Chameleon Screening in the Jordan Frame: 1D Radial Solutions, Effective Mass Scale, and Connection to Scale-Dependent Emergent Gravity

We present a detailed numerical study of the chameleon screening mechanism in the Unified Chameleon Scalar-Tensor Gravity (CSTG) model. By solving the static, spherically symmetric Klein-Gordon equation for a Gaussian matter distribution, we obtain the radial profiles of the scalar field φ (r), the effective mass mₑff (r), and the post-Newtonian parameter γ (r). The scalar field is found to decrease in the interior of dense objects, while the effective mass grows by orders of magnitude, confirming the chameleon effect. The resulting γ remains extremely close to unity, satisfying Solar System constraints. We further establish a direct analogy between the chameleon screening length λₑff ≡ mₑff^-1 and the interaction range λ in discrete relational network models, where Newtonian scaling was recently shown to emerge at a critical scale separation L/λ ≈ 4. This connection suggests that the chameleon mechanism may be understood as a continuous realisation of scale‑dependent emergent gravity, unifying two previously independent lines of research.