We prove that a large class of algebras in the framework of root quantum cluster algebras has a maximal degree structure of central simple algebras and Cayley-Hamilton algebras in the Procesi sense. We show that all roots of the unity upper quantum cluster algebra are of maximum degree and obtain an explicit formula for its reduced trace. Under mild assumptions, within each such algebra we construct a canonical central subalgebra isomorphic to the underlying upper cluster algebra such that its pair is a Cayley-Hamiltonian algebra. Its complete Azumaya locus has been shown to contain copies of the underlying cluster $\mathcal{A}$ species. Both results are evidenced by the broader generality of the intersection of mixed quantum tori on subcollections of seeds. Furthermore, we prove that all unary subalgebras of the roots of the unit quantum torus are Cayley-Hamilton algebras, and classify those of maximal degree. Any intersection of them on a subset of seeds is also proved to be a Cayley-Hamilton algebra. Previous approaches to construct maximum order relied on filtering and homology methods. We use a new method based on cluster algebra.
