Assumption on weights in threshold circuits
Given $w_1,\ldots,w_n,t$, let $X$ be the set of assignments such that $\sum_i w_i x_i \geq t$, and consider the linear program with variables $z_1,\ldots,z_n$ and the $2^n$ constraints
$$
\begin{align*}
\sum_i z_i x_i \geq +1 &amp; (x_1,\ldots,x_n) \in X \\
\sum_i z_i x_i \leq -1 &amp; (x_1,\ldots,x_n) \notin X
\end{align*}
$$
This linear program (with a constant objective function) is feasible (since $z_i = w_i/t$ is a solution), and so has a solution $z_1,\ldots,z_n$ which is a vertex. As such, it is the solution of a system of $n$ equations of the form
$$
\sum_i z_i x_i = \pm 1.
$$
Cramer's rule gives each $z_i$ as a ratio of two $n \times n$ determinants of matrices of $0,\pm 1$ entries; in fact the denominator is the same. Each numerator $N_i$ and the common denominator $D$ is at most $n!$ in magnitude, so multiplying everything by $D$, we get an integral solution $N_1,\ldots,N_n,D$, where all the quantities are at most $n! \leq n^n = 2^{n\log n}$ in magnitude.