An identity for the sum of residues related to Goldbach partitions

Abstract In this paper, we introduce the Goldbach varphi function, \varphi_{k}:\mathbb{Z}/p_{k}\#\mathbb{Z}\rightarrow\mathbb{R}, which provides a normalized representation of the remainder term in a double sieve of dimension \omega=2. We establish a fundamental identity connecting this function to the count of sifted residues m_{k}(2n), defined as the number of integers q\in(0,2n) such that both q and 2n-q are coprime to the primorial p_{k}\#. By exploiting the symmetry of residue classes in primorial rings, we demonstrate that the variation of \varphi_{k} remains strictly bounded. We prove that for a sifting depth k\geq12, the density-weighted signal consistently dominates the accumulated discrepancy for all even integers within the primorial period. Consequently, we establish that m_{k}(2n)>2 for 2n such that p_{k}p_{k-1}<2n