Algebraic Bounds on Chameleon Screening and Universal-Coupling Limits for Satellite Tests of Gravity

Version 4. 0. 0 Update (The Universal-Coupling Closure Synthesis): This publication establishes theoretical claim discipline by structurally differentiating exact algebraic solutions from leading-order phenomenological approximations, resolving the observational constraints of terrestrial chameleon screening for the MICROSCOPE mission and thus establishes that the Lambert-W₀ derivation is the explicit tool that proves that unextended universal coupling cannot yield a non-zero signal. Exact Lambert-W0 Formulation: Derives the density-dependent minimum exactly in closed form using the principal branch of the Lambert-W₀ function. Strict dimensional compliance is preserved via the normalized argument z (). Explicit Perturbative Control: Demonstrates that for a fiducial terrestrial core density and =1, the dimensionless expansion parameter is extraordinarily small (xc 8. 7 10^-31), mathematically guaranteeing the conformal exponential operates deep within the perturbatively controlled linear regime. Leading-Order Thin-Shell Reduction: Derives the terrestrial thin-shell factor using a clearly labeled leading-order approximation, stating all geometric and kinematic assumptions explicitly: static spherical symmetry, constant interior density, thin-shell regime (R_/R_ 1), and long exterior Compton wavelength. Monotonic Exterior-Density Bound: Proves the scalar field minimum is monotonically decreasing with ambient density. This replaces uncontrolled, mathematically unclosed atmospheric profile patches with a rigorous upper bound on Earth's screening factor (_ 2. 1 10^-7) evaluated safely at the cosmological vacuum limit. The Universal-Coupling Null Signal: Decisively proves that if Platinum and Titanium test masses are both unscreened and couple through the same universal dimensionless coupling, the MICROSCOPE differential observable strictly vanishes at leading order (₌₈₂ = 0). We formally demonstrate that generating a non-zero differential signal necessitates either extending the matter sector beyond the universal-coupling action or computing residual finite-size composition dependencies not contained in standard macroscopic scalar-tensor treatments. Files Included: main. pdf / main. tex: The definitive PRD-compliant manuscript executing the exact vs. leading-order separation. chameleonᵤniversalcouplingᵥ4. py: Open-source Python toolkit (v4. 0. 0) generating exact boundary matching plots, monotonic exterior-density upper bounds, perturbative control scales, and the universal-coupling null differential signal.