= High-speed passage expansion of a localized trap
{title2=$\Delta x=\sum_{m\geq1}v_T^{-m}\int(F/\zeta)^m dy$}
If the maximum force-induced speed is smaller in magnitude than $v_T$, expand the passage denominator as a uniformly convergent <geometric series>. For a localized conservative well with equal outside potential levels, the first force moment vanishes and displacement starts at order $v_T^{-2}$. An odd force has no cubic moment, giving the next correction at order $v_T^{-4}$. Compact force support alone does not guarantee a vanishing first moment or positive displacement.
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