Finite-energy hard celestial current algebra from the Banerjee--Mandal--Sahoo dipole Ward identity in QED

We use the Banerjee–Mandal–Sahoo dipole-current Ward identity for the one-loop logarithmic soft-photon theorem as input and determine its finite-energy action on Mellin-difference hard currents. The commutator with such hard currents has a scheme-independent hard-hard residue that survives every one-particle redefinition. With the meromorphic continuation stated explicitly below, a two-particle Plancherel transform identifies this residue with an analytic two-particle primary module, and the coefficient map is a hard-current one-cocycle. The cocycle defines a minimal filtered abelian extension. It has a canonical two-particle primitive and integrates to an affine action. For scalar hard legs, the fixed-leg operator agrees coefficient by coefficient with the symmetry-governed long-range logarithmic tower of Choi, Kadhe, and Puhm. Applied to a tree-level scalar-QED photon-exchange block, the finite-energy analysis determines the logarithmic two-particle coefficient functional from the ordinary hard amplitude and the Banerjee–Mandal–Sahoo ordered-pair soft kernel. This gives a finite-energy relation between the Banerjee–Mandal–Sahoo dipole-current Ward identity and the exponentiated long-range celestial OPE.