Emergent Gravity at the Critical Point: From Discrete Relational Networks to Chameleon Cosmology

We demonstrate that Newtonian gravity emerges as a collective phenomenon from a critical regime in an underlying relational network, characterised by the ratio L/λ≈4L/λ≈4 between the system size LL and the interaction range λλ. Using the Yukawa potential as a minimal continuous description, we identify a robust transition window 3≲L/λ≲63≲L/λ≲6 where the effective force law closely approximates the 1/r21/r2 behaviour. This critical regime is naturally realised in the Chameleon Scalar–Tensor Gravity (CSTG) model, where the density‑dependent Compton wavelength λeff=1/meffλeff=1/meff plays the role of λλ. The observable universe today satisfies LH/λeff≈4.2LH/λeff≈4.2, placing it precisely at this threshold. We show that L/λ=4L/λ=4 acts as a global dynamical attractor: numerical minimisation of the pure field energy leads to spontaneous convergence to this exact value without external fine‑tuning. The framework naturally alleviates the S8S8 tension, predicts a distinctive crossover in the growth rate fσ8(z)fσ8(z) at z≈0.6z≈0.6, and implies a stochastic gravitational‑wave background of amplitude ΩGW≲10−10ΩGW≲10−10 in the frequency band f∼10−16 Hzf∼10−16Hz, accessible to LISA and pulsar timing arrays. Furthermore, the Hubble tension is resolved as a direct consequence of the evolution of LH/λeffLH/λeff from 4.04.0 at recombination to 4.24.2 today, yielding an 8.3%8.3% increase in H0H0 with no additional free parameters. Our results thus unify local gravity tests, the suppression of large‑scale structure growth, and the expansion history into a single, falsifiable emergent‑gravity paradigm.