= Symmetric cluster weight at the self-dual parameter
{title2=$\phi(\omega)\propto(\sqrt q)^{k(\omega)+k(\bar\omega^*)}$}
The <planar cluster-count identity> rewrites $o(\omega)+2k(\omega)$ as $|V|-1+k(\omega)+k(\bar\omega^*)$. 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.
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