= Solution
Expand the two central second-difference operators. The given stencil identity yields
$$
\Gamma_9u=h^2\Delta u+\frac{h^4}{12}\Delta^2u+h^6\left[\frac{u_{xxxxxx}+u_{yyyyyy}}{360}+\frac{u_{xxxxyy}+u_{xxyyyy}}{72}\right]+O(h^8).
$$
Consequently its unnormalized local residual is $\Gamma_9u-h^2f=(h^4/12)\Delta f+O(h^6)$. \b[The normalized Laplacian approximation has leading error $(h^2/12)\Delta f$ and is second order.] These Taylor statements require the indicated smooth <derivatives>; boundary accuracy is a separate matter.
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