High-speed passage expansion of a localized trap ($\Delta x={\sum }_{m\ge 1}{v}_T^{-m}\int (F/\zeta {)}^{m}dy$)

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.