= Cubic endpoint-to-saddle transition
{title2=$\delta=O(\nu^{-2/3})$}
For a <Laplace integral> with local phase $\phi(t;\delta)=\delta t+bt^3/3+\cdots$, $b>0$, an interior minimum approaches the endpoint as $\delta\uparrow0$. The <distinguished limit> is $t=O(\nu^{-1/3})$, $\delta=O(\nu^{-2/3})$. With $s=(\nu b)^{1/3}t$, its leading local integral is
$$
(\nu b)^{-1/3}I(-\delta\nu^{2/3}b^{-1/3}),\qquad I(x)=\int_0^\infty e^{xs-s^3/3}\,ds.
$$
The <cubic Laplace transition integral> joins the ordinary endpoint estimate for positive $\delta$, the cubic endpoint estimate at zero, and the interior <Laplace's method> estimate for negative $\delta$. Separate fixed-parameter formulas fail to be uniform when the minimum is within its own width of the endpoint.
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