The integer point enumerator of one irrational translate of P is a complete invariant

For a full-dimensional rational polytope $P\subset {\mathbb{R}}^d$ and a real dilation parameter $t>0$, the integer point enumerator is defined by $L_P(t):=\mathrm{\mid }tP\cap {\mathbb{Z}}^d\mathrm{\mid }$. We determine exactly which translation vectors $\mathbf{y}=(y_1,\dots ,y_d)\in {\mathbb{R}}^d$ have the property that the single translated counting function $t⟼L_{P+\mathbf{y}}(t)$, with $t\in {\mathbb{Q}}_{>0}$, uniquely determines $P$ among all full-dimensional rational polytopes in ${\mathbb{R}}^d$. The necessary and sufficient condition is that $1,y_1,\dots ,y_d$ be linearly independent over $\mathbb{Q}$. In particular, we may use the explicit algebraic vector ${\mathbf{y}}^{\ast }:=(2^{1\mathrm{/}(d+1)},2^{2\mathrm{/}(d+1)},\dots ,2^{d\mathrm{/}(d+1)})$ in every dimension $d$. The sufficiency proof recovers the primitive facet inequalities from isolated discontinuities of the counting function, while necessity follows from an affine-unimodular obstruction.