Graph-Schur Stability and Certified Rank Selection for Dipole–Quadrupole–Octupole BBGKY Correctors

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.