I'm not even past page 2 yet, but what I'm seeing is gibberish.

You start by claiming that Cantor's Theorem is wrong because of an "assumed relationship" between A and P(A) in the arbitrary mapping used. Based on your attempted counterexample, you seem to think that this incorrectly-assumed relationship is that we can't assume P(A) to be the power set of A instead of some other power set. That's not an assumption; that's what writing "P(A)" means.

Your "counterexample" uses a bijection g from O to P(E), and says this breaks Cantor's Theorem since there are no odd numbers in any subset of P(E). Your justification for this is that P(O) is equivalent to P(E). But the reason they're equivalent is that there's a second bijection h:P(E) -> P(O). Your counterexample doesn't even apply to Cantor's Theorem, since you're not talking about a function A->P(A).

For your argument to have any bearing on the theorem, you'd have to examine the composition h∘g - but then your counterexample isn't a counterexample at all.

I haven't gotten any farther than that yet, but I don't have high hopes. If you don't understand what Cantor's Theorem is even saying, you are definitely not going to be able to disprove it.