Analytic and Conditional Resolution Framework for Goldbach’s Conjecture: From Mirror Prime Densities to Two-Lemma Reduction

This paper presents a unified analytic–conditional framework for resolving Goldbach’s Conjecture. The first part develops an analytic demonstration based on the mirror-law symmetry of prime densities around E/2 and the overlap of symmetric windows derived from the Prime Number Theorem. This analytic structure shows that the density of primes on both sides ofE/2 necessitates the existence of at least one symmetric prime pair for every sufficiently large even integer E. The second part formulates Goldbach’s Conjecture as an equivalent conditional theorem requiring the validity of only two explicit lemmas: a local symmetry lemma (Lemma C) and a global overlap lemma (Lemma S). Appendix 8 provides a mathematical derivation of Lemma C, while Appendix 8B establishes a partial reduction strategy for Lemma S. Appendix 9 identifies an explicit threshold E₀ beyond which the analytic overlap is guaranteed. The resulting hybrid manuscript gives both (1) an analytic justification for the structure underlying Goldbach’s identity, and (2) a precise blueprint for reducing the conjecture to two explicit, finitely verifiable lemmas. This combined approach significantly narrows the gap between heuristic,analytic, and conditional pathways to a full resolution.