  Let $\dlap$ be the discrete Laplace operator acting on functions (or rational
matrices) $f:\mathbf{Q}_L\to\mathbb{Q}$, where $\mathbf{Q}_L$ is the two
dimensional lattice of size $L$ embedded in $\mathbb{Z}_2$. Consider a rational
$L\times L$ matrix $\mathcal{H}$, whose inner entries $\mathcal{H}_{ij}$
satisfy $\dlap\mathcal{H}_{ij}=0$. The matrix $\mathcal{H}$ is thus the
classical finite difference five-points approximation of the Laplace operator
in two variables. We give a constructive proof that $\mathcal{H}$ is the
restriction to $\mathbf{Q}_L$ of a discrete harmonic polynomial in two
variables for any $L>2$. This result proves a conjecture formulated in the
context of deterministic fixed-energy sandpile models in statistical mechanics.
