Coupled Coprimality and Conditional Goldbach for Primorials

We isolate a simple but structurally significant property of primorial integers, which we term coupled coprimality. For a primorial N=pk#, any integer a coprime to N has a mirror partner N−a that is automatically coprime to the same set of primes. This induces a canonical pairing of the reduced residue system modulo N and yields a large, symmetric pre-sieved workspace for potential Goldbach representations. We formalize this observation and quantify the resulting structure, including a deterministic initial segment of guaranteed prime summands and an amplification of the Hardy–Littlewood singular series for primorials. These features explain why primorials empirically admit unusually many Goldbach partitions and isolate the remaining analytic obstacle to proving Goldbach’s conjecture in this context.