We extend Binary Emergent Gravity (BEG — companion paper) to the wide binary star regime (d_bin = 500–30 000 AU), where gravitational accelerations are in the range g_N ~ 0.05–500 a₀. BEG predicts a gravitational boost γ_g = G_eff/G_N ≈ 1.077, constant across all separations — a signature qualitatively distinct from MOND (which predicts no effect in the high-acceleration regime where g_N > a₀). A theorem of invariance establishes that this boost arises exclusively from the entropic cross-term, not from any radial potential modification. Comparison with Gaia DR3 Bayesian orbit measurements (Chae 2025) shows agreement at <2σ. The discriminating test — Γ at g_N > 10 a₀ with σ_Γ < 0.005 — requires ~600 pairs with 3D velocities, accessible to Gaia DR4. Companion paper to BEG Article 1.