  In the first part of this paper, we derive the general replica equations that
describe infinite-dimensional hard spheres at any level of replica symmetry
breaking (RSB) and in particular in the fullRSB scheme. We show that these
equations are formally very similar to the ones that have been derived for spin
glass models, thus showing that the analogy between spin glasses and structural
glasses conjectured by Kirkpatrick, Thirumalai, and Wolynes is realized in a
strong sense in the mean field limit. We also suggest how the computation could
be generalized in an approximate way to finite dimensional hard spheres. In the
second part of the paper, we discuss the solution of these equations and we
derive from it a number of physical predictions. We show that, below the
Gardner transition where the 1RSB solution becomes unstable, a fullRSB phase
exists and we locate the boundary of the fullRSB phase. Most importantly, we
show that the fullRSB solution predicts correctly that jammed packings are
isostatic, and allows one to compute analytically the critical exponents
associated with the jamming transition, which are missed by the 1RSB solution.
We show that these predictions compare very well with numerical results.
