This preprint develops a finite-depth certification theory for multipole BBGKY correctors. The main algebraic result is a Graph-Schur lower bound that reduces positivity of a cumulative weighted Gram matrix to a spectral-radius condition on a normalized finite coupling graph. The manuscript introduces inductive rank ladders, one-step ladder extension criteria, weighted certification refinements, and a classification of certification regimes including uniform, harmonic, product-decay, tail-decoupled, and finite-band cases. The analysis is finite-depth: all ladder statements are made at prescribed terminal levels and no unconditional bounded inverse is claimed for the infinite-rank limit. The manuscript also includes manufactured diagnostics and source material for reproducing the finite coupling-law and response-cell certification tables.
Dmytro Panasenko (Sun,) studied this question.