Algebraic Bounds on Perturbative Control and Solver Stiffness in Scalar–Tensor Screening Mechanisms

We examine perturbative control in chameleon-type scalar–tensor screening models and derive explicit algebraic bounds linking the conformal-coupling expansion parameter to macroscopic thin-shell observables. Starting from the scalar–tensor action we derive the static field equation and obtain the exact stationary condition for the effective potential minimum, including a Lambert W closed-form solution. The perturbative breakdown scale is connected to a detectability threshold derived from thin-shell suppressed fifth-force observables. The resulting master inequality provides a single algebraic condition relating the scalar potential scale, matter densities, gravitational potential of the source body, and experimental sensitivity. We also discuss numerical conditioning in discretized boundary-value solvers, clarifying which solver-complexity claims apply only to multidimensional discretizations rather than strictly one-dimensional radial reductions.