Parafermi Algebra and Interordinality

Starting from the observation that for neighboring orders $p=2^{n}-1, p'=2^{n+1}-1$ of the well-known Green's representations of parafermi algebra there exists a specifiable interordinal relationship, matrices with similar properties are constructed. They exhibit intrinsic Catalan structure and sine-like and cosine-like substructures, dubbed interordinal and intraordinal respectively, that seemingly have a bearing on Euclidean geometry and applications via Nebe kissing numbers.