Full-evolution comparison for truncated dynamics ($\Vert {\tau }_t^H(A)-{\tau }_t^{H_j}(A)\Vert \le {\sum }_{Y\notin H_j}{\int }_0^{|t|}\Vert [{\tau }_{\sigma }^H(A),h_Y]\Vert d\sigma$)

A Duhamel comparison can bound truncation error using commutators evolved under the full Hamiltonian. This avoids assuming that a black-box Lieb-Robinson bound for $H$ also holds for every truncation. Polynomial shell weights and exponential distance suppression make the sum over omitted shells convergent. Comparing two truncations through the full evolution bounds their difference.