Refinements of the Ternary Goldbach Theorem: Representations with Constrained Pair Sums

This preprint studies a constrained variant of the ternary Goldbach problem in which the shifted pair summ=p1+p2+σm = p_1 + p_2 + =p1+p2+σ, with fixed σ∈{±1}σ ∈ \{± 1\}σ∈{±1}, is required to satisfy an additional multiplicative condition, whilen=p1+p2+p3n = p_1 + p_2 + p_3n=p1+p2+p3 remains a classical ternary prime representation. The paper establishes three results of increasing strength: an unconditional result in which the pair sum is an almost-prime, an asymptotic formula under the Bombieri–Vinogradov theorem, and a conditional theorem in which the pair sum itself is prime, assuming either the Generalized Riemann Hypothesis together with a bilinear dispersion input, or an averaged Elliott–Halberstam–type hypothesis.