= Solution
The primary vorticity <Jacobian determinant> vanishes because $\nabla^2\psi=-s\psi$. The quadratic <temperature> <Jacobian determinant> is
$$
J(\psi,b\cos\alpha x\sin\pi z)=\frac{\alpha\pi ab}{2}\sin2\pi z.
$$
The mean <temperature> correction therefore obeys $\dot c=-4\pi^2c-\alpha\pi ab/2$. Its <Jacobian determinant> with the primary <streamfunction> is
$$
J(\psi,c\sin2\pi z)=\alpha\pi ac\cos\alpha x(\sin3\pi z-\sin\pi z).
$$
Projecting on the retained primary modes gives, with $D=s+Q\pi^2/s$,
$$
\dot a=-\sigma Da+\sigma R\alpha b/s,\quad
\dot b=-sb+\alpha a+\alpha\pi ac,\quad
\dot c=-4\pi^2c-\alpha\pi ab/2.
$$
For a nonzero steady roll, $b=(s^2+Q\pi^2)a/(R\alpha)$ and $c=-\alpha ab/(8\pi)$. Substitution in the second equation gives, within this truncation,
$$
\boxed{a^2=\frac{8(R-R_c)}{s^2+Q\pi^2},\qquad c=-\frac{R-R_c}{\pi R}.}
$$
The positive denominator ensures branches only on the supercritical side for every $Q\ge0$. To verify <dynamical stability> as well as existence, the critical right <eigenvector> is $(1,\alpha/s)$; normalizing the left <eigenvector> gives linear <eigenvalue> derivative
$$
K=\frac{\sigma\alpha^2}{s(s+\sigma D)}>0.
$$
Quadratic slaving gives $c=-\alpha^2a^2/(8\pi s)$, and projection of its feedback gives the cubic amplitude equation
$$
\boxed{\dot a=K\left[(R-R_c)a-\frac{s^2+Q\pi^2}{8}a^3\right]+\cdots.}
$$
Its nonzero branches are stable in the center direction, while all other <eigenvalues> remain negative at a simple onset. <Reflection> about the mid-width sends $a\to-a$, so the bifurcation is \b[a supercritical <pitchfork bifurcation> for all $Q$], in the generic case of a single critical roll mode.
The <cubic saturation of quasistatic magnetoconvection> result is identical in <Fourier truncation> and perturbation theory at the orders needed here. At quadratic order the only forced mode is precisely the retained horizontally uniform $\sin2\pi z$ <temperature> mode. At cubic order its feedback projects on the primary mode with the same coefficient. The generated third vertical harmonic is orthogonal to that primary projection and contributes only to higher-order corrections. Thus both methods give the same linear threshold and cubic solvability condition; the truncated finite-amplitude branch is not an exact solution of the full PDE at all amplitudes.
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