There's a gap in the proof that X and Y cooperate. You may know how to close it, but if it's possible it's not obvious enough so the extra steps should be added to the article. More importantly, if it can't be closed the theorem might not be true.

The gap: We hypothesize that statement S is provable in (system of X). Therefore X will Cooperate. This guarantees that T is true, by definition, but not that Y will prove that T is true. Presumable Y can recreate the proof of S being true, but it cannot conclude that X will cooperate unless it also can prove that X will prove it.

I cannot see how to resolve this without stepping out of a specific formal system, which would make Lob's Theorem unusable.

Am I missing something?