This is very interesting stuff. However, I hate to tell you that it has little or nothing to do with the famed Fermat's Last Theorem. That states that no integer solutions exist that solve when . The formulation of this theorem is extremely specific, as by simply adding a fourth term, , there are plenty of solutions for many higher values of .
This is much closer to Hilbert's Tenth problem, which asks whether there is a general method of determining if a Diophantine equation has solutions. Unfortunately, the answer is no. That means that by changing or altering a single thing in a previously known Diophantine problem, you create an entirely new problem whose solution has nothing to do with the seemingly similar original problem.
So no, your question probably cannot somehow be tied to Fermat. As for your conjecture, that no multigrade can be dotted by any vector and made into a new multigrade higher than two, you may do better to recruit the help of a computer program to accrue some evidence for such a claim, before attempting to prove it. Counterexamples tend to be easier to find than proofs.