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Intrinsically cold particle dark matter inevitably creates halos with sharpdiscontinuities in projected surface density caused by the projection of foldcatastrophes onto the sky. In principle, these imperfections can be detectedand measured with gravitational lensing through discontinuities in imagemagnification and image structure. Lens solutions are discussed for the mostcommon universal classes of discontinuities. Edges caused by cold particlessuch 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 bystellar and quasi-stellar sources which show sudden changes in brightness oreven sudden disappearances (sometimes within hours) as edges are crossed.Images of extended objects such as edge-on galaxies or jets can show suddenbends at an edge, or stretched, parity-inverted reflection symmetry about asharp line. Observational strategies and prospects are briefly discussed.Comment: 9 pages, AASTeX. Final version, with explanatory figure added, to be  published in the Astrophysical Journa

Gravitational Lensing by Cold Dark Matter Catastrophes