Almost-exponential locality of filtered Hamiltonian terms ($\Vert A^{(Z,d)}\Vert \le O(\Vert h_Z\Vert d^{\alpha +1}{e}^{-cd/(\log (cd){)}^2})$)

In a telescoping local-shell decomposition, split the spectral filter integral at a time proportional to the shell distance. A Lieb-Robinson bound controls short times, while the two-sided almost-exponential spectral filter tail controls long times. Polynomial shell-weight growth then gives the displayed decay. The estimate is asymptotic at large distance; small shells retain the elementary operator-norm bound. The evenization of a nonnegative bandlimited filter can supply the two-sided tail from a one-sided existence statement.