= Logarithmic overlap creates a switchback term
If a stretched variable is $x=\epsilon r$ and an <outer expansion> contains $\epsilon^2\log x$, its expression in the fixed-$r$ region is $\epsilon^2\log r+\epsilon^2\log\epsilon$. The parameter logarithm must therefore be present in the <inner expansion>, often multiplying a homogeneous solution selected by a boundary condition. An ansatz containing only integer powers with parameter-independent coefficients misses this <switchback term>. The same reasoning applies to other powers of $\epsilon$ and other stretched-coordinate scalings.
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