A Minimal Duality Estimate for the Surface-Code Threshold under Nearest-Neighbor Correlated Errors

We apply the single-equation duality criterion to the square-octagonal random-bond Ising model recently obtained from an exact error-edge map for a surface code with nearest-neighbor correlated errors. The calculation is performed for the minimal cell after the error-edge reduction. For the symmetric case $p_1=p_2=p_3=p$, this gives $p_c=0.0288427147$, in close agreement with the reported numerical threshold of about $3\mathrm{\%}$.