We show that the onset and rise of QCD minijets provide the dynamical mechanism behind the appearance of a soft edge in pp collisions around CERN intersecting storage rings (ISR) energies, and thus such a soft edge is built in our minijet model with soft gluon resummation. Here the model is optimized for the LHC at root s = 7, 8 TeV and predictions made for higher LHC and cosmic ray energies. Further, we provide a phenomenological picture to discuss the breakup of the total cross section into its elastic, uncorrelated, and correlated inelastic pieces in the framework of a one-channel eikonal function.Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)