Covering Smooth Numbers to Uncover Goldbach and Lemoine Representations

We develop a finite-kernel “forcing” framework for near-critical constrained representations in the Goldbach and Lemoine settings. For even N we consider the targets MG(N ) = N (Goldbach) and ML(N ) = N + 1 (Lemoine), together with the near-critical smoothness rules YG(N ) = (N/4)+1, YL(N ) =(N/5)+1. For each system ⋆ ∈ {G, L} we introduce explicit bounded kernel weights ωs and shifts s ∈ K⋆, and study kernel-weighted complement–smooth correlation contrasts ∆C⋆(N ; s) = C⋆(N +2; s) − C⋆(N ; s). A deterministic conservation identity writes the cover deficit as D⋆(N, Y ) =N + C⋆(N ; Y ) − 2A⋆(N, Y ), where A⋆ varies slowly with N . A window forcing lemma shows that a single major-arc “swing” in ∆C⋆(N ; s) with enough margin forces D⋆ > 0 at some point, yielding a constrained ⋆–representation. We prove unconditional Lipschitz bounds for the slow terms, and record exact mod-p identities for the local factors driving the singular-series swing, reducing the remaining analytic task to formally deriving lower bound major arc estimates for a fixed finite family of kernels. The complementary minor-arc contribution is controlled by restriction estimates for smooth exponential sums. Computations are consistent with the required major-arc dominance and indicate that small prime channels dominate the observed variation.