= Solution
The linearized complex equation couples a <perturbation> and its conjugate, so a single complex <Fourier mode> by itself is not a closed <perturbation> ansatz. Equivalently use real <amplitude> and <wave phase> <perturbations>, with <Fourier modes> paired to make each real field. Writing $A=A_0+\alpha+i\beta$ and $B=A_0^2$ gives
$$
\alpha_T=4\alpha_{XX}+(\mu+9\hat sB-50B^2)\alpha,\qquad
\beta_T=4\beta_{XX}+(\mu+3\hat sB-10B^2)\beta.
$$
A real <amplitude> mode of slow <wavenumber> $\ell>0$ therefore becomes neutral when
$$
\boxed{\mu-4\ell^2+9\hat sA_0^2-50A_0^4=0.}
$$
For a nonzero uniform state the <equilibrium point> condition eliminates $\mu$, leaving <amplitude> growth $6\hat sB-40B^2-4\ell^2$. The <wave phase> mode growth is simply $-4\ell^2$, so it cannot give a nonzero-wavenumber bifurcation from these undetuned states.
The concave quadratic has maximum
$$
\max_{B\ge0}(6\hat sB-40B^2)=\frac{9\hat s^2}{40},\qquad B=\frac{3\hat s}{40}.
$$
Thus a nonzero uniform state needs $\ell\le3\hat s/(4\sqrt{10})$ to have a modulation bifurcation. The slow domain length is $\varepsilon^2L$, so $\ell_n=2\pi n/(\varepsilon^2L)$. If its smallest positive <wavenumber> already exceeds this bound, none can become neutral. Since $s=\varepsilon^2\hat s$, the conclusion is
$$
\boxed{L<\frac{8\pi\sqrt{10}}{3s}\ \Longrightarrow\
\text{no modulation bifurcation of a nonzero uniform state}.}
$$
This is the <modulation cutoff of a cubic-quintic uniform pattern>. If the bound is reversed, candidate squared amplitudes are $B=(3\hat s\pm\sqrt{9\hat s^2-160\ell^2})/40$. They lie on the lower branch; the upper branch is already damped in <amplitude> and cannot undergo this stationary modulation bifurcation.
The word “nonzero” is necessary in the final claim. At $A_0=0$, the <equilibrium point> exists independently of $\mu$, and the mode growth is $\mu-4\ell^2$. It becomes neutral at $\mu=4\ell_n^2$ for every finite domain. For example, take $\hat s=1$, $\varepsilon^2L=20$: then $L=20/\varepsilon^2<8\pi\sqrt{10}/(3s)$, but the zero state has a first-sideband threshold at $\mu=\pi^2/25$. Therefore the printed assertion, if read to include the trivial state, is false. The derived domain-size bound is the correct statement for nonzero pattern branches.
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