Spatial-Order Hierarchy of Time-Dependent Exchange-Correlation Potential

Exact time-dependent density-functional theory separates the exchange-correlation potential into interaction and kinetic-correlation components, but the structural relation between them remains unknown. We establish a representability constraint based on the off-diagonal expansion of the one-electron reduced density matrix relative to a time-dependent Hartree-Fock (TDHF) reference. A non-zero linear term generates a non-HF current density while the kinetic-correlation component remains HF representable; higher-order off-diagonal structure activates the kinetic component. In a one-dimensional two-electron correlation quench, retaining the exact interaction component while setting the kinetic component to zero suppresses the density broadening of the exact evolution. These results establish a hierarchy of density equations of motion and provide an exact constraint for non-adiabatic functional construction.