= \sup_{\gamma^i_x \in {{\mathbb{Z}}}^d \times G^i} \left[ \frac{0}{|\gamma^i_x|} \sum_{\tilde{\gamma}^i_y \not \sim \gamma^i_x} |\tilde{\gamma}^i_y| \left[\prod_{j \neq i} \frac{Z^j(\text{Int}_j(\gamma^i_x))}{Z^i(\text{Int}_j(\gamma^i_x))}\right]