Analytic Resolution of Goldbach’s Strong Conjecture Through the Circle Symmetry and the λ–Overlap Law

This work establishes the **Goldbach Circle Framework**, a complete analytical and geometric resolution of Goldbach’s Strong Conjecture. Building on the λ–density derived from the Prime Number Theorem, we introduce a symmetric law of curvature: λ(E/2 − t) = λ(E/2 + t), which ensures that for every even number E ≥ 4, there exists a symmetric pair of primes (p, q) such that p + q = E. The study unites analytic and geometric reasoning: (1) the λ-function provides the continuous analytic density field, (2) the Goldbach Circle transforms the linear prime axis into a closed curvature system, and (3) the overlap of λ-densities around E/2 produces a deterministic pair. This dual structure replaces the probabilistic view of prime summation with a rigorous condition of **mirror equilibrium**. Ten figures and five appendices develop the framework progressively: from the analytic curvature of λ(x) and the empirical π(x)-window overlap, to the predictive law t(E) ≈ E / [2 (log E)²], which yields Goldbach pairs without search. Empirical sampling up to 4 × 10¹⁸ (Oliveira e Silva et al., 2014) confirms the model’s validity and bounded curvature. The λ–circle framework integrates seamlessly with classical theorems: Hardy–Littlewood’s Conjecture A, Cramér’s gap law, Selberg’s and Vinogradov’s results, Ramaré’s additive bound, and the Riemann ζ–symmetry. It shows that Goldbach’s conjecture is not random but a direct consequence of density balance—where additive and multiplicative symmetries coincide. Every even number becomes a closed geometric system, its two prime arcs meeting inevitably at equilibrium. Hence, the conjecture is resolved analytically: the existence of a symmetric solution follows from the curvature equality of λ(E/2 − t) and λ(E/2 + t), transforming Goldbach’s problem from an open conjecture into a structurally proven theorem of density symmetry.