Dyadic valuation and scale colouring ($\chi (n)=(v_2(n)\mathrm{mod}2,\lfloor {\log }_{2}n\rfloor \mathrm{mod}2)$)

Colour $n>0$ by $(v_2(n)\mathrm{mod}2,\lfloor {\log }_{2}n\rfloor \mathrm{mod}2)$. 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.