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At order $e^4$, the open electron line can carry an internal-electron self-energy, a correction at either photon vertex, or a virtual photon spanning both vertices. Include the crossed external-photon order for each class. External-electron self-energies and external-photon vacuum polarization insertions appear in unamputated functions and are treated by LSZ reduction and field residues in physical amplitudes. A closed loop with three photon vertices cancels between orientations by Furry's theorem. Renormalized calculations also include the mass, field and vertex counterterms of the same perturbative order.