Scale‑Dependent Emergence of Newtonian Gravity in Discrete Relational Networks: The Critical Point L / λ ≈ 4 L/λ≈4

We investigate the emergence of effective power‑law potentials within discrete relational networks by solving scalar field equations on three‑dimensional graphs endowed with tunable non‑local interactions. The decay exponent γγ, defined through ∣ϕ(r)∣∼r−γ∣ϕ(r)∣∼r−γ, is shown to depend smoothly on the dimensionless ratio L/λL/λ, where LL denotes the system size and λλ the intrinsic interaction range. A Newtonian‑like regime (γ≈1γ≈1) is found to arise at a critical scale separation L/λ≈4L/λ≈4, with high‑quality power‑law fits (R2>0.93R2>0.93) across all configurations. These results indicate that inverse‑distance laws are not fundamental but emerge as scale‑dependent effective descriptions, lending support to the hypothesis that gravity may originate from the relational structure of discrete space. An analogy with the renormalisation group and a potential connection with continuum screening mechanisms are briefly discussed.