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We consider the theory of the glass phase and jamming of hard spheres in the large space dimension limit. Building upon the exact expression for the free-energy functional obtained previously, we find that the random first order transition (RFOT) scenario is realized here with two thermodynamic transitions: the usual Kauzmann point associated with entropy crisis and a further transition at higher pressures in which a glassy structure of microstates is developed within each amorphous state. This kind of glass–glass transition into a phase dominating the higher densities was described years ago by Elisabeth Gardner, and may well be a generic feature of RFOT. Microstates that are small excitations of an amorphous matrixseparated by low entropic or energetic barriersthus emerge naturally, and modify the high pressure (or low temperature) limit of the thermodynamic functions.