An identity for the sum of residues related to Goldbach partitions

In this paper, we define a new arithmetic function φₖ, which we shall call the Goldbach varphi function, defined as φₖ:Z/pₖ#Z. We establish a fundamental identity that connects the values of this function with the local count of Goldbach partitions mₖ(2n) . By exploiting the symmetry of residue classes modulo the (k+1)-th prime and utilizing the structural expansion of the primorial ring, we prove that φₖ(2n) remains strictly bounded in the critical range pₖpₖ₋₁<2n 2 can be expressed as the sum of two primes, thus providing a formal proof of Goldbach's conjecture.