I think there are two problems with your answer (neither of which is the reason given for deleting it, before, but that's understandable given problem #1).
The first problem is that your argument is really hard to follow. The second problem (if I have in fact successfully figured out what you mean) is that it doesn't resolve the paradox.
It's hard to follow for two reasons. The first, and more important, is that it doesn't state clearly how it supposedly resolves the paradox.
The reason why the paradox feels paradoxical is that it seems that the prisoner is able to prove that if the judge is telling the truth then there's no day on which they could be hanged unexpectedly -- and yet, then they get hanged and they didn't expect it, so it turns out the judge was telling the truth.
In order to resolve the paradox, you need to show that something is wrong with the reasoning described in the paragraph above. Maybe your answer does that, or at least tries to, but you never say explicitly what step in the apparently-paradoxical reasoning it finds a problem with.
The second problem is that it's laid out as a single paragraph and lacks "signposts" indicating how its steps fit together.
I don't know whether I have made my argument clear, above, so let me make it more concrete by offering what I think is a version of your argument that is easier to follow. I am not 100% sure I'm not distorting the argument because, as mentioned above, it's hard to be sure exactly what the argument is.
The point at which the "paradox" supposedly arises is where the prisoner, having proved that if the judge is telling the truth then a hanging on any day would be expected, then gets dragged from his cell on (say) Wednesday to a hanging he didn't in fact expect -- so that the judge was truthful after all.
But I say that his actual hanging wasn't unexpected. After all, the prisoner had already proved that if not already hanged by Tuesday, he must be hanged on Wednesday. And then, having not been hanged by Tuesday, he is hanged on Wednesday. That's the very opposite of "unexpected". He had an actual proof that it must happen!
(It's true that he also had a proof that he wouldn't be hanged on Wednesday, but so what? I say that if you have a proof that X will happen, then when X happens it is not "unexpected", no matter what other things you also have proofs of.)
So there isn't an unexpected hanging at the end after all, the prisoner was correct to conclude that there wouldn't be one, and there is no paradox, just a judge saying something that (as the prisoner predicted) turned out to be false.
As I say, I'm not 100% sure that the above is a presentation of the same argument as you are making; but I think it probably is, and at any rate I think it's much easier to follow, because it says exactly what it finds wrong with the "paradoxical" reasoning, and makes its argument explicitly.
Having done my best to clarify your argument, I don't find it convincing and don't think it does in fact resolve the paradox.
Your argument depends on a particular notion of what it means for something to be "unexpected" or a "surprise". I don't think that notion matches how these words are actually used in practice, nor is there anything specific to the unexpected-hanging scenario that calls for a nonstandard understanding of the term.
(To be concrete: you say "the hanging on Wednesday is defined not to be a surprise hanging"; no, it isn't. Not by how the word "surprise" is usually used; not by anything in the statement of the paradox.)
Yes, sure, the prisoner had a proof that if the judge was telling the truth (and if the prisoner knew that the judge was telling the truth) and he hadn't been hanged by Tuesday then he must be hanged by Wednesday. So far, so good. But the judge doesn't say "when we knock on your door at noon you will not have a proof that we were going to", he says "when we knock on your door at noon you will be surprised", and those two things are not the same, and simply declaring (as you do) that they are the same doesn't make them so. The story tells us that the prisoner is in fact surprised when the knock-on-the-door actually comes, and nothing in the story makes that difficult for us to believe.
In this case, indeed, the prisoner has a proof of that proposition. But he also has proofs (with the same hypotheses) of a bunch of other things, which add up to a contradiction. As a matter of psychology, when you prove a bunch of mutually contradictory things, it is perfectly possible to then be surprised when one of them turns out to be true. As a matter of mathematics, when you make some assumptions and use them to prove a bunch of mutually contradictory things, the correct conclusion is not that all those things are true, it's that something was wrong with your assumptions.
(In this case, it turns out that the prisoner should drop the assumption that he knows that what the judge said is true, or maybe the assumption that if he can prove that something will happen then he will be unsurprised when it happens; the first of those assumptions turns out to be untrue because he ends up not confident that what the judge said is true, and the second turns out to be untrue because he "proves" several things that turn out to be mutually contradictory.)