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 m_eff(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 λ_eff ≡ m_eff⁻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.