This preprint presents a finite-block spectral certification method for density-selected BBGKY response correctors. The unconditional layer is a three-block graph-Schur theorem: a cumulative rank-fifteen dipole–quadrupole–octupole Gram matrix is coercive whenever the diagonal response blocks have spectral gaps and the normalized off-diagonal coupling graph has spectral radius below one. The BBGKY-facing layer is deliberately conditional. The abstract block matrix is realized as an observable-dual Gram matrix of finite multiple response tests, and closure estimates are stated under explicit cutoff, response-map, residual-tail, source, and scale-gate hypotheses. This separation keeps the graph-Schur result a finite-dimensional spectral theorem while making the kinetic-response application model-dependent and checkable. The manuscript also introduces a hysteresis-controlled certified selector over the finite-rank menu 3, 8, 15. A reproducible manufactured benchmark on the three-dimensional torus illustrates the certification pipeline using explicit trigonometric response cells, a generated 15 x 15 Gram certification table, a trajectory-derived selector table, and one diagnostic figure. This is the author-submitted manuscript version associated with the submission to Mathematical Methods in the Applied Sciences, manuscript ID 1622935. The preprint was made available after journal submission for transparency.
Dmytro Panasenko (Sun,) studied this question.