A code $X$ is not primitivity preserving if there is a primitive list $\ws \in \lists X$ whose concatenation is imprimitive. We formalize a full characterization of such codes in the binary case in the proof assistant Isabelle/HOL. Part of the formalization, interesting on its own, is a description of $\{x,y\}$-interpretations of the square $xx$ if $\abs y \leq \abs x$. We also provide a formalized parametric solution of the related equation $x^jy^k = z^\ell$