When I edited my first post, I said I made some mistakes. You said I was wrong and that I didn't make a mistake, I just worded it wrong. My reply: wording things wrong is a mistake.Can you restate this? I don't quite understand what you just said. (This isn't a joke, as I have better things to do than to make fun of people, if that's what you are thinking).
I don't understand how that didn't make sense unless it was my spelling.
Also, when I said that not all permutation parities can be solved with 3-cycles, I was talking about these cases:
I'm now aware after reading your post that this would be an OLL parity because they take and odd number of slice moves to solve. I still think of this as permutation because the edges are in fact flipped correctly.
When I read that all permutation parities are solvable with 3-cycles, I included these and made the statement that the single edge flip parity could be solved with 3-cycles because it was possible to convert it to one of these cases using a 3 cycle.