= Solution
\b[The two versions of <partition regularity> are equivalent.] If the row is <partition regular> on all positive integers, pull a colouring of the even positive integers back by $n\mapsto2n$. A <monochromatic> solution $\mathbf y$ for this pulled-back colouring gives the even solution $\mathbf x=2\mathbf y$ in the original colouring, since the equation is homogeneous.
Conversely, restrict any colouring of all positive integers to its even members. A <monochromatic> solution supplied by regularity on the even integers is already a valid solution in the full domain. This argument is an instance of <scaling invariance of homogeneous partition regularity> and does not require a new coefficient criterion.
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