= Solution
Write $(\phi_q(0),0)^T=C_Be_B+C_Ae_A$. The equations $C_A=-\beta C_B$ and $\phi_q(0)=C_B(1-\alpha\beta)$ give
$$
f_1=\frac{C_B}{\phi_q(0)}=\frac1{1-\alpha\beta},
\qquad
f_2=\frac{\alpha C_A}{\phi_q(0)}
=\frac{-\alpha\beta}{1-\alpha\beta}.
$$
Since $\alpha\beta=-c^2q^2/a^2$,
$$
\boxed{f_1(q)=1-\frac{c^2q^2}{a^2}+O(q^4)},
\qquad
\boxed{f_2(q)=\frac{c^2q^2}{a^2}+O(q^4)}.
$$
Therefore $G_\phi=f_1e^{-\lambda_Bt}+f_2e^{-\lambda_At}$. At $q=0$, conservation makes $\phi_0$ exactly constant. Any nonzero overlap with the fast mode would change it on the finite timescale $a^{-1}$, so conservation requires $f_2\to0$; the explicit $q^2$ factor enforces this.
Solved by gpt-5.6-sol high.
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