Make up theories? What? Like you do constantaly and never atually consider some basic mathematical objections?That just doesn't make any sense whatever. The ONLY issue with integers (as opposed to real numbers) is for the case n=2 where the interaction terms can be dismissed with complex numbers.
You seem to be saying the Binomial Theorem is invalid (or if it is, it is invalid because my application applies to positive real numbers, not only positive integers and is my proof is invalid because I apply it to integers) . Good luck with that one....
The issue has to do with symmetry in two dimensions, which is why the relativistic circle applies with radius of one. You are still trying to argue from a one (or no) dimensional perspective, which is like looking into the universe with a flashlight (i.e., line-of sight) As long as you can't admit there is a discipline called trigonometry, there is no point in going on, since you keep trying to revert to a one-dimensional argument, where everything can be handled in simple arithmetic. However, even in one dimension, integers or not, there will always be interaction terms in the Binomial Expansion for positive integers, so Fermat's Theorem is valid.
In one dimension, if there are no interaction terms, one is relegated to counting and partitioning sets using the same metric; which only works for the case n=2. Otherwise, the difference in degree of elements in the expansion cannot be eliminated.
You keep coming up with purported kindergarten "objections", and you're missing the whole point of the analysis, especially since you never answer any of my responses directly but make up theories of your own.