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\section{The Wu-Bradley model.}  \label{sec:isat-nn}  A model of interacting SAT with both nearest-neighbour interaction and site interactions can be defined as follows. Consider the set of bond-avoiding paths $\mathcal T_n$ of length $n$. Given a SAT $\psi_n \in \mathcal T_n$, we associate an energy $-\varepsilon_t$ every time the path visits the same site more than once, as in ISAT. Additionally, we define a \emph{contact} whenever there is a pair of sites that are neighbours on the lattice but not consecutive on the walk, as in ISAW. We associate an energy $-\varepsilon_c$ with each contact.