Covering Smooth Numbers to Uncover Goldbach and Lemoine Representations

We prove that every odd integer N ≥ N0,L is a sum of a prime and twice a prime (Lemoine), and that every even integer N ≥ N0,G is a sum of two primes (Goldbach), where N0,L and N0,G are explicit computable thresholds obtained from explicit prime-count estimates in arithmetic progressions. The argument proceeds in four stages: (i) an exact involutive (Goldbach) or affine (Lemoine) organization of candidate summands into partner-closed domains with monotone overlap counts; (ii) a near-critical reduction showing that representation failure in a near-critical window forces the existence of a prime exceeding a smoothness threshold; (iii) a deterministic triple-modulus channel decomposition (moduli 15, 77, and 1155) via exact inclusion–exclusion; and (iv) closure by a partner-admissibility squeeze combined with modulus-specific arithmetic-progression bounds of Bennett–Martin–O’Bryant–Rechnitzer and overlap corrections, yielding strict inequalities above computable thresholds. For Lemoine, the one-front modular decomposition yields the threshold N0,L = 2×10^8. For Goldbach, the full modulus-specific density analysis yields N0,G = 6 ×10^16. Combining these analytic results with published computational verifications up to 10^13 in the odd case (Juhasz, et al.) and up to 4 × 10^18 in the even case (Oliveira e Silva, et al.) yields the conjectures for all remaining integers. The analytic component requires no unproven hypotheses (e.g. GRH).