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We present deep VLT and HST observations of the nearest examples ofcalcium-rich 'gap' transients -- rapidly evolving transient events, with aluminosity intermediate between novae and supernovae. These sources arefrequently found at large galactocentric offsets, and their progenitors remainmysterious. Our observations find no convincing underlying quiescent sourcescoincident with the locations of these transients, allowing us to rule out anumber of potential progenitor systems. The presence of surviving massive-starbinary companions (or other cluster members) are ruled out, providing anindependent rejection of a massive star origin for these events. Dwarfsatellite galaxies are disfavoured unless one invokes as yet unknown conditionsthat would be extremely favourable for their production in the lowest masssystems. Our limits also probe the majority of the globular cluster luminosityfunction, ruling out the presence of an underlying globular cluster populationat high significance, and thus the possibility that they are created viadynamical interactions in dense globular cluster cores. Given the lack ofunderlying systems, previous progenitor suggestions have difficulty reproducingthe remote locations of these transients, even when considering solelyhalo-borne progenitors. Our preferred scenario is that calcium-rich transientsare high velocity, kicked systems, exploding at large distances from theirnatal site. Coupled with a long-lived progenitor system post-kick, thisnaturally explains the lack of association these transients have with theirhost stellar light, and the extreme host-offsets exhibited. Neutron star --white dwarf mergers may be a promising progenitor system in this scenario.

The progenitors of calcium-rich transients are not formed in situ