  Jammed packings' mechanical properties depend sensitively on their detailed
local structure. Here we provide a complete characterization of the pair
correlation close to contact and of the force distribution of jammed
frictionless spheres. In particular we discover a set of new scaling relations
that connect the behavior of particles bearing small forces and those bearing
no force but that are almost in contact. By performing systematic
investigations for spatial dimensions d=3-10, in a wide density range and using
different preparation protocols, we show that these scalings are indeed
universal. We therefore establish clear milestones for the emergence of a
complete microscopic theory of jamming. This description is also crucial for
high-precision force experiments in granular systems.
