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 30.We prove the following theorem by elementary modular arithmetic alone requiring no unproven hypothesis and no computation.Theorem. Let n≡8 (mod 30) with n≥38. Then every Goldbach pair (p,q) with p+q=n and p,q prime satisfies p≡q≡1 (mod 6). Furthermore, for any n≡28 (mod 30) within n≥28, every Goldbach pair (p,q) of n satisfies p≡q≡2 (mod 3) which forces both p+10 and q+10 to be divisible by 3 and therefore composite.As a consequence, no Goldbach pair of any n≡28 (mod 30) can produce a Goldbach pair of n+10 via the shift (p,q)↦(p+10,q+10).We then investigate the coupled pairs (n,n+20) where n≡8 (mod 30) observing that n+20≡28 (mod 30) always. For such a coupled pair, the shift (p,q)↦(p+10,q+10) maps a Goldbach pair of n to a Goldbach pair of n+20 automatically in terms of the sum since (p+10)+(q+10)=n+20 provided both p+10 and q+10 are prime.We define the shift-propagation count,R(n)=#{p≤n/2 ∶p prime,n-p prime,p+10 prime,n-p+10 prime}and present the following conjecture supported by extensive computation.Conjecture. For every even integer n≡8 (mod 30) with n≥38, we have R(n)≥1. That is, at least one Goldbach pair of n always shifts by +10 to produce a Goldbach pair of n+20.We verify this conjecture computationally for all 33,332 values of n≡8 (mod 30) in the range38≤n≤999,980 finding zero exceptions. The minimum value R(n)=1 occurs only at n=128 across this entire range and the average value of R(n) grows consistently with the scale of n from an average of 2.00 at the smallest values to an average of 197.69 across the full range to 10^6.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.