= Translation-covariant phase expansion
{title2=$\mathcal D_Y=\partial_Y+\phi_Y\partial_\xi$}
Let $p=\phi_Y$, $r=\phi_{YY}$ and $w_2=C_1(\xi)r+C_2(\xi)p^2$. Two applications of the <chain rule> give $\mathcal D_Y^2w_2=C_1\phi_{YYYY}+2(C_1'+C_2)p\phi_{YYY}+(C_1'+2C_2)r^2+(C_1''+5C_2')p^2r+C_2''p^4$. In a <Fredholm solvability condition>, the fourth term generally contributes to the coefficient of $\phi_Y^2\phi_{YY}$. It cannot be omitted merely because the fast-coordinate correction functions have been denoted by $C_j(x)$.
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