Chebyshev Bias in Goldbach Partitions: Empirical Evidence and a Dirichlet Character Correction to the Hardy–Littlewood Formula

The first large-scale empirical study of Chebyshev bias in Goldbach partitions. Analyzing 2,495,001 even integers up to 5,000,000, we find that numbers n ≡ 1 mod 3 have 0.26% more Goldbach representations than n ≡ 2 mod 3, with p = 2.0×10⁻¹⁹ (t-test) and p = 4.07×10⁻²⁰⁴ (sign test). The bias propagates through the (2,2) prime pair channel and decays as n⁻¹/². A Dirichlet character correction A₃χ₃(n)n¹/²/(ln n)² with A₃ = 1.23×10⁻⁵ is proposed. Null results for mod 5 and mod 7 are reported. Comprehensive robustness checks included. Interactive prediction tool (CLI + Streamlit) with 0.04% mean error provided. Full dataset and code available.