= Endpoint asymptotics after odd Gaussian cancellation
If an odd amplitude multiplies an even Gaussian phase on an interval asymmetric about its interior minimum, the symmetric central integral cancels exactly. The surviving leading term can be exponentially small and comes from the nearer unmatched endpoint. For example,
$$
\int_{-2}^1\sin t\,e^{-\lambda t^2}\,dt
=-\int_1^2\sin t\,e^{-\lambda t^2}\,dt
\sim-\frac{\sin1}{2\lambda}e^{-\lambda}.
$$
An interior <Laplace method> expansion alone yields zero to every algebraic order and does not capture this term.
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