  The positions of multiple images in galaxy lenses are related to the galaxy
mass distribution. Smooth elliptical mass profiles were previously shown to be
inadequate in reproducing the quad population. In this paper, we explore the
deviations from such smooth elliptical mass distributions. Unlike most other
work, we use a model-free approach based on the relative polar image angles of
quads, and their position in 3D space with respect to the Fundamental Surface
of Quads. The FSQ is defined by quads produced by elliptical lenses. We have
generated thousands of quads from synthetic populations of lenses with
substructure consistent with $\Lambda$CDM simulations, and found that such
perturbations are not sufficient to match the observed distribution of quads
relative to the FSQ. The result is unchanged even when subhalo masses are
increased by a factor of ten, and the most optimistic lensing selection bias is
applied. We then produce quads from galaxies created using two components,
representing baryons and dark matter. The transition from the mass being
dominated by baryons in inner radii to being dominated by dark matter in outer
radii can carry with it asymmetries, which would affect relative image angles.
We run preliminary experiments using lenses with two elliptical mass components
with nonidentical axis ratios and position angles, perturbations from
ellipticity in the form of nonzero Fourier coefficients $a_4$ and $a_6$, and
artificially offset ellipse centers as a proxy for asymmetry at image radii. We
show that combination of these effects is a promising way of accounting for
quad population properties. We conclude that the quad population provides a
unique and sensitive tool for constraining detailed mass distribution in the
centers of galaxies.
