Symmetric cluster weight at the self-dual parameter ($\varphi (\omega )\propto (\sqrt{q}{)}^{k(\omega )+k({\bar{\omega }}^{\ast })}$)

The planar cluster-count identity rewrites $o(\omega )+2k(\omega )$ as $|V|-1+k(\omega )+k({\bar{\omega }}^{\ast })$. At the self-dual parameter of the random-cluster model, $p/(1-p)=\sqrt{q}$, so its weight is $(\sqrt{q}{)}^{o(\omega )+2k(\omega )}$ up to a configuration-independent factor. Absorbing $(\sqrt{q}{)}^{|V|-1}$ into normalization proves the symmetric weight, with the unbounded dual face counted.