  When the source in a 4-image gravitational lens lies close to a "fold"
caustic, two of the lensed images lie close together. If the lens potential is
smooth on the scale of the separation between the two close images, then the
difference between their fluxes should approximately vanish, Rfold =
(F1-F2)/(F1+F2) \approx 0. Violations of this "fold relation" in observed
lenses are thought to indicate the presence of structure on scales smaller than
the separation between the close images. We study the fold relation and find it
to be more subtle and rich than was previously realized. The degree to which
Rfold can differ from zero for realistic smooth lenses depends not only on the
distance of the source from the caustic, but also on its location along the
caustic, and on the angular structure of the lens potential. Thus, it is
impossible to say from Rfold alone whether observed flux ratios are anomalous
or not. Instead, we must consider the full distribution of Rfold values that
can be obtained from smooth potentials that reproduce the separation between
the two close images and the distance to the next nearest image. We analyze the
generic and specific features of this distribution, and then show that 5 of the
12 known lenses with fold configurations have flux ratio anomalies: B0712+472,
SDSS 0924+0219, PG 1115+080, B1555+375, and B1933+503. Combining this with our
previous analysis revealing anomalies in 3 of the 4 known lenses with cusp
configurations, we conclude that at least half (8/16) of all 4-image lenses
that admit generic, local analyses exhibit flux ratio anomalies indicative of
small-scale structure. The fold and cusp relations do not reveal the nature of
the implied small-scale structure, but do provide the formal foundation for
substructure studies. [Abridged]
