If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
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Fortunately, H will never find your argument because it is not a correct proof. You rely on hidden assumptions of the following form (given informally and symbolically):
where #φ denotes the Gödel number of the proposition φ.
Statements of these form are generally not provable. This phenomenon is known as Löb's theorem - featured in Main back in 2008.
You use these invalid assumptions to eliminate the first two options from Either H returns true, or false, or loops forever. For example, if H returns true, then you can infer that "FF halts on input FF" is provable, but that does not contradict FF does not halt on input FF.
I'm very confused. Of course if φ is provable then it's true. That's the whole point of using proofs.