Failure of integer-degree harmonic dimension comparison

Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ³ has at least $(k+2)^2$ independent harmonic functions of growth at most k, exceeding the Euclidean count $(k+1)^2$. The metric may depend on k.