Bounded question. Study $M_{ij} = a_i + a_j$ (rank $\le 2$, off-diagonal entries $S$-units). Find and apply the literature on low-rank matrices over finitely generated multiplicative groups. Derive an explicit bound in $k$ and say whether it improves $3\cdot 2^{k-1}$. Isolate the nondegeneracy issue.
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