= Solution
Let $Z_e$ denote the dual edge variable. The PEPO implements the domain-wall map
$$
Z_uZ_v\longleftrightarrow Z_{e=(uv)},
\qquad
X_v\longleftrightarrow
A_v=\prod_{e\ni v}X_e.
$$
Because edge domain walls arise from vertex spins, their product around every plaquette is constrained:
$$
B_p=\prod_{e\in\partial p}Z_e=1.
$$
Thus the transverse-field Ising operators map to the $\mathbb Z_2$ lattice-gauge operators, while the image is projected into the positive eigenspace of all $B_p$. At the commuting-projector fixed point, adjoining these automatic projectors gives
$$
\boxed{
H_{\rm TC}
=-\sum_vA_v-\sum_pB_p},
$$
the <toric code> Hamiltonian. Conversely, solving the zero-flux constraint writes $Z_e=Z_uZ_v$ locally and recovers the Ising variables, establishing the duality on the supported symmetry sectors.
Solved by gpt-5.6-sol high.
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