  A local ring $R$ is regular if and only if every finitely generated
$R$-module has finite projective dimension. Moreover, the residue field $k$ is
a test module: $R$ is regular if and only if $k$ has finite projective
dimension. This characterization can be extended to the bounded derived
category $\mathsf{D}^f(R)$, which contains only small objects if and only if
$R$ is regular.
  Recent results of Pollitz, completing work initiated by
Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete
intersections: $R$ is a complete intersection if and only if every object in
$\mathsf{D}^f(R)$ is proxy small. In this paper, we study a return to the world
of $R$-modules, and search for finitely generated $R$-modules that are not
proxy small whenever $R$ is not a complete intersection. We give an algorithm
to construct such modules in certain settings, including over equipresented
rings and Stanley-Reisner rings.
