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The Galactic Core is, obviously, the center of the galaxy. However, there may exist two galactic cores - the one of the TrueSpace galaxy (the Milky Way) and the one of the HyperSpace galaxy corresponding to it.
Regardless of the relation between the two cores, we only have information as to the HyperSpace galactic core, which is the direction of the Precursor departure aboard great, circular spaceships similar to the Mark II — and the direction indicated by the Rainbow Worlds arrow.