Critical Scaling of Ollivier-Ricci Curvature in Discrete Relational Networks and an Exploratory Cosmological Crossover

We investigate the emergence of Newtonian-like scaling in discrete relational networks and explore, as a proof of concept, a possible mapping to a chameleon scalar-tensor gravity (CSTG) model. High-precision simulations of Watts-Strogatz small-world networks (N = 300 to 1500 nodes) show that the average Ollivier-Ricci curvature scales as ⟨κ⟩ ∝ a^-2. 53±0. 04, where a is an effective scale factor identified with the inverse edge density. This topological scaling implies an effective relational energy density ρᵣed ∝ a^-2. 5. In the continuum limit, we embed this scaling into the CSTG action with conformal coupling ξ = 1/6 and a runaway potential V (ϕ) = Λ⁵/ϕ. We define a criticality order parameter η (z) = LH/λₑff - 4, where LH is the physical Hubble horizon (modulated by the conformal coupling) and λₑff is the Compton wavelength of the chameleon field. Under the hypothesis mₑff² ∝ ρᵣed, we solve the full coupled background equations in the Jordan frame. The resulting zero crossing of η (z) occurs at z ≈ 0. 23, marking a transition from enhanced to suppressed gravity at low redshift. Solving the linear growth equations with spectral network attenuation (derived from the deep Ricci well, Γ = 0. 709) yields predictions for fσ₈ (z) that are very close to the ΛCDM baseline. A diagonal χ² analysis gives Δχ² ≈ 0. 2 for 12 degrees of freedom (p ≈ 0. 41), indicating that the network-induced scaling does not significantly affect structure growth under the small-correction hypothesis Ωᵣed0 = 0. 005. The model satisfies Solar System constraints via the chameleon screening mechanism and offers a falsifiable scalar breathing mode in gravitational waves as a future test. We discuss the main limitations of the approach and outline necessary steps toward a more rigorous derivation.