as the value of $m_\mu(x)$ can be thought of as a “distance" from the point $x$ to the plane $\partial {\mathcal{H}}_e^+$. Rather than prove that $m_\mu$ stays a bounded distance from an affine function, it turns out to be sufficient for homogenization to show roughly that, for some exponent $\alpha>0$ and a deterministic constant $\overline m_\mu(e)$, we have $$\label{}
\sup_{ x\in B_r\cap {\mathcal{H}}_e^+} \frac1r \left| m_\mu (x) - \overline m_\mu(e) (x\cdot e) \right| = O\left( r^{-\alpha} \right) \quad \mbox{as} \ r \to \infty \quad \mbox{with high probability.}$$ In other words, we may soften the requirement that $m_\mu$ be a bounded distance from a plane by allowing the permitted distance to depend on the distance from the boundary plane $\partial {\mathcal{H}}_e^+$ and by only checking points in a bounded set. The heart of the paper is the proof of this estimate, which is stated precisely in Proposition \[p.planarquantitative\] and proved in Sections \[s.fluctuations\] and \[s.means\]. The argument is naturally split into two steps: first we show by a concentration argument, exploiting the finite range dependence of the coefficients, that the random fluctuations of $m_\mu(x)$ are at most of order $O((x\cdot e)^{\frac12})$. Then we argue that the means ${\mathbb{E}}\left[ m_\mu(x) \right]$, which 