Alfredo Luis Acosta Padilla

We refine the constant tracking in the Hardy-Littlewood circle method so that the binary Goldbach conjecture is reduced to the interval 4 ≤ n ≤ 2.0 × 10¹⁹. Our two principal ingredients are: A) A point-wise minor-arc bound |S(α)| ≤ 1.2 n/(log n)¹² obtained via an L²-Bessel estimate, an improved inverse large sieve, and the choice y = n^0.32 in Vaughan's identity. B) A fully explicit major-arc evaluation. The constants are tabulated and every inequality is justified in the appendices, allowing an independent audit. Only 1.6 × 10¹⁹ even integers remain to be checked by computer.