  A model independent Levy expansion method is introduced to describe nearly
Levy shaped squared moduli of Fourier transforms. We apply this method to
precisely characterize the most recent elastic scattering data of proton-proton
collisions at $\sqrt{s} = 13$ TeV. The results reveal a substructure of
protons, which is found to be significantly larger and darker in high energy
proton-proton collisions at the TeV scale, as compared to a rather faint and
apparently overlooked sub-structure, that we also identify in elastic
proton-proton scattering at the ISR energy range of $\sqrt{s} = 23 - 62$ GeV.
