When Hypatia or the insurgents talk in reference to Eden, they mean the Earth. Singularity exists in a far-flung future in which human life on its mother soil is an idea belonging more to myth than history. Although Hypatia keeps detailed records of Eden's history, these exist in files so rarely accessed and referenced that when she encrypted everything from 2010 to the present, no one really noticed. No one among the insurgency is able to tell player characters in concrete terms how long ago humanity left Earth or what happened to it. Rather, they speak of being "cast out," and that man has wandered the stars ever since.
Eden, being a religious concept, is rarely brought up by the everyman, with one exception: Hypatia will define the 24 hour cycle which Sacrosanct follows as following an Eden cycle. NPCs in Singularity do not use our time units, although Hypatia is gradually adjusting the auto-translator to accommodate unit conversion.