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

Version 1.3 further clarifies the normalization, conductor analysis, and coefficient-uniformity aspects of the proposed Goldbach-type dispersion framework. Goldbach counting convention and diagonal normalization The sharp Goldbach counting function is now explicitly normalized so that Goldbach representations are counted as unordered pairs, with p₁ ≤ p₂. Under the parametrization p₁ = M − d, p₂ = M + d, this convention corresponds to d ≥ 0. The possible diagonal representation d = 0, occurring when M is prime, is separated explicitly. The smoothing function is smooth, compactly supported, and supported on the positive half-line (0, ∞). Consequently, the smoothed correlation involves only positive shifts and introduces no double counting arising from the interchange of p₁ and p₂. Uniformity of the bilinear Kloosterman core The bilinear Kloosterman estimate is now stated explicitly to be uniform with respect to divisor-bounded coefficient sequences. The proof uses only the corresponding L² bounds and does not rely on primality properties of the intermediate variables, nor on cancellation arising from the signs of the Heath–Brown coefficients. The decisive saving comes instead from complete modular oscillation after conductor factorization and conductor counting. Conductor mechanism The conductor-level clarifications introduced in the previous versions remain part of the framework. In particular, the small-gcd extension of the CRT recombination uses the compatibility condition g = gcd(r,z), g ∣ ℓN. The conductor counting associated with the lifted shift parameter Δ♯ is reduced to divisor counting for the fixed integer B = ℓΔ♯, with the exceptional case ℓΔ♯ = 0 assigned to the degenerate ledger. Averaged inverse-conductor estimate The manuscript also retains the deterministic averaged inverse-conductor estimate Σ(r ∼ R²) Σ(z ∼ R) Dc(r,z)⁻¹ᐟ² ≪ R³Rᵒ⁽¹⁾, where Dc(r,z) = gcd(ℓN,r) · gcd(ℓΔ♯,z). 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. Parity obstruction The discussion of the parity obstruction has also been made more explicit. The Heath–Brown decomposition is parity-blind by itself, but the saving in the proper central rank 2 block is obtained only after dispersion, subtraction of the local main term, Poisson summation, CRT recombination, double reciprocity, and the resulting complete Kloosterman oscillation. The argument therefore does not attempt to distinguish primes from almost-primes through parity-sensitive sieve weights. The analytic gain arises instead from the transformed complete modular phase. Summary of Version 1.3 Version 1.3 is primarily a clarification and normalization update. It makes fully explicit the Goldbach counting convention, the diagonal/off-diagonal separation, the conductor mechanism, and the coefficient-uniformity of the bilinear Kloosterman core.