Four-neighbour mean expansion (${\mathrm{mean}}_{h}u=u+\frac{1}{4}{h}^2\Delta u+O(h^4)$)

Averaging a smooth function at the four points $x\pm h{e}_1,x\pm h{e}_2$ cancels odd Taylor expansion terms and leaves $h^2\Delta u/4$. The fourth-order correction is $h^4(u_{1111}+u_{2222})/48$. Equality to the center value at every sufficiently small radius therefore forces $\Delta u=0$ at that point, but does not establish harmonicity throughout a neighbourhood.