This preprint develops a normalized Schur-coercivity framework for rank-eight dipole–quadrupole block Gram matrices arising in cutoff-regularized density-selected BBGKY closure. The manuscript separates theorem-level finite-dimensional Schur/Gram certification from conditional model-dependent BBGKY closure hypotheses. The main theorem layer includes normalized Schur coercivity, rank-eight block-Gram certification, observable-dual Gram realization, and a torus-compatible periodic benchmark. The conditional layer explains how the certified rank-eight tensor corrector may be used under explicitly stated BBGKY closure assumptions. The record includes the audit-corrected manuscript v0.6.1, a supplementary polynomial response-cell benchmark, LaTeX sources, a verification script for the periodic benchmark matrix entries, output logs, metadata files, and SHA256 checksums. No external experimental datasets are used; all verification material is synthetic/manufactured and generated from the stated formulas.
Dmytro Panasenko (Sat,) studied this question.