The $\alpha\beta\gamma$-ansatz for random-cluster correlations

Let $G$ be any graph and $0\leq q\leq1$. Let $\mathcal{M}_{ef}(q)$ be the random-cluster polynomial associated with distinct edges $e,f$, and let $A_{\beta,\gamma}$ be the set of $\alpha$'s compatible with $\beta$ and $\gamma$. The $\alpha\beta\gamma$-ansatz. The polynomial $\mathcal{M}_{ef}(q)$ can be written as

$$\frac{\mathcal{M}_{ef}(q)}{q^2}=\sum_{\beta,\gamma}\mathbf{x}^{\beta}\mathbf{x}^{\gamma}Q_{\beta,\gamma},$$

where $Q_{\beta,\gamma}$ is a positive semidefinite quadratic form in the variables $\{\mathbf{x}_{\alpha}\mid\alpha\in A_{\beta,\gamma}\}$. Furthermore, setting $q=0$ and taking lowest-degree terms in $Q_{\varnothing,\varnothing}$ gives the classical UST formula in the source's theorem. This ansatz proposes a structural positivity decomposition for random-cluster correlations and connects its lowest-degree specialization with the uniform spanning tree formula. The supplied text gives no resolution status.