Curvature-Induced Localization Towers

We investigate a non-minimally coupled scalar field in a Randall-Sundrum background and study the interplay between curvature-induced effective mass shifts and gravitational backreaction. Working within a weak-backreaction expansion, we derive explicit higher-order corrections to both the scalar profile and the warped geometry. The analysis reveals a recursive structure in which curvature corrections generate successive localization contributions. Starting from a leading exponential scalar profile, the coupled Einstein-scalar system produces a hierarchy of higher-order terms. We explicitly compute the first nontrivial levels of this hierarchy and demonstrate a formal recursive closure of the perturbative tower. We further show that the resulting bulk branch admits compatible endpoint data on a finite Randall-Sundrum interval through generalized non-minimal junction conditions. The construction establishes the existence of a mathematically consistent perturbative localization tower within the Einstein-scalar system. The broader curvature-bootstrap viewpoint is discussed as an interpretive framework that organizes the feedback relation between curvature, effective mass, localization, and backreaction. The existence and stability of nontrivial fixed-point solutions remain open problems.