Microscopic Pair-Correlation Criteria for Cutoff-Regularized Singular BBGKY Defect Channels

This preprint develops a microscopic pair-correlation framework for cutoff-regularized Coulomb-type BBGKY defect channels. A smooth radial cutoff separates the transition shell, an active mesoscopic annulus, and the macroscopic projected-response region. On the active annulus the paper uses an amplitude formulation F_epsilon = u_epsilon^2, where the radial amplitude belongs to a uniform shell-admissible class allowing correlation-hole behavior without imposing a cutoff-independent positive lower bound. The second-hierarchy defect current is decomposed, in a tested distributional sense, into macroscopic projected, shell-pair, pair-residual, three-body, and stress components. The scalar-pair-generated defect class is closed in the calibrated dual norm, while three-body residues outside this class are detected by annihilating test functionals. A model angular example illustrates how non-radial shell components can have positive distance from scalar radial pair corrections. The manuscript formulates basis-dependent leakage conditions preventing shell-scale residuals from contaminating the macroscopic Gram/Riesz certificate and provides sufficient conditions for convergence of the selected defect identity in a fixed tested distributional channel. No cutoff-free Coulomb closure, trace-norm propagation of chaos, full many-body derivation, or complete Bogoliubov fluctuation theorem is claimed.