Differentiating of the karyon tilings

We consider the universal $d$-dimensional karyon tilings $\mathcal{T}(\mathbf{m}, v)$. Its parameters, the weight vector $\mathbf{m}$ and the star $v$, belong to the dual module space $\triangle^d \times \triangle^d$ that is the direct product of two $d$-dimensional simplexes. The star $v$ defines the geometry of the parallelepipeds $T_{0}, T_{1}, \ldots, T_{d}$, which the tiling $\mathcal{T}(\mathbf{m},v)$ consists of, and the weight vector $\mathbf{m}$ sets the local rules and frequency distribution of the parallelepipeds in the tiling. Knowing the parameters $\mathbf{m}, v$, by the local algorithm $\mathcal{A}$ anyone can construct the whole tiling $\mathcal{T}(\mathbf{m},v)$. It is proved that the differentiation of the karyon tiling $\mathcal{T}(\mathbf{m},v)\rightarrow \mathcal{T}^{\sigma}(\mathbf{m}, v)$ is equivalent to some explicitly defined elementary transformation of the centered unimodular basis $\mathbf{u}$.