Covering Smooth Numbers and Uncovering Lemoine Representations: A Conditional Window Method

We examine the covering properties of smooth numbers and their complements in the interval [1,N] with respect to the pivot M = N + 1. In a “near-critical” regime with smoothness bound Y = ⌊N/5⌋ + 1, we establish an unconditional framework that reduces the existence of constrained Lemoine representations M = p + 2q (with both primes p, q > Y ) to verifying a single analytic estimate on major arc contributions in the Hardy-Littlewood circle method. Our approach synthesizes three unconditional components: (i) a window forcing technique showing that bounded neighborhoods must contain at least one covering failure, and thus at least one Lemoine representation, when doubly-smooth counts oscillate sufficiently, (ii) a deterministic 3-channel lower bound yielding macroscopic periodic oscillations in the doubly-smooth count, and (iii) Lipschitz bounds on smooth-number counts under local perturbations. We prove that if the major-arc singular series exhibits the expected separation of local factors at small primes, then constrained Lemoine witnesses exist with density tending to 1 among odd integers. Under stronger pointwise (or averaged) minor-arc control, we obtain periodic failures of the Y-smooth two-term cover C(N,Y)—and the corresponding presence of Lemoine witnesses—reflecting explicit modular structure.