We present computational evidence supporting the following conjecture: For every even integer n >= 1000, there exists a Goldbach partition n = p + q (with p <= q) such that the smaller prime p satisfies: p < sqrt(n) * (ln n)^2 AND p is a quadratic residue modulo the smallest prime factor of n/2.. An exhaustive search over 50,000 cases found no counterexample. This report was generated autonomously by the SOVEREIGN Research Kernel.