= Brauer configuration for a one-row zero-sum equation
Suppose nonzero integer coefficients have a nonempty zero-sum subset $I$, and fix $i_0\in I$. Put $s=|a_{i_0}|$ and $C=\sum_{j\notin I}a_j$. A <Brauer progression theorem> configuration containing a sufficiently long <monochromatic> progression of step $d$ and the point $sd$ gives a positive solution: take every outside variable equal to $sd$, and the inside variables from the progression with offsets $\lambda_i$, where $\lambda_{i_0}=-\operatorname{sgn}(a_{i_0})C$ and the other offsets are zero. Shifting all inside offsets by the same integer keeps their weighted sum unchanged, because $\sum_{i\in I}a_i=0$. This removes negative offsets while preserving the equation.
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