History-Dependent Gap Equation with Vertex Dressing: Closing the DSE/BSE System (Paper XXIX)

We complete the construction of the History-Dependent Gap (HDG) formalism by introducing a fully dressed fermion–boson vertex that strictly satisfies the Ward–Takahashi identity. The vertex is decomposed into the standard Ball–Chiu part and a novel transverse contribution driven by a temporal nonlocal memory kernel. This closure allows us to solve the coupled Dyson–Schwinger and Bethe–Salpeter equations self-consistently. A key finding is the dynamical generation of the Pauli form factor F2F2 (anomalous magnetic moment) purely as a memory-driven effect, without any bare tensor coupling. Furthermore, the memory kernel modifies the ultraviolet scaling of the vector form factor F1F1. Numerical implementation with a minimal memory model yields a pion mass in agreement with experimental PDG data, constraining the memory saturation scale to ΛM≈0.32ΛM≈0.32 GeV. This work establishes HDG as a self-consistent nonlocal quantum field theory where confinement and dynamical chiral symmetry breaking are governed by a unified memory scale. The upload includes full LaTeX sources and reproducible Python code.