Analytic Demonstration of Goldbach’s Conjecture through the λ-Overlap Law and Symmetric Prime Density Analysis

This study introduces a unified analytical framework, the λ-Overlap Law, which provides a deterministic proof of Goldbach’s Strong Conjecture. The approach derives directly from the Prime Number Theorem and the explicit inequalities of Dusart, establishing that for every even integer E ≥ 4, there exist two primes p and q satisfying p + q = E. The method defines the prime-density kernel λ(x) = 1/(x ln x) and demonstrates that its mirrored forms λ1(E/2 − t) and λ2(E/2 + t) necessarily intersect within a finite interval proportional to (ln E)². This intersection guarantees the existence of at least one symmetric prime pair for every E. The paper distinguishes intuitive heuristic representations (such as the rabbit-motion and circle analogies) from the formal analytical derivation based on covariance, overlap integrals, and continuity arguments. Empirical validation for 106 ≤ E ≤ 10¹8 confirms the analytic predictions, while the geometric λ-circle model illustrates the inherent symmetry of prime distributions. The resulting formulation unifies probabilistic, analytic, and geometric interpretations into a self-consistent proof framework, positioning λ symmetry as a fundamental principle governing additive properties of primes.