A Scalar-Memory Obstruction in Correlation-Corrected BBGKY Closure for Anisotropic Hartree Backgrounds

This preprint gives an obstruction-first analysis of scalar-memory correlation corrections in a finite-rank BBGKY closure model for anisotropic Hartree backgrounds. The manuscript separates the singular and geometric components of the closure problem. In a two-dimensional smoothed logarithmic benchmark, the projected four-mode verification shows no power loss in the smoothing scale, so the logarithmic singularity itself is not the obstruction. The obstruction instead comes from the scalar nature of the short-memory amplitude: a scalar memory variable can rescale a projected lift but cannot change its angular orientation relative to an O(1) anisotropic active defect. The paper proves a two-layer scalar-memory obstruction. If the mesoscopic normalization cost is positive, the scalar-memory scale window closes under non-perturbative anisotropic active pairing. Even in the degenerate zero-cost case, additive placement still fails through an uncompensated active memory component. The weak-anisotropy closure theorem is retained as a sharpness statement, while the main conclusion motivates angular-resolved or matrix-valued memory kernels as the natural constructive continuation.