Curvature-Activated Extensions of Einstein Gravity: Nonlinear Dynamics, Perturbations, and Cosmology

We present a curvature-activated effective extension of Einstein gravity where nonlinear geometric corrections regulate high-curvature regimes while preserving a consistent General Relativity (GR) limit. Formulated within a covariant effective field theory framework, the model introduces a scalar sector coupled to a bounded activation function $F(R)$, ensuring that additional gravitational dynamics decouple in low-curvature environments. We derive the complete gravitational field equations from the action principle and establish mathematical consistency through the contracted Bianchi identities.A rigorous 3+1 ADM decomposition is performed, and the Hamiltonian constraint structure is analyzed to confirm the closure of the constraint algebra and the stability of the dynamical degrees of freedom. We explicitly derive the linear perturbation equations for scalar, vector, and tensor sectors, demonstrating stable propagation of gravitational waves on cosmological backgrounds. Furthermore, we examine black-hole interior geometries, showing how the activation mechanism regulates the growth of curvature invariants to potentially resolve classical singularities. Our results suggest that curvature-activated dynamics provide a robust, ghost-free framework for addressing the high-curvature limits of Einstein gravity within an effective field theory paradigm.