  We prove that, independent of the choice of a lens model, the total signed
magnification always sums to zero for a source anywhere in the four-image
regions of swallowtail, elliptic umbilic, and hyperbolic umbilic caustics. This
is a more global and higher-order analog of the well-known fold and cusp
magnification relations, in which the total signed magnification in the
two-image region of the fold, and the three-image region of the cusp, are both
always zero. As an application, we construct a lensing observable for the
hyperbolic umbilic magnification relation and compare it with the corresponding
observables for the cusp and fold relations using a singular isothermal
ellipsoidal lens. We demonstrate the greater generality of the hyperbolic
umbilic magnification relation by showing how it applies to the fold image
doublets and cusp image triplets, and extends to image configurations that are
neither. We show that the results are applicable to the study of substructure
on galactic scales using observed quadruple images of lensed quasars. The
magnification relations are also proved for generic 1-parameter families of
mappings between planes, extending their potential range of applicability
beyond lensing.
