2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63027Using some adaptations of the adjacency decomposition method \cite{CR} and the program {\it cdd} (~\cite{Fu}), we compute the first computationally difficult cases of convex cones of $m$-ary and oriented analogs of semi-metrics and cut semi-metrics, which were introduced in \cite{DR2} and \cite{DP}. We considered also more general notion of $(m,s)$-super-metric and corresponding cones. The data on related cones - the number of facets, of extreme rays, of their orbits and diameters - are collected in Table \ref{tab:MainLovelyTable}. We study also criterion of adjacency for skeletons of those cones and their duals. Some families of extreme rays and operations on them are also given.22 pages, 3 figuresMetric GeometryCombinatoricstext