couplings. Formally, it is because of the emergent ${\mathbb{Z}}_2 \times {\mathbb{Z}}_2$ symmetry. Physically, it is because $\pm 2\pi$ vortices are linearly confined by the bulk (see Fig. \[fig:conf\]). Further, a gauge symmetry breaking term ($K\sum_{\ell\in\partial E}A_{\ell}$) can also be studied using duality [@Balian] and has no effect on the $\sigma$’s in the limit $J_2,J_1\to\infty$.

The SPT perspective {#sec:SPTpersp}
-------------------

As discussed in section \[sec:AKLT\] the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ two-dimensional classical topological paramagnet can be related to the imaginary time partition function of a $1+1D$ quantum SPT phase. In this section we provide support for the analogous statement in $3D$, proving that all of the above models are in the same universality class as the Euclidean time partition function of certain $2+1D$ quantum SPTs. We will show this by analyzing the responses to gauge fluxes, or equivalently, the statistical phases obtained by braiding flux excitations.

As starting point we perform a gauge-to-Ising duality transformation on the bulk [@Balian] trading $A$’s for spins $\tau$’s on the vertices of the dual lattice, resulting in an equivalent bulk theory with weights: $$\begin{aligned}
  \label{eq:Atau}
  \prod_{p\in P_L}(\tanh J_2)^{\frac{1-\tau_k\tau_l}{2}}
  \prod_{p\in P_R}(\tanh J_1)^{\frac{1-\tau_k\tau_l}{2}}(\tau_k\tau_l)^{\omega_p}\, ,\end{aligned}$$ where $kl$ is the link dual to $p$. The term $\prod_{p \in P_R}(\tau_{k}\tau_{l})^{\omega_p}$ is in 