Emergent Gravity at the Critical Point L/λ = 4: Finite‑Size Scaling, Chameleon Screening, and Falsifiable Predictions from Discrete Networks to CSTG Cosmology

We strengthen the unified model of emergent gravity (discrete relational networks + Chameleon Scalar-Tensor Gravity) by providing an analytical estimate of the scaling relation γ = ½(L/λ), a comprehensive finite-size scaling analysis, and quantitative observational predictions. Using simulations of Watts–Strogatz networks with up to N∼10⁴ nodes, we show that the critical attractor L/λ = 4 is approached as N → ∞, with an extrapolated infinite-size value L/λ = 3.894 ± 0.009 (deviation 2.6%). We compute the growth rate fσ₈(z), the scale-dependent matter power spectrum P(k), the stochastic gravitational wave background Ω_GW, the time variation of G, and discuss how the discrete nature of the network avoids curvature singularities. A new appendix validates numerically the conjecture that local clustering maps to Ollivier–Ricci curvature (Pearson correlation r → 0.964 for N=500), supporting the conformal coupling ξ = 1/6. All predictions are presented with realistic uncertainty ranges and are falsifiable with upcoming surveys (DESI, Euclid, LISA, SKA).