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The factorization of parton distribution functions (PDFs) is a fundamental principle in perturbative calculations for hadron colliders. However, a formal proof has only been established for the inclusive Drell-Yan cross section, where the cancellation of Glauber gluons plays a crucial role. These Glauber gluons also lead to factorization violation in space- like splittings, rendering them, in general, process-dependent and the reconciliation of PDF factorization with this remains an open question. In this talk, I examine the gap-between-jets cross section at hadron colliders, which is sensitive to space-like collinear splittings and thus to factorization-breaking effects. I analyze the low-energy dynamics of these cross sections, where Glauber phases spoil collinear cancellations, leading to (“super-leading”) double-logarithmic behavior. Using a method-of-regions (MoR) analysis, I identify three-loop contributions from perturbative Glauber-gluon exchanges that restore PDF factorization below the gap veto scale.