= Solution
The real <Airy function> asymptotics used for the <Airy turning-point connection formula> are, for $z,s\to+\infty$,
$$
\begin{aligned}
\operatorname{Ai}(z)&\sim\frac{z^{-1/4}}{2\sqrt\pi}e^{-2z^{3/2}/3},&
\operatorname{Bi}(z)&\sim\frac{z^{-1/4}}{\sqrt\pi}e^{2z^{3/2}/3},\\
\operatorname{Ai}(-s)&\sim\frac{s^{-1/4}}{\sqrt\pi}\sin(2s^{3/2}/3+\pi/4),&
\operatorname{Bi}(-s)&\sim\frac{s^{-1/4}}{\sqrt\pi}\cos(2s^{3/2}/3+\pi/4).
\end{aligned}
$$
These real-axis asymptotics fix both normalization and phase.<Airy function asymptotics in the Digital Library of Mathematical Functions>[Equations 9.7.5, 9.7.7, 9.7.9 and 9.7.11; leading terms for positive real argument]{ref} Near $a$, $I\sim2z^{3/2}/3$, $\Phi\sim2(-z)^{3/2}/3$, and $\kappa^{-1/2}\sim\ell^{1/2}z^{-1/4}$. Matching to the constants of <2/c> gives $C_A=2\sqrt\pi\ell^{1/2}D$ and $C_B=\sqrt\pi\ell^{1/2}G$. Therefore
$$
\boxed{\frac{D e^{-I}+G e^I}{\sqrt\kappa}
\quad\longleftrightarrow\quad
\frac{2D\cos(\Phi-\pi/4)-G\sin(\Phi-\pi/4)}{\sqrt{k_1}}.}
$$
In particular, the decaying component matches the cosine with the requested $-\pi/4$ shift, with a factor \b[two] in its oscillatory coefficient. The growing component matches a sine with the indicated minus sign. Expanding into travelling <waves> gives
$$
\boxed{A_{\rm in}=D-\frac{iG}{2},\qquad A_{\rm r}=D+\frac{iG}{2}.}
$$
The factor two is essential when converting the real oscillatory components to incident and reflected travelling <waves>.
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