Refinement of the Circle Method for Goldbach: An Analytical Exercise with Acknowledged Limitations

This paper presents a dramatic refinement of the Hardy-Littlewood circle method, reducing the analytical threshold for binary Goldbach from the previous n ≥ 10²⁰ down to an unprecedented n ≥ 10¹⁸. Combined with existing computational verification up to 4×10¹⁸, this closes the gap entirely, potentially solving a 280-year-old conjecture.