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.

Second, we prove an equivalence: for 2 ≤ H ≤ p_{k+1}², Goldbach representations of N with a summand below H correspond bijectively to primes in (N-H, N-2], so that #{p < H : p, N-p both prime} = π(N-2) - π(N-H). The window-restricted primorial Goldbach problem is therefore identical to a prime-gap problem at N, and the least Goldbach summand of N coincides with the lesser Fortunate number of N whenever the latter is below p_{k+1}². The forced-primality mechanism underlying this equivalence is due to Čejchan, Křížek, and Somer; we sharpen their lower-side threshold from p_{k-1}² to p_{k+1}² and upgrade their pointwise theorems to a bijective counting identity.

Third, we prove an invisibility theorem: every truncated sieve functional of level D < p_{k+1}, and every sieve supported on moduli dividing P(z) with z ≤ p_k, is constant on the complement set {N-p}. In particular the truncated sifted count X_R(z) equals π_S identically. Decomposing general moduli into a smooth sector (divisors of N) and a rough sector (coprime to N), we evaluate the smooth sector deterministically — π_S(d, N mod d) = 0 for every 1 < d | P(z), with a closed form for the resulting square sum — and show the rough sector is empty below the Barban–Davenport–Halberstam range for windows of length O(p_{k+1}).