Inequalities on Six Points in a $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ Space

We establish a family of inequalities that hold true on any $6$ points in any $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ space. We prove that the validity of these inequalities does not follow from any properties of $5$-point subsets of $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ spaces. In particular, the validity of these inequalities does not follow from the $\mathrm{C}\mathrm{A}\mathrm{T}(0)$ $4$-point condition.