Einstein Dynamics from Operator-Level Vacuum Correlation Closure

This preprint presents an operator-level derivation of Einstein dynamics within a local metric effective field theory universally coupled to stress–energy. The argument does not invoke entanglement entropy, thermodynamic extremality, or loop-induced curvature terms. Instead, it proceeds from the universal ultraviolet singularity structure of Lorentz-invariant two-point functions, the linear response of quantum observables under metric perturbations, and the Raychaudhuri focusing equation for null congruences. We show that consistency between microscopic correlation response and geometric area variation requires equality of null–null projections of the Ricci and stress–energy tensors at leading order in a derivative expansion. Stress–energy conservation and the contracted Bianchi identity then yield the Einstein field equation up to a cosmological constant and higher-curvature corrections. The gravitational coupling scales with ultraviolet correlation density and the effective number of microscopic degrees of freedom. The result is interpreted as a consistency theorem within effective field theory. This work is presented as a standalone theoretical analysis and is intended for journal submission.