Sifted Symmetry and Sieve Structures of Primorial Goldbach Complements

(Substantially revised) Let N = pₖ# be the k-th primorial. We prove an exact pre-sieving property: for every prime p > pₖ, the complement N-p is coprime to every prime q ≤ pₖ. 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) = ∏₏ ≤ ₙ 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₊+₁², 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₊+₁². The forced-primality mechanism underlying this equivalence is due to Čejchan, Křížek, and Somer; we sharpen their lower-side threshold from p₊-₁² to p₊+₁² and upgrade their pointwise theorems to a bijective counting identity. Third, we prove an invisibility theorem: every truncated sieve functional of level D < p₊+₁, and every sieve supported on moduli dividing P (z) with z ≤ pₖ, is constant on the complement set N-p. In particular the truncated sifted count XR (z) equals πS identically. Decomposing general moduli into a smooth sector (divisors of N) and a rough sector (coprime to N), we evaluate the smooth sector deterministically — πS (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₊+₁).