Cross-Section Bootstrap: Unveiling the Froissart Amplitude

We derive a universal bound on the integrated total scattering cross-section at \emph{finite} energies, expressed in terms of a single low-energy coefficient constrained by the non-perturbative S-matrix Bootstrap. At high energies, the bound is compared with proton-proton scattering data; at low energies, with numerical bootstrap results obtained by directly maximizing the cross-section. We conjecture that the amplitude saturating the cross-section at high energies lies at a strongly-coupled corner of the allowed space of low-energy parameters. This universal amplitude exhibits a rising total cross-section, a shrinking elastic differential cross-section with multiple diffractive minima, and a surprisingly rich spectrum of resonances aligning along Regge trajectories, including Pomeron-like and daughter trajectories, as well as unusual ``singular" trajectories in the forward limit which appear deeply interconnected with Froissart growth. Remarkably, the eikonal representation reveals that the scattering is localized within an annular region that slowly expands with energy, challenging the traditional ``disk" diffraction picture. Our results open the door to theoretical and phenomenological studies of \emph{soft} high-energy hadronic scattering via the S-matrix Bootstrap.