2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/186176We predict $σ_{p{-}\rm air}^{\rm prod}$, the proton--air inelastic production cross section, at $pp$ center-of-mass energies $2\le\sqrt s \le 100000$ TeV, using high energy predictions from a saturated Froissart bound parameterization of accelerator data on forward $\bar pp$ and $pp$ scattering amplitudes, together with Glauber theory. The parameterization of the $\bar pp$ and $pp$ cross sections incorporates analyticity constraints and unitarity, allowing accurate extrapolations to ultra-high energies. Our predictions are in excellent agreement with cosmic ray extensive air shower measurements, both in magnitude and in energy dependence4 pages, 4 figures, 1 table, manuscript in .TEXHigh Energy Physics - PhenomenologyAstrophysicstext