Algebraic Bounds on Perturbative Control in Chameleon Scalar–Tensor Screening and Implications for MICROSCOPE

This work derives the density–dependent minimum of the chameleon scalar field in closed form using the principal branch of the Lambert-W₀ function. The solution propagates the exact stationary condition directly into the thin-shell screening factor and the observable differential acceleration measured by the MICROSCOPE satellite. The analysis shows that the MICROSCOPE observable depends on microscopic model parameters only through the environmental screening configuration of the terrestrial density profile. As a result, the satellite constraint restricts a degenerate combination of microscopic parameters rather than uniquely determining them. The derivation demonstrates how the exact Lambert-W₀ stationary solution propagates through the thin-shell formalism to the macroscopic observable without invoking weak-field approximations. The closed-form Lambert-W₀ stationary solution also allows the screening transition to be expressed directly in terms of the ambient matter density. By inverting the stationary condition, the local matter density can be related to the scalar control parameter that governs the screening behavior. This relation defines a characteristic screening density scale that can be mapped onto the cosmological density field commonly used in structure-formation analyses. The corresponding overdensity threshold provides a direct bridge between the chameleon screening transition and the dark-matter overdensity hierarchy of the cosmic web. In this framework the scalar field becomes suppressed in high-density regions such as galaxy halos, while remaining active in lower-density environments including filaments and voids. This mapping makes it possible to compare the screening transition directly with large-scale structure observables used in modern cosmological surveys.