A MODULAR STRUCTURE THEOREM FOR GOLDBACH REPRESENTATIONS AND A COMPUTATIONAL STUDY OF SHIFT-PROPAGATING PRIME PAIRS IN THE ARITHMETIC PROGRESSION n≡8 (mod 30).

We study a structural property of Goldbach representations which are expressions of even integers as sums of two primes within two specific arithmetic progressions modulo . We prove the following theorem by elementary modular arithmetic alone requiring no unproven hypothesis and no computation. Theorem. Let with . Then every Goldbach pair with and prime satisfies . Furthermore, for any within , every Goldbach pair of satisfies which forces both and to be divisible by and therefore composite. As a consequence, no Goldbach pair of any can produce a Goldbach pair of via the shift . We then investigate the coupled pairs where observing that always. For such a coupled pair, the shift maps a Goldbach pair of to a Goldbach pair of automatically in terms of the sum since provided both and are prime. We define the shift-propagation count, and present the following conjecture supported by extensive computation. Conjecture. For every even integer with , we have . That is, at least one Goldbach pair of always shifts by to produce a Goldbach pair of . We verify this conjecture computationally for all values of in the range finding zero exceptions. The minimum value occurs only at across this entire range and the average value of grows consistently with the scale of from an average of at the smallest values to an average of across the full range to We present the modular structure theorem with complete proof, state the conjecture precisely and provide full computational verification. We make no claim of proving Goldbach’s conjecture. We propose that this modular structure and the coupled pair phenomenon may serve as a foundation for future analytic work toward Goldbach’s conjecture.