Double Reciprocity and a Bilinear Kloosterman Estimate in a Goldbach-Type Dispersion Framework

Version 1.4 further clarifies the exponent bookkeeping, Poisson-frequency structure, and final consistency checks of the proposed Goldbach-type dispersion framework. Poisson zero and non-zero frequencies The discussion following the four-form structure after dispersion has been expanded to distinguish more explicitly between the zero-frequency and non-zero-frequency contributions after Poisson summation. The zero-frequency term is assigned to the local main contribution, while the dispersed minor contribution is carried by the remaining non-zero Poisson phases, except for rank-degenerate or diagonal configurations, which are assigned to the degenerate ledger. Normalization of the exponent bookkeeping The ledger summary now includes an explicit normalization check for the proper central rank 2 block. The central estimate is not formulated as an isolated raw bound of the form M^alpha attached to a single asymmetric Heath-Brown subblock. Instead, the exponent bookkeeping is carried out in the R,Q-normalization. In this normalization, the conductor-counting and bilinear insertion give D_central << R^(13/2+o(1)), and the fourth-root return from the dispersion variance gives |T_min| << Q D_central^(1/4) R^o(1) << R^(29/8+o(1)) = Q^2 Q^(-3/16+o(1)). Thus the proper central rank 2 contribution is measured against Q^2 = D, not against an isolated raw M^alpha scale. Final consistency check The final consistency audit has been expanded with a check excluding any hidden M^0.85-type obstruction in the closure of the proper central rank 2 block. The manuscript does not use such a bound as an intermediate target. Any reformulation in terms of a raw M^alpha scale must specify the precise relations between M, Q, R, D, the variance normalization, and the return from the quadratic dispersion object to the original linear minor contribution. These additions do not change the architecture of the proof. They are intended to make explicit the normalization already used in the central block, to separate the local main term from the dispersed non-zero Poisson contribution, and to prevent dimensionally misleading comparisons between isolated Heath-Brown subblocks and the final o(D) criterion.