= Solution
The three fixed-$a$ estimates below cease to be uniform as the minimum of the phase approaches the endpoint. For the <cubic endpoint-to-saddle transition>, write
$$
a=1+l\nu^{-2/3},\qquad t=2^{1/3}\nu^{-1/3}s.
$$
The <Taylor series> of the <hyperbolic sine> gives, for bounded $s$ and fixed $l$,
$$
\nu(a\sinh t-t)=2^{1/3}ls+\frac{s^3}3+O(\nu^{-2/3}).
$$
The cubic term controls the tail, so localization of the <Laplace integral> gives the uniform leading formula
$$
\boxed{A_\nu(\nu+l\nu^{1/3})\sim2^{1/3}\nu^{-1/3}I(-2^{1/3}l),\qquad I(x)=\int_0^\infty e^{xs-s^3/3}\,ds.}
$$
Here $I$ is the <cubic Laplace transition integral>, equal to $\pi\operatorname{Hi}$ in terms of the <Scorer Hi function>.
To recover the endpoint regime, let $l\to+\infty$. Scale $s=v/(2^{1/3}l)$ in $I$; the cubic term becomes negligible and $I(-2^{1/3}l)\sim(2^{1/3}l)^{-1}$. Therefore
$$
A_\nu\sim\frac{\nu^{-1/3}}l=\frac1{\nu(a-1)},
$$
which agrees with part (i) in the overlap $1\ll l\ll\nu^{2/3}$. At $l=0$, the <Gamma integral> gives $I(0)=3^{-2/3}\Gamma(1/3)$, recovering part (iii).
For $l\to-\infty$, set $M=-2^{1/3}l>0$. The exponent $Ms-s^3/3$ has its maximum at $s=\sqrt M$, with second <derivative> $-2\sqrt M$. Thus <Laplace's method> gives
$$
I(M)\sim\sqrt\pi M^{-1/4}e^{2M^{3/2}/3},
$$
and the transition formula becomes
$$
A_\nu\sim2^{1/4}\sqrt\pi\nu^{-1/3}(-l)^{-1/4}\exp\left[\frac{2\sqrt2}3(-l)^{3/2}\right].
$$
For $a=1-\eta$ with $\eta\downarrow0$, part (ii) has
$$
\operatorname{arcosh}(1/a)-\sqrt{1-a^2}=\frac{2\sqrt2}3\eta^{3/2}+O(\eta^{5/2}),\qquad (1-a^2)^{-1/4}\sim(2\eta)^{-1/4}.
$$
Putting $\eta=(-l)\nu^{-2/3}$ reproduces both the exponential and its prefactor. For relative agreement of these leading exponentials, one may use the overlap $1\ll-l\ll\nu^{4/15}$, which makes $\nu\eta^{5/2}\to0$. Thus the same transition integral connects all three regimes.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-336-cubic-transition.png]
{title=The cubic endpoint-to-saddle transition}
{description=Direct numerical integration of the original phase approaches the same cubic transition function as the large parameter increases. Negative transition parameter places the minimum inside the interval; positive parameter leaves an ordinary endpoint minimum.}
{height=430}
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