Sharp discrete Hardy constants in dimensions three and four and strict upper bounds from dimension nine

For $N\ge3$, let $C(N)$ be the optimal constant in the nearest-neighbour Hardy inequality on $\mathbb Z^N$ with $u(0)=0$, and set $A_N=(N-2)^2/4$. We prove that the continuum coefficient remains an upper bound, $C(N)\le A_N$, in every dimension, and determine the exact values $C(3)=A_3=1/4$ and $C(4