  Intrinsically cold particle dark matter inevitably creates halos with sharp
discontinuities in projected surface density caused by the projection of fold
catastrophes onto the sky. In principle, these imperfections can be detected
and measured with gravitational lensing through discontinuities in image
magnification and image structure. Lens solutions are discussed for the most
common universal classes of discontinuities. Edges caused by cold particles
such as condensed axions and thermal WIMPs are very sharp, respectively about
$10^{-12}$ and $10^{-7}$ of the halo scale. Their structure can be resolved by
stellar and quasi-stellar sources which show sudden changes in brightness or
even sudden disappearances (sometimes within hours) as edges are crossed.
Images of extended objects such as edge-on galaxies or jets can show sudden
bends at an edge, or stretched, parity-inverted reflection symmetry about a
sharp line. Observational strategies and prospects are briefly discussed.
