Nontransitivity of braid group actions on exceptional bases in $K_0(\mathbb{P}^n)$ for $n=4,6$

Let $X=\mathbb{P}^n$, where $p=n+1$ is prime. Consider the orbit of the standard numerical exceptional basis in $K_0(X)$ under mutations, sign changes, and isometries of the Euler form. We construct an invariant of this orbit: for every basis from the orbit, the square of rank of all basic vectors a