The Goldbach Problem for Primorials: Singular Series, Type II Sums, and LPF Structure

Let p_k# denote the k-th primorial. We develop a complete analytic program for the binary Goldbach problem restricted to the primorial family: conditionally proving that G(p_k#) ≥ 1 (every primorial is the sum of two primes) for all k ≥ 3 under either of two independent hypotheses.

Building on the pre-sieving and variance results of [Ross2026a], we apply Vaughan's identity to decompose the Goldbach sum Σ(p_k#) = Σ_{p ∈ S} Λ(p_k# - p) into Type I and Type II components, where S = [p_{k+1}, λp_{k+1}] is the pre-sieved window. We prove the singular series identity S(p_k#) = (p_k# / φ(p_k#)) · C_2^(k) by three independent methods (local densities, Ramanujan sums, and Vaughan main-term analysis), and establish S(p_k#) ≫ log p_k.

Type I errors are controlled unconditionally via the BDH machinery, with a quantitative improvement from a restricted large-sieve proposition saving a factor C_k = (3/π²) ∏_{p ≤ p_k} p/(p+1) ~ A / log² p_k over the standard constant. We formulate Hypothesis H(k) — a D_k-restricted, weighted bilinear minor-arcs hypothesis — and prove that H(k) ⟹ G(p_k#) ≥ 1. We show H(k) is implied by Elliott–Halberstam at any θ > 1/2, and that it is strictly weaker than the standard Elliott–Halberstam conjecture, sitting precisely at the level of the Weak Bilinear Minor-Arcs Hypothesis restricted to D_k-moduli. We assess four potential unconditional approaches to the Type II sum (restricted large sieve, Friedlander–Iwaniec analogy, Kloosterman/Weil bounds, GL(3) spectral theory) and identify the precise obstruction in each case.

As an alternative to H(k), we develop a structural approach via least-prime-factor (LPF) allocation of the rough complement set, establishing that a Buchstab-type upper bound on individual LPF class sizes implies G(p_k#) ≥ (1 - e^{-γ}) π_S ≈ 0.4385 π_S for large k. The paper concludes with two explicit open problems whose resolution would yield an unconditional proof.