In the Modular Entropic Gravity (MEG) framework, gravitational screening occurs where the baryonic modular perturbation overwhelms the vacuum entropy mode. The screening threshold is governed by the lock equation, 8πGρ₀ℓ₀² = c²ζ_SM, which relates Newton's constant to the coherence length ℓ₀, the screening density ρ₀, and the Standard Model vacuum modular variance ζ_SM. Inverting the lock equation gives G to 7% of the CODATA value (Devlin 2026, doi:10.5281/zenodo.19946249). This paper extends that determination to a derivation through a single integer k = 3. Two new results make this possible. First, the vacuum modular variance ζ_SM = (2.0 ± 0.3) × 10⁻⁶, previously computed from SM stress-tensor correlators, is derived as 2α₁⁶ = 2/(k²+1)⁶ from the Z₃ cyclic tunnelling at the coherence scale. At the coherence scale ℓ₀, the coarse-grained entropy field must be single-valued over the coarse-graining volume: the net Z₃ winding number must be zero. Single-step and two-step tunnelling leave the field in the wrong sector and average to zero under Z₃ symmetry. Only the full Z₃ cycle — three sequential tunnelling events returning to the starting sector — survives. The amplitude for the cycle is α₁³; the probability is α₁⁶. Two orientations (clockwise and counterclockwise, corresponding to the two generators of Z₃) give ζ_SM = 2α₁⁶. The structure is directly analogous to the Wilson loop in gauge theory: only closed paths contribute to the long-distance physics. Second, the coherence length is derived from the de Sitter modular structure, ℓ₀ = R_dS e^{-(4π+1/2)} ≈ 11.0 kpc (matching SPARC to 0.1%), where the correction δ = 1/2 arises from the zero-point action of the s-wave mode at the de Sitter horizon (Devlin 2026, doi:10.5281/zenodo.20573964). The resulting derivation chain is: O → SO(8) → k = 3 → α₁ = 1/10 → ζ_SM = 2α₁⁶ → lock equation → G, with G = c²α₁⁶/(4πρ₀ℓ₀²) ≈ 6.2 × 10⁻¹¹ m³ kg⁻¹ s⁻² (−7% from CODATA). Every dimensionless quantity in the chain is determined by k = 3. The two dimensionful inputs (H₀ and ρ₀) are G-free observations. The screening density ρ₀ remains the single empirical anchor; its derivation (the Density Threshold Theorem) is the last open question of the MEG constants programme. No other framework in the literature produces a numerical value of Newton's constant from a structural combination of a gauge-theoretic integer, particle physics vacuum content, and G-free astrophysical observations. The paper includes: the lock equation derivation chain (5-step from PLES through Lindhard self-energy to Newtonian matching); a detailed independence analysis showing ρ₀ is G-free (exact cancellation in the spherical inversion) and ℓ₀ has three independent determinations (galactic, de Sitter, SM classicality); a full sensitivity analysis; the 55-galaxy SPARC validation (mean χ²_red = 0.51); comparison to prior approaches (Verlinde, Jacobson, Klinkhamer); and an honest assessment identifying the Density Threshold Theorem as the remaining frontier.