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

We present a comprehensive analysis of the unified model of emergent gravity, where Newtonian scaling arises from a critical scale separation L/λ=4L/λ=4 in discrete relational networks and is realized in Chameleon Scalar‑Tensor Gravity (CSTG). Using analytical estimates based on the Yukawa potential, we derive the linear relation γ=12(L/λ)γ=21(L/λ) between the effective decay exponent and the system size–interaction range ratio. Extensive simulations of adaptive Watts–Strogatz networks (up to N≈40 000N≈40000 nodes) demonstrate that the critical attractor is approached as N→∞N→∞, with an extrapolated infinite‑size value L/λ=3.894±0.009L/λ=3.894±0.009 — within 2.6%2.6% of the exact critical point. A weighted linear fit confirms the 1/N1/N scaling with χ2/dof=0.65χ2/dof=0.65. The CSTG background evolution yields a characteristic crossing in the growth rate fσ8(z)fσ8(z) near z≈0.6z≈0.6, alleviating the S8S8 tension. A direct χ2χ2 comparison with DESI Y1+KiDS‑1000 data gives Δχ2=−11.4Δχ2=−11.4 in favour of CSTG over ΛΛCDM. The scale‑dependent matter power spectrum predicts an excess of 4.2±0.5%4.2±0.5% at k=0.1 h Mpc−1k=0.1hMpc−1 and z=0z=0. The stochastic gravitational wave background is estimated as ΩGW∼10−13ΩGW∼10−13 to 10−1110−11, and the cosmological variation of the gravitational constant is G˙/G∼(2−4)×10−13 yr−1G˙/G∼(2−4)×10−13yr−1. All predictions include realistic uncertainties and are falsifiable with upcoming surveys (DESI, Euclid, LISA, SKA). The discrete network picture also suggests a natural avoidance of curvature singularities. This work consolidates the emergent gravity paradigm into a quantitatively testable framework, bridging discrete relational networks and continuum cosmology through the universal critical exponent L/λ=4L/λ=4.