Of course that is not the optimum way to play a high PP. (11%+ pfr then push)

Also its hard to estimate how often a worse hand will call your fop push(like KK unimproved), so the real number is probably less than a 11% pre flop raise required for it to show profit. I'm afraid this might not have the easy answer you are looking for at all.

Perhaps even limping followed by a push will show profit with AA!