Hidden-variable encoding

Replace a finite many-variable relation $R(X_1,\dots ,X_k)$ by one auxiliary variable $Y$ whose domain is the allowed tuples of $R$, together with binary constraints equating each tuple coordinate to its corresponding $X_i$. Any original satisfying assignment extends to exactly its allowed tuple, and any satisfying encoded assignment projects to a solution of $R$. The tuple domain can be large; the transformation changes arity, not necessarily representation size efficiently when $R$ is succinct.