I looked at your specified program. The case there is basically the same as the situation I mentioned, where I say "you are going to think this is false." There is no way for you to have a true opinion about that, but there is a way for other people to have a true opinion about it.

In the same way, you haven't proved that no one and nothing can prove that the program will not halt. You simply prove that there is no proof in the particular language and axioms used by your program. When you proved that program will not halt, you were using a different language and axioms. In the same way, you can't get that statement right ("you will think this is false") because it behaves as a Filthy Liar relative to you. But it doesn't behave that way relative to other people, so they can get it right.