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. ; Comment: 11 pages, 7 figures