Discrete Unique Continuation on Simplex

For integers $N\ge0$ and $n\ge2$, let \[ Δ_N^{(n)} =\left\{α\in\mathbb Z_{\ge 0}^n: α_1+\cdots+α_n=N\right\}. \] We study discrete unique continuation for functions on this lattice simplex. For an integer $R\ge1$, let $g:Δ_{nR}^{(n)}\to\mathbb R$ satisfy \[ \sum_{i=1}^n g(β+e_i)=0, \qquad β\inΔ_{nR-