Complete Characterization of Overlap Structure in Goldbach Representations: The Modulo 3 Phenomenon and Deterministic Unit Deviation

We present the first systematic computational investigation of the overlap between primesappearing in binary Goldbach representations of 2n and ternary representations of n. Forodd n ≥ 7, we define the overlap count Sn as the number of primes appearing in both typesof representations.Through exhaustive computation up to N = 10^7 and targeted verification at N = 10^8, weestablish a near-perfect correlation (R = 1.0 to machine precision) between Sn and r2(2n),the number of binary Goldbach representations. Remarkably, 98.83% of cases exhibit exactequality Sn = r2(2n).We provide a complete characterization of all deviations: they occur exclusively withinthe residue class n ≡ 0 (mod 3) and follow a deterministic rule Sn = r2(2n) − 1 withoutexception across 67,979 verified instances spanning three orders of magnitude (N =10^5, 10^6, 10^7). The probability of this occurring by chance is less than 10^−20,000.Furthermore, we discover a phase transition between N = 10^6 and N = 10^7 whereoutlier distribution shifts from mixed residue classes (8% in n ≡ 0 (mod 3) at N = 10^6) tocomplete concentration (100% at N = 10^7). Within the deviation class, we observe perfectequipartition among residues n ≡ 9, 15, 21 (mod 30) (33.4% each, δ < 0.1%).The error decay follows a power law with exponent β ≈ 1.87 (R^2 = 0.945), and weestablish a strong connection to Hardy-Littlewood singular series (ρ = 0.99949). Thesefindings reveal previously unknown structural constraints in additive prime representationsand suggest deep connections between binary and ternary Goldbach problems.Keywords: Goldbach conjecture, additive number theory, prime representations, computationalnumber theory, modular arithmetic, phase transitions, singular seriesMSC2020: 11P32, 11N05, 11Y16