Discrete Unique Continuation on Simplex

Abstract

For integers $N\ge0$ and $n\ge2$, let \[ \Delta_N^{(n)} =\left\{\alpha\in\mathbb Z_{\ge 0}^n: \alpha_1+\cdots+\alpha_n=N\right\}. \] We formulate a discrete unique-continuation problem on this lattice simplex. Given an integer $R\ge1$, consider a function $g:\Delta_{nR}^{(n)}\to\mathbb R$ satisfying the complete oriented-simplex relations \[ \sum_{i=1}^n g(\beta+e_i)=0, \qquad \beta\in\Delta_{nR-1}^{(n)}, \] where $e_i$ is the $i$th standard basis vector. We prove that a nonzero value at the balanced point forces the support-cardinality estimate with optimal growth exponent: if $g(R,\ldots,R)\neq 0$, then $|\operatorname{supp}(g)|\ge c_n R^{\lceil n/2\rceil}$. Here $c_n>0$ depends only on $n$. The key input is a \emph{Pascal uncertainty principle}. After factorial normalization, the simplex relations become a single directional differential equation. A nonzero balanced coefficient then produces a monomial whose relevant facet-chart exponents are all large, while the tensorized Pascal uncertainty principle prevents the coefficient supports in all partially shifted affine charts from being simultaneously sparse. Comparing those charts with two coordinate facets and summing over disjoint derivative shells gives the lower bound. Explicit constructions show that the exponent $\lceil n/2\rceil$ is optimal. The proof was obtained through human-guided discovery and exploration with the assistance of GPT-5.6 Sol.