Projected Channel Renormalization for Cutoff-Regularized Coulomb BBGKY Residuals: Conditional Dynamic Estimates and Contrast Budgets

We formulate a conditional projected-channel renormalization theorem forcutoff-regularized Coulomb BBGKY residuals. The starting point is the spectralobstruction that a similarity transform Z−1W Z cannot attenuate a non-zeroHermitian matrix witness in full operator norm. The replacement mechanismstudied here is a non-similarity observation channel: in the minimal model, aPauli-diagonal completely positive trace-preserving channel acts on an M2(C)internal layer and damps declared contrast observables while preserving thetrace sector. The main result is a conditional dynamic gate theorem: if theresidual source, comparison defect, derivative budget, interaction-transportcommutator, and dual-orbit attenuation assumptions are supplied by a concretecutoff model or by an upstream higher-correlation gate, then the projectedwitness component is controlled with an explicit error budget. The theoremis deliberately conditional; it is not a cutoff-free Coulomb closure, not apropagation-of-chaos theorem in trace norm, and not a derivation of thesource budget from the full BBGKY hierarchy. Its contribution is to isolate theexact price of an E6-type operator-valued renormalization: contrast damping,observable-comparison loss, derivative compatibility, and source-budget controlmust all be paid separately.