Lemma 2 — the optimum of $\min x^{\mathsf T}Dx$ over $[0,1]^m$ is $0$ or at most $-2^{-L}$

Let $D$ be an integer symmetric matrix of order $m$, $Q(x) = x^{\mathsf T}Dx$, and let $L$ be the encoding size of $D$. Consider the QP (8):

Its optimal value is either $0$ or at most $-2^{-L}$. Equivalently, either

or there is an $x \in [0,1]^m$ with

The lemma is the precision bound of the reduction: a quadratic form with integer coefficients that dips below zero on the unit box does so by an amount with only polynomially many bits, which is what lets the reduction subtract a tiny $\varepsilon$ without changing the answer.

Formalization Note The optimum of (8) exists (the box is compact) and is at most $Q(0) = 0$, so "the optimum is $0$ or $\le -2^{-L}$" is stated as the disjunction above, without sInf. $L$ is Schrijver's encoding size (encSize), since the paper does not define "size". $D$ is taken symmetric: the paper's §4 says "as before, let $D$ be an integer square symmetric matrix", and the LCP (9) used in the proof is the optimality system of (8) only for symmetric $D$.