Certified Effective-Rank Channel Dictionaries for Observation-Dual BBGKY Residuals

We develop a deterministic effective-rank certification theorem for channel dictionaries used to control observation-dual BBGKY residuals. The setting is an N-dependent Hilbert observation layer equipped with finite retained channel dictionaries and frame bounds. The main estimate separates three quantities that must be controlled simultaneously: the off-dictionary tail, the retained channel coefficient budget, and the reconstruction cost given by the inverse square root of the lower frame bound. In certified form, the residual is bounded by the sum of a certified dictionary-tail bound and a frame-cost-weighted certified coefficient/error budget. Here the certified tail bound controls the part of the residual not represented by the retained dictionary, the certified coefficient bound controls the retained channel component, and the certified error budget collects statistical, discretization, cutoff, and algorithmic certification errors. The paper also gives a spectral-threshold selection rule, a rate-competition model for choosing the effective rank, perturbation-stability estimates, and interfaces with scale admissibility, witness-directed feedback, observation comparison, and higher-correlation transfer. The result is a sufficient certification gate: it does not prove cutoff-free Coulomb BBGKY closure, trace-norm propagation of chaos, or physical sufficiency without a separate observation-comparison theorem.