Goldbach-like Additive Closure Under Global Phase Correlations: Numerical Evidence at Fixed Density

The Goldbach conjecture is commonly interpreted as a consequence of the average density of prime numbers. However, density-based arguments alone do not account for the remarkable robustness of additive closure observed in numerical data. In this work, we investigate Goldbach-like additive closure in several prime-inspired random models while explicitly controlling for global density. We show that purely local constraints, such as hard-core exclusion, and purely congruential filters significantly degrade additive closure at fixed density. In contrast, we introduce a minimal global phase correlation mechanism, implemented as a weak non-local bias depending on a cumulative log-periodic phase, which restores full additive closure across a robust range of parameters. These results indicate that additive closure is governed not only by density but also by the global correlation structure of the underlying set. The present work is numerical and exploratory in nature and does not claim to model or explain the distribution of prime numbers.