out to be. The proof of self-contradiction simply follows the prisoner's reasoning. This line of attack seems to have been first rigorously formalized by Fitch, "A Goedelized formulation of the prediction paradox", can't find the full text online. And that's all there is to it.