I didn't really read your proof, because it's late and I'm tired, but there is no Banach-Tarski construction in 1 or 2 dimensions (whether or not you like the a

I didn't really read your proof, because it's late and I'm tired, but there is no Banach-Tarski construction in 1 or 2 dimensions (whether or not you like the axiom of choice). The third dimension is crucial. So if your proof doesn't somehow intrinsically rely on "n >= 3", it can't be right.