Master equation for screening in luminal Horndeski gravity

Determining the active screening mechanism from a general scalar-tensor Lagrangian remains a challenging problem. As a diagnostic tool, we present a systematic study of nonlinear cosmological perturbations in luminal Horndeski theories. Working in the 𝛼 basis on a flat Friedmann-Lemaître-Robertson-Walker background, we derive and organize the full set of unapproximated second-order perturbation equations and systematically apply the quasistatic and weak-field limits. We find that second-order effects modify only the scalar-field equation. We derive, for static and spherically symmetric configurations, a master screening equation that classifies the nonlinear operators driving screening, recovering the Vainshtein mechanism and the onset of chameleon screening. We also identify a novel candidate regime, which we term Phaedrus screening, characterized by a screening radius that scales linearly with the source mass. For each mechanism, we derive analytical and numerical solutions and clarify the conditions under which they activate. Two new publicly available software packages are introduced: (i) xAlpha, a Mathematica package to compute and organize perturbation equations in scalar-tensor theories, and (ii) escut, a python module to solve the nonlinear scalar equation. In many cases, these tools enable the identification of the active screening type directly from a luminal Horndeski Lagrangian.