Emergent Relational Geometrodynamics: Dynamical Self-Organization of Newtonian Scaling in Adaptive Networks with Curvature Constraints

We introduce a dynamical extension of the Emergent Relational Geometrodynamics program in which the interaction range of each node depends adaptively on its local connectivity, thereby mimicking the chameleon screening mechanism of scalar‑tensor gravity. The network evolves through local rewiring moves that minimise a structural energy functional combining field‑induced tension with a curvature penalty that penalises low clustering. Numerical simulations demonstrate that the system spontaneously self‑organises into a configuration where the effective decay exponent of the scalar field approaches the Newtonian value γ≈1γ≈1 without any externally imposed interaction scale. The mean ratio L/⟨λ⟩L/⟨λ⟩ stabilises precisely at the critical value 44 identified in earlier static network models, while the curvature term successfully prevents dimensional collapse and preserves a three‑dimensional effective geometry. These findings indicate that Newtonian gravity can arise as a dynamical attractor in discrete relational systems and provide a direct conceptual link with the chameleon mechanism in the continuum.