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Apply Haar twirling conditional expectation to the complement of a neighbourhood around the initial operator support. A Lieb-Robinson bound on commutators with arbitrary complement-supported unitaries then bounds the approximation error without summing over sites. For a bound proportional to $e^{-\mu d}(e^{2kst}-1)$, choosing $v=4ks/\mu$ gives an error at most $\mu vt|X|\Vert A_X\Vert e^{-\mu l/2}$ outside radius $vt+l$.

Let $R=X(l)$ and $C$ be its complement, with Hilbert-space dimension $d_C$. Use Haar twirling conditional expectation to defineThis is an operator supported on $R$. The normalization $1/d_C$ is essential: the partial trace alone would not fix an operator already supported on $R$.

Subtract the integrand from $A_X(t)$ and use the operator norm triangle inequality. Since $A-UA{U}^{†}=[A,U]U^{†}$ and every unitary operator has norm one, the assumed Lieb-Robinson bound, applied to a unitary supported on the entire complement, givesThe factor $\min (|X|,|C|)$ was bounded by $|X|$; there is no sum over sites and hence no volume-dependent prefactor.

ChooseFor $t,l\ge 0$, $e^{2kst}-1\le 2kst{e}^{2kst}$, and thereforeThus $A_{X(l)}(t)=A_R(t)$ is the desired Lieb-Robinson localization by Haar twirling. The displayed choice assumes the usual positive constants $\mu ,s$; if the interaction bound has $s=0$, the error is identically zero and any positive $v$ works. For negative times the same argument uses $|t|$.