Sifted Symmetry and 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 complement N-p is coprime to every prime q ≤ p_k. We then establish a pointwise symmetry principle: whenever m | N, one has gcd(N-x, m) = gcd(x, m) for every integer x. Consequently coprimality with P(z) = ∏p ≤ z p and, more finely, the least-prime-factor value lpf = r (whenever P(r) | N) are preserved exactly under x ↦ N-x on arbitrary sets, with no stability hypothesis; for general N an explicit deterministic error bound J(I) · 2^π(z) holds on unions of J(I) intervals. We organize these statements as a trichotomy that separates the trivial (involution-stable) case from the genuinely primorial (pointwise) case. On the line x+y = N, the divisibility conditions q | x and q | y select the same points whenever q | N. This geometric coincidence also explains the single forbidden residue class at such primes in the Goldbach singular series.