= Finite separating-functional proof of the columns condition
After clearing denominators in a <matrix>, for every disjoint pair of index sets $I,B$ with $B$ nonempty and $\sum_{i\in B}a_i$ outside the <linear span> of the $I$-columns, choose an integer <linear functional> vanishing on those columns but not on the indicated sum. Choose a <prime number> dividing none of the finitely many nonzero evaluations. The <last nonzero digit coloring> for this prime, applied to a <monochromatic> positive solution, groups coordinates into blocks by <P-adic valuation>. Applying the appropriate functional and reducing modulo the prime shows that each block sum lies in the <linear span> of all earlier columns. This proves necessity in <Rado's theorem> without an asymptotic limiting argument.
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