Random very loose packings

We measure the number ?(?) of mechanically stable states of volume fraction ? of a granular assembly under gravity. The granular entropy S(?)=log?(?) vanishes both at high density, at ???rcp, and a low density, at ???rvlp, where ?rvlp is a new lower bound we call random very loose pack. ?rlp is the volume fraction where the entropy is maximal. These findings allow for a clear explanation of compaction experiments and provide the first first-principle definition of the random loose volume fraction. In the context of the statistical mechanics approach to static granular materials, states with ?<?rlp are characterized by a negative temperature.