= Solution
For the <second-harmonic feedback in a long-wave convection amplitude equation>, at $\mu=2$ the linear steady operator is $L_0=-(1+\partial_x^2)^2$. Its <eigenvalue> on $e^{inx}$ is $-(1-n^2)^2$, so the critical <Fourier modes> are $n=\pm1$. More generally the linear <dispersion relation> is $\lambda(k)=-1+\mu k^2-k^4$, giving first onset at $\mu=2,k=1$.
At order $\epsilon^2$, the <weakly nonlinear expansion> gives
$$
L_0\Theta_2=s(\Theta_{1x}^2)_x.
$$
With $\Theta_1=Ae^{ix}+\overline A e^{-ix}$,
$$
\Theta_{1x}^2=-A^2e^{2ix}-\overline A^{\,2}e^{-2ix}+2|A|^2.
$$
The constant part disappears under differentiation. Inverting $L_0$ on the second harmonic, whose <eigenvalue> is $-9$, gives
$$
\boxed{\Theta_2=\frac{2is}{9}A^2e^{2ix}+\mathrm{c.c.}}
$$
up to a critical-harmonic correction absorbed into the definition of $A$. At order $\epsilon^3$,
$$
0=L_0\Theta_3-\mu_2\Theta_{1xx}
-2s(\Theta_{1x}\Theta_{2x})_x+(\Theta_{1x}^3)_x.
$$
The coefficient of $e^{ix}$ in the last three terms is respectively
$$
\mu_2A,\qquad \frac{8s^2}{9}|A|^2A,\qquad -3|A|^2A.
$$
The <Fredholm solvability condition> requires their sum to vanish, because $L_0$ annihilates the critical <Fourier mode>. For a nonzero amplitude,
$$
\boxed{|A|^2=p(s)\mu_2,\qquad p(s)=\frac9{27-8s^2}.}
$$
This requires $\mu_2/(3-8s^2/9)>0$. The cubic amplitude coefficient is positive for $s^2<27/8$, yielding a small-amplitude branch in a <supercritical bifurcation>; it is negative for $s^2>27/8$, yielding a <subcritical bifurcation> branch in this leading approximation. At \b[$s^2=27/8$], the cubic coefficient vanishes and the quoted relation is singular: higher-order nonlinear terms and a different detuning balance are needed. One must not assert a finite $p$ there.
If a term $\epsilon\mu_1$ were included in $\mu$, order $\epsilon^2$ would also contain $-\mu_1\Theta_{1xx}=\mu_1\Theta_1$. The quadratic nonlinearity produces only the zeroth and second harmonics, with the zeroth differentiated away, so it cannot balance a first-harmonic contribution at that order. Solvability would give $\mu_1A=0$. Hence \b[a nontrivial critical-mode expansion forces $\mu_1=0$] and first balances detuning against cubic amplitude effects at order $\epsilon^3$.
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