Early type galaxies,ellipticals and S0s,have projected central density brightness profile logarithmic slopes,γ,ranging from about 0 to 1.We show that γ is strongly correlated with a cores coarse grain phase density,Q0 =ρ/σ3.The γ-luminosity correlation is much weaker.Q0 also serves to separate the distribution of power-law profiles,γ>0.5 from nearly flat profiles,γ<0.3,although there are many intermediate slopes at intermediate Q0 in a volume limited sample.The transition phase density separating the two profile types is approximately 0.003 M⊙ pc-3 km-3 s3,which is also where the relation between Q0 and core mass shows a change in slope,the rotation rate of the central part of the galaxy increases,and the ratio of the black hole to core mass.We consider primordial core models,globular cluster inspiral buildup and core scouring mechanisms for creating the core slope– core mass relation,finding that all are partially successful.Globular cluster inspiral models have quantitative predictions that the data support,but no single model yet completely explains the correlations.