Scale and Cutoff Admissibility Gates for Post-Obstruction BBGKY Residuals

This paper proves a conditional scale-admissibility meta-theorem for cutoff-regularized Coulomb-type residuals in a fixed-level BBGKY hierarchy. The result is formulated in the observation-dual testing topology supplied by the companion Gate E3 framework. Its purpose is not to derive a microscopic BBGKY transfer estimate from first principles, but to convert any verified cascade budget into an explicit admissible relation between particle number, cutoff scale, hierarchy depth, and observation resolution. The input is an observation-dual residual decomposition into cascade, cutoff-observation separation, tail, and model-error budgets. Under the default worst-case Sobolev cascade envelope, the theorem identifies a scale window in which the observation scale exponent is smaller than the cutoff exponent, and the cutoff exponent remains below the critical hierarchy-dependent cascade threshold. Within this window, the tested residual vanishes up to the prescribed tail and model errors, for fixed hierarchy depth and fixed finite time horizon. The paper also records a general interface formulation in which the worst-case cascade exponent is replaced by an externally supplied cascade exponent. This makes the result reusable for higher-correlation or refined-transfer estimates. The theorem is explicitly not a cutoff-free Coulomb closure theorem and does not prove propagation of chaos in trace norm, energy norm, or any stronger microscopic topology.