Background: Goldbach’s strong conjecture states that every even integer E ? 4 can be written as the sum of two primes. This work presents the “Goldbach Circle” as a unified geometric–analytic model designed to predict where symmetric prime pairs typically occur around the midpoint E/2.
Methods: A smooth prime-density proxy, called the lambda-density, is introduced as lambda(x) = 1 / (x ln x). Symmetric candidates are parameterized by an offset t around the midpoint: p = E/2 ? t and q = E/2 + t. The Goldbach Circle maps the interval [0, E] to a circle with diameter E and uses symmetry about E/2 to define an overlap window of half-width ?(E). The model proposes that ?(E) grows on the order of (ln E)^2 with an empirically stabilized constant K. The framework is supported by a clear separation between (i) analytic symmetry of the density field, (ii) geometric translation on the circle, and (iii) empirical verification through sampled computations.
Results: The model yields a practical prediction mechanism: Goldbach pairs tend to be localized within a narrow symmetric window around E/2 whose scale is consistent with logarithmic-square growth. Figures 1-9 provide the full geometric definition, the lambda-symmetry mechanism, the shrinking angular separation as E grows, and global error statistics, including distributional summaries. Figures 10-11 give estimate f the ôverlap zone.
Conclusion: The Goldbach Circle is presented as an asymptotic predictive law with strong empirical support. Claims are stated with explicit scope: the analytic–geometric structure explains concentration near E/2 for large E, while universal validity for all E is treated as a conjectural extension supported by computation and known verification records.