if $\tilde{\sigma}_{i;p}$ is long range ordered then $\tilde{\sigma}_{i;p'}$ must be disordered as it is equal to a power of $\tilde{\sigma}_{i;p}$ times a non-trivial power of the disordered variable $\mu_i$. Lastly, we justify the nomenclature topological. By this we mean that an interface between two distinct admissible phases would contain some form of long range or quasi long range order. Consider such an interface between a $p$ topological phase and a $p'$ topological phase. This scenario can be engineered by setting $K_2=0$ and $\tilde{\sigma}_p$ ordered on one side of the interface, and $K_1=0$ and $\tilde{\mu}_{p'}$ ordered on the other. On the interface these two order parameters will leak and so $(\mu_i \sigma^p_i)(\mu_i \sigma^{p'}_i)^{-1} = \sigma^{p-p'}_i$ would be ordered. Notably the latter, and only the latter, is a local order parameter and thus we have shown the existence of 1D long range order on such interfaces

Three dimensions
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Next we wish to generalize the above construction to $3D$. In $2D$ we attached closed polygons to domain walls of the spins. Turning to $3D$, polygons on the dual lattice appear naturally in ${\mathbb{Z}}_2$ gauge theories, where they correspond to discrete flux lines. However domains walls become $2D$ objects, and we instead look 