In my original version of the post I had intended to argue for a geometric mean, which was sort of a joke, as it only works where f(x) is a constant k*x. So the

In my original version of the post I had intended to argue for a geometric mean, which was sort of a joke, as it only works where f(x) is a constant k*x. So the statement "A false resolution follows, first to spot the problem..." appears in the post. Upon seeing this an enterprising and intelligent person may have clicked "reply" While writing this I realized that inverting the definition of A gave a different resolution based loosely on operator theory, and posted that instead. A couple minutes later I realized the mistake and changed the beginning to refer to the first to spot "a" problem. The argument you've presented is good, but you have attacked the well-foundedness of the problem itself rather than the logic I've applied. I'm not sure if you get the cookie.