A Coulomb Gate Theorem for Cutoff-Regularized BBGKY Defects: Gate Topology, Coupled Scaling, and Dual-Witness Obstructions

This paper develops a conditional gate framework for cutoff-regularized Coulomb-type BBGKY defects in an observable-dual topology. It does not claim unconditional cutoff-free BBGKY closure, trace-norm propagation of chaos, or a complete microscopic cancellation theorem. The construction separates the problem into a model gate topology, coupled admissible triples, a scale-consistency meta-contract, macro and adaptive finite-rank budgets, a microscopic correction gate assumption, higher-order residual budgets, and a dictionaryrelative dual-witness obstruction alternative. The main theorem states that verified module estimates imply vanishing of the regularized defect in the selected observable topology. A finite-window toy instantiation is included to show that the algebraic gate is not merely formal: concrete exponent choices can pass all budget inequalities without asserting any physical Coulomb closure. The derivative bookkeeping for the explicit cutoff and the mollified-cutoff variant are stated so that cutoff-loss exponents, projector constants, and micro-correction constants remain visible in later model instantiations. Conversely, failure of the gate identifies precise targets for new short-scale estimates, enlarged dictionaries, radial or angular obstruction witnesses, topology-cost accounting, or a separate compactness-andidentification argument. The purpose is therefore not to finish the Coulomb endpoint, but to state exactly which estimates would be sufficient and which residual channels remain mathematically exposed.