= Dyadic valuation and scale colouring
{title2=$\chi(n)=(v_2(n)\bmod2,\lfloor\log_2 n\rfloor\bmod2)$}
Colour $n>0$ by $(v_2(n)\bmod2,\lfloor\log_2n\rfloor\bmod2)$. No increasing infinite sequence can make all its sums $x_i+x_j$ and $x_i+2x_j$, $i<j$, one colour. If its <2-adic valuations> are bounded, two terms of equal valuation and equal odd part modulo four give pair sums of different valuation parity. If they are unbounded, a later term divisible by a sufficiently large power of two prevents adding a fixed earlier term from crossing either dyadic boundary; the two resulting sums have adjacent logarithmic scale indices.
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