it will be useful to generalize this approach to the discrete symmetry case which is more experimentally relevant. In this work we address the question of what restrictions, analogous to symmetry protection, should be imposed on a classical system under which it supports robust classical topological phases (CTPs). The first requirement is to consider systems invariant under a group $G$ and a local constraint whose defects carry elements of another group $G'$. (More details about defects can be found in Appendix \[App:Constraint\].) One example would be a gauge theory with gauge group $G'$ and defects being monopoles. The second requirement is that these phases must be short range correlated in the bulk and in particular must not break the symmetry spontaneously. The third is that they must confine defects of the constraints into neutral pairs (see Appendix \[App:Constraint\] for a precise definition). We refer to phases which obey the above restrictions as “admissible phases”.

Interestingly, we find that given a dimension $d$, and the groups $G$, and $G'$ as above, there are many inequivalent admissible phases. As standard, two phases are deemed equivalent if a continuous deformation from one to another is possible without crossing a critical point. By continuous 