Finite-Dictionary Bridge Certification for Projected BBGKY Residuals

This preprint develops a finite-dictionary certification framework for projected BBGKY residuals. The method constructs response coordinates, a finite response matrix, and a projected correction, then separates coefficient-level and response-weighted budgets. A scale-safe Gronwall gate gives conditional closure when the declared response-margin, projection-tail, dynamic-replacement, source/remainder, and scale-safe gates are verified. The complementary obstruction theorem records the failure modes of the finite-dictionary method class: non-small projection tail, coefficient-dynamic obstruction, response-amplified coordinate obstruction, source/remainder obstruction, and singular Gronwall multipliers. The manuscript includes Schur/Riesz response-margin criteria, rank-two and finite Schur examples, a deterministic PASS/OBSTRUCTION audit routine, and an outlook on time-dependent dictionaries and nonstationary coefficients. This preprint is method-level and conditional. Its conclusions apply when the declared finite-dimensional gates are verified. Related companion manuscripts address specific model instances and separate asymptotic questions; the present work does not duplicate active journal submissions. In particular, the rank-two benchmark is used only as a model-verification example, and Gaussian large-window/effective-rank asymptotics are treated as separate companion-work territory.