2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/186131Ordinary fracture functions, describing hadrons production in the deep inelastic scattering target fragmentation region, are generalized to account for the production of hadrons in arbitrary number, thus offering a renewed framework for dealing with QCD initial state radiation. We also propose a new jet-like observable which measures beam remnants and low-$p_{\perp}$ scattering fragments and derive its QCD evolution equations by using Jet Calculus. Possible implications for semi-inclusive deep inelastic scattering and hadron-hadron reactions are shortly discussed.10 pages, 5 figures, revtex4High Energy Physics - Phenomenologytext