Fixed-3 Decomposition and a Parallel Modular Structure for Goldbach Representations

This paper presents a structural approach to the relationship between the ternary and binary forms of Goldbach’s problem. The starting point is the proven ternary Goldbach theorem, which states that every odd integer greater than 5 can be represented as the sum of three primes. We investigate a distinguished subclass of such representations in which the prime 3 is fixed as one summand: N=3+p+q, where N is odd and p,q are primes. Subtracting the fixed prime 3 gives N-3=p+q, an even integer represented as the sum of two primes. The paper develops this construction through a modular classification of odd and even integers modulo 6. Odd integers are partitioned into the classes G_1,G_3,G_5, while even integers are partitioned into G_2,G_4,G_6. For primes greater than 3, only G_1 and G_5 are available, and their pairwise sums generate the three even residue classes: G_1+G_1→ G_2,_5+G_5→ G_4,_1+G_5→ G_6. This provides a complete modular classification of the possible residue classes of binary prime sums. The paper further observes that, with 3 held fixed, the transformation N↦ N-3 maps the consecutive odd integers 9,11,13,… onto the consecutive even integers 6,8,10,…, without omission. The central research question is whether the modular structure together with the fixed-3 construction can be used to establish, for every odd N≥9, the existence of primes p,q satisfying N=3+p+q. If this universal existence statement is established, then every even integer E≥6 would satisfy E=p+q, yielding the strong Goldbach conjecture. The purpose of this paper is therefore to formalize the proposed mechanism, identify the exact logical point requiring proof, and provide a framework that can be independently tested, extended, or refuted.