Maximum deconstructibility in module categories

We prove that Vop\v{e}nka’s Principle implies that for every class $\mathfrak{X}$ of modules over any ring, the class of $\mathfrak{X}\mathbf{\text{-Gorenstein}}\mathbf{\text{Projective}}\mathbf{\text{modules}}$ ($\mathfrak{X}\mathbf{\text{-}}$) is a special precovering class. In particular, it is not possible to prove (unless Vop\v{e}nka’s Principle is inconsistent) that there is a ring over which the $\mathbf{\text{Ding}}\mathbf{\text{Projectives}}$ ($$) or the $\mathbf{\text{Gorenstein}}\mathbf{\text{Projectives}}$ ($$) do not form a precovering class (\v{S}aroch previously obtained this result for the class $\mathcal{G}\mathcal{P}$, using different methods). The key innovation is a new “top-down” characterization of \emph{deconstructibility}, which is a well-known sufficient condition for a class to be precovering. We also prove that Vop\v{e}nka’s Principle implies, in some sense, the maximum possible amount of deconstructibility in module categories.