Weakly Nonlocal Curvature-Screened Quadratic Gravity: Ghost Suppression, Complete Variational Structure, and Quantum-Geometric Flow

We formulate a weakly nonlocal extension of quadratic gravity in which ultraviolet gravitational propagation is dynamically suppressed through a curvature-dependent geometric screening function and an exponential nonlocal form factor. The theory is constructed from a generally covariant variational principle and analyzed at the levels of metric variation, propagator structure, ghost suppression, ultraviolet consistency, effective dispersion relations, Bianchi identities, and geometric renormalization flow. Unlike ordinary local quadratic gravity, the present framework replaces polynomial higher-derivative operators by entire nonlocal operators, thereby avoiding additional propagator poles while preserving infrared Einstein gravity.