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

For $N\ge 3$, let $C(N)$ be the optimal constant in the nearest-neighbour Hardy inequality on ${\mathbb{Z}}^N$ with $u(0)=0$, and set $A_N=(N-2{)}^2\mathrm{/}4$. We prove that the continuum coefficient remains an upper bound, $C(N)\le A_N$, in every dimension, and determine the exact values $C(3)=A_3=1\mathrm{/}4$ and $C(4)=A_4=1$. In higher dimensions we show $C(N)<A_N$ for $N=9,10$ and obtain the explicit bound $$C(N)\le 3N-\sqrt{N^2+8N-8}<2N(N\ge 3),$$ which lies below $A_N$ from dimension eleven onward. The low-dimensional equalities follow from shifted radial supersolutions, angular convexity, and a discrete ground-state representation. We also show that this shifted-power mechanism cannot work at the continuum coefficient from dimension five onward. Dimension nine is treated by a Gaussian Rayleigh–Ritz construction combined with exact Jacobi theta-function estimates, while the higher-dimensional bounds are obtained through finite-dimensional orbit compressions. Finally, we derive positive spatial remainders in dimensions three and four and a spectral consequence for the associated discrete Schrödinger operators in the high-dimensional regime.