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

Version 3 update. This release incorporates several clarifications and internal consistency checks in the analytic derivation of perturbative control bounds for chameleon scalar–tensor screening models relevant to the MICROSCOPE satellite equivalence-principle experiment. Updates in this version include: • Explicit justification of the Lambert-W principal branch used in the exact stationary solution. • Verification of dimensional consistency for the stationary condition x^{2}e^{x}=\Lambda^{5}\beta/(\rho M_{\rm Pl}). • Numerical regime evaluation confirming that the master perturbative control parameter satisfies \Pi \approx 4.5\times10^{-35}, demonstrating that the MICROSCOPE configuration lies well within the perturbatively controlled screened regime. • Minor structural and clarity improvements to the manuscript. No changes were made to the underlying analytic derivations or physical conclusions relative to previous versions.