Scalable methods for networked multi-agent reinforcement learning let each agent plan using only a small neighborhood of the agent graph. This works only when the system is value-local, meaning a perturbation at one agent affects the long-run value at another agent weakly when the two are far apart. In the average-reward setting, the standard way to certify locality is the Dobrushin row-sum bound on a single matrix $C^π$ that captures how each agent's next state depends on each other agent's current state. To make this matrix easy to work with, prior work bounds it by a supremum over joint actions. The resulting bound is independent of the policy, but it is loose whenever the policy never picks the worst-case action. We split $C^π$ into pieces that separately track environment sensitivity and policy sensitivity, $C^π\preceq E^{\mathrm s}+E^{\mathrm a}Π(π)$, where $E^{\mathrm s}$ measures how the next state moves with the current state, $E^{\mathrm a}$ measures how it moves with the current action, and $Π(π)$ measures how reactive the policy is to changes in state. The spectral radius of $H^π:= E^{\mathrm s}+E^{\mathrm a}Π(π)$ then controls the decay of the average-reward Poisson solution, and the spectral certificate $ρ(H^π)<1$ is strictly weaker than the row-sum condition $\|H^π\|_\infty<1$ on the same matrix and applies in regimes where policy-independent action-supremum bounds used in prior Dobrushin-style work cannot. For temperature-$τ$ softmax policies we get $Π(π)\le L/(2τ)$, so the softmax temperature directly controls locality. We use this decay result to give a deterministic oracle guarantee for a block-coordinate KL-proximal policy-improvement template whose truncation bias decays exponentially in the message-passing radius $κ$.