The Goldbach Circle: A Unified Geometric and Analytic Law for Predicting Prime Paris

For nearly three centuries, Goldbach’s Strong Conjecture has resisted both analytic proof andcomplete computational verification. In this work a new perspective is introduced—the*Goldbach Circle*—which unites analytic number theory, geometric symmetry, and empiricalregularity into a single predictive law. Every even number E is represented as a circle ofdiameter E, with its midpoint E / 2 serving as the axis of mirror symmetry. The analyticprime-density function λ(x) = 1 / (x ln x) defines two mirrored density fields,λ₁(E / 2 − t) and λ₂(E / 2 + t). Their continuous equality determines a bounded overlapwindow Δ(E) ≈ K (ln E)², with K ≈ 0.1, in which at least one symmetric pair of primes(p, q) exists such that p + q = E and |p − E / 2| = |q − E / 2| = t.Geometrically, this corresponds to two arcs on the circle whose angular separationφ = 2 t / E = 2 K (ln E)² / E shrinks inversely with E, ensuring persistent intersection.The resulting *Predictive Goldbach Law*, p = E / 2 − K (ln E)², q = E / 2 + K (ln E)²,links analytic continuity, geometric curvature, and empirical stability acrosstested domains up to 10¹⁸.The model transforms Goldbach’s problem from an additive conjecture into a symmetry theorem:prime distribution behaves as a continuous field of mirror equilibrium. The analytic andgeometric forms converge to the same invariant, revealing that Goldbach’s Conjecture is thenatural expression of the constant curvature of the prime field.