Curvature Saturation in Scalar–Gauss–Bonnet Gravity: Complete Variational Derivation, Constant-Curvature Branches, Stability Analysis, and Effective Field Theory Regime

We present a complete derivational construction of a four-dimensional scalar–Gauss–Bonnet gravity theory exhibiting curvature saturation through a constant-curvature branch. All metric and scalar variations are performed explicitly, and the Lovelock structure guarantees second-order field equations. Linearization around Minkowski space yields two tensor polarizations and one massive scalar mode without higher-derivative instabilities. For maximally symmetric geometries, the Gauss–Bonnet invariant reduces to �, leading to a closed algebraic system governing constant-curvature solutions. Stability conditions, hyperbolicity of the field equations, effective field theory validity bounds, and weak-field consistency are derived in detail. The construction is interpreted as a controlled effective extension of general relativity and does not claim ultraviolet completion.