Spectral Geometry of Fourth-Order Scalar Fields: Energy Bounds, Interaction Kernels, and Stability

This paper introduces a spectral framework for curvature localization based on a fourth-order elliptic operator. An associated energy functional is shown to be bounded below, and stability is characterized by a sharp spectral threshold analogous to the Breitenlohner–Freedman bound. Localized curvature structures are modeled as eigenmode superpositions whose interactions are governed by a Green’s function kernel admitting a double-channel screened decomposition. The system admits angular momentum sector decomposition under symmetry, enabling sector-wise stability analysis and the definition of spectral invariants. A causal constraint on spectral mixing is derived, and a correspondence with AdS-type curvature dynamics is outlined. The framework provides a unified operator-based description of curvature interaction and stability.