This version 1. 2 incorporates further clarifications concerning the conductor mechanism, the lifted shift parameter, and the role of the parity obstruction in the proposed Goldbach-type dispersion framework. It builds on the conductor-level clarifications introduced in Version 1. 1. In particular, the small-gcd extension of the CRT recombination is made explicit through the compatibility condition (g= (r, z) N), and the conductor counting associated with the lifted shift parameter (^) is treated by divisor counting for the fixed integer (B=^), with the exceptional case (^=0) assigned to the degenerate ledger. This version also adds a deterministic averaged inverse-conductor bound. WritingDc (r, z) = (N, r) (^, z), the manuscript records the estimateₑ ₑℂₙ ₑDc (r, z) ^-1/2 R³R^o (1). This makes explicit that the averaged conductor loss is controlled by divisor counting alone and does not rely on a probabilistic genericity assumption on the lifted shift parameter. Finally, the discussion of the parity obstruction has been expanded. The revised section explains that the Heath--Brown decomposition is parity-blind by itself, but that the decisive saving in the proper central rank (2) block is obtained only after dispersion, subtraction of the local main term, Poisson summation, CRT recombination, and double reciprocity. The transformed object is a complete reciprocal Kloosterman phase rather than a classical sieve weight acting on primes and almost-primes. These additions are intended to make the central conductor mechanism more transparent and to clarify why the parity obstruction, while fundamental for classical sieve mechanisms, is not the operative obstruction in the reciprocal Kloosterman phase used in the present framework.
Mohamad TAGHLOBI (Sat,) studied this question.