Sieve Structures of Primorial Goldbach Complements

Let N = p_k# be the k-th primorial. We prove an exact "pre-sieving" property: for every prime p > p_k, the complementary integer N-p is automatically coprime to every prime q ≤ p_k. Equivalently, primorial complements eliminate all local obstructions coming from primes ≤ p_k. We then develop a rigorous symmetry principle for sifted sets under the involution x ↦ N-x. For any z ≥ 2 and any ψ-stable set I ⊆ [1, N-1], we compare the counts of integers in I coprime to P(z) = ∏_{p ≤ z} p with the corresponding counts for N-x. If P(z) | N (in particular, when N = p_k# and z ≤ p_k), the symmetry is exact; in general we obtain an explicit deterministic error bound of size O(J(I) · 2^π(z)). A corresponding least-prime-factor symmetry theorem shows exact balance in LPF distributions for primorials. Finally, we give a rigorous Bombieri–Vinogradov/BDH mean-square bound controlling a truncated sifted complement statistic X_R(z) in a natural sieve range. For primorials, the moduli d | P(z) all divide N, making the discrepancies E_S(d, N mod d) fully explicit and deterministic: each equals -π_S(1,0)/φ(d). This yields an explicit formula for the variance sum V(N, z, R). Computational verification for N = 2,310 confirms the exact LPF symmetry predicted by the corollary on LPF primorial exactness for integers in the sieve window, and illustrates the structural gap between this sieve-level balance and prime-level behavior.