Let \dlap be the discrete Laplace operator acting on functions(or rational matrices) f:QL→Q,where QL is the two dimensional lattice of size Lembedded in Z2. Consider a rational L×L matrix H, whose inner entries Hij satisfy \dlap\mathcal{H}_{ij}=0. The matrix H is thus theclassical finite difference five-points approximation of theLaplace operator in two variables. We give a constructive proofthat H is the restriction to QL of adiscrete harmonic polynomial in two variables for any L>2. Thisresult proves a conjecture formulated in the context ofdeterministic fixed-energy sandpile models in statisticalmechanics