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 art congruent to $1$ modulo $p$. For $\mathbb{P}^4$ and $\mathbb{P}^6$, we present explicit numerical exceptional bases violating this congruence; hence the action is not transitive. We also establish a stronger obstruction modulo $p^2$ and use Polishchuk's theorem to show that no vector of the constructed bases is the class of an exceptional object. Finally, we prove that for every prime $p$, the category $D^b(\operatorname{Coh}\mathbb{P}^{p-1}_{\mathbb C})$ does not contain a pair of mutually orthogonal exceptional objects.