Verlinde's (2016) emergent gravity framework — equation 7.40 — is derived for a single isolated spherically symmetric mass. No extension to gravitationally bound two-body systems existed in the literature. We derive such an extension, termed Binary Emergent Gravity (BEG). The superposition of two displacement fields in the elastic de Sitter medium introduces an entropic cross-term ∇u₁·∇u₂ generating a scale-dependent effective mass M_eff(r) with a three-zone structure governed by the transition radius r_trans = √(d₁ × d₂) and the holographic invariant α × n = ln(L_H/d_bin). Applied to Sirius A/B: r_trans = 9.41 AU, binary compensation factor f_comp = 0.200, periapsis precession excess ΔΦ_BEG − ΔΦ_GR ∈ [15.6, 27.8] μas/yr (S/N 32–58 over 50 yr with Gaia), period correction ΔT ~ 0.3–9.2 days for a companion at 0.5–1.3 AU from Sirius B. Extended to wide binaries (500–30 000 AU): gravitational boost γ_g ≈ 1.077, constant across all separations — distinct from MOND, compatible with Gaia DR3 (Chae 2025) at <2σ. All predictions are parameter-free and falsifiable with Gaia DR4. Part of the SSEG series (Papers I–V on solar system kinematics and emergent gravity).