We present a model-independent analysis of the experimental data on the ratio $X$ between the elastic and total cross-sections from $pp$ and $bar{p}p$ scattering in the c.m. energy interval 5 GeV - 8 TeV. Using a novel empirical parametrization for t
hat ratio as a function of the energy and based on theoretical and empirical arguments, we investigate three distinct asymptotic scenarios: either the black-disk (BD) limit or scenarios above and below that limit. Our analysis favors a scenario below the BD, with asymptotic ratio $X = 0.36 pm 0.08$.
Using a novel representation for the Martins real-part formula without the full scaling property, an almost model-independent description of the proton-proton differential cross section data at high energies (19.4 GeV - 62.5 GeV) is obtained. In the
impact parameter and eikonal frameworks, the extracted inelastic overlap function presents a peripheral effect (tail) above 2 fm and the extracted opacity function is characterized by a zero (change of sign) in the momentum transfer space, confirming results from previous model-independent analyses. Analytical parametrization for these empirical results are introduced and discussed. The importance of investigations on the inverse problems in high-energy elastic hadron scattering is stressed and the relevance of the proposed representation is commented. A short critical review on the use of Martins formula is also presented.
