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Write $s={\alpha }^2+{\pi }^2$ and use modes satisfying all boundary conditions,The quadratic temperature distortion $c$ is retained for the saturation calculation; it does not enter the linear onset. Substitution of the primary modes givesThe determinant vanishes whenThe trace is $-[s+\sigma (s+Q{\pi }^2/s)]<0$. Thus no conjugate eigenvalues can cross the imaginary axis away from zero, and the initial instability is stationary, not oscillatory. Indeed for $R\ge 0$ the discriminant is $[\sigma (s+Q{\pi }^2/s)-s{]}^2+4\sigma R{\alpha }^2/s>0$. This is quasistatic vertical-field magnetoconvection: there is no independent magnetic evolution mode that could produce a magnetic-thermal Hopf bifurcation instability.

For a fixed box the other allowed horizontal wavenumbers are $n\pi /L$, and the actual first threshold is the minimum of their corresponding $R_c$ values. Vertical index $m$ replaces ${\pi }^2$ by $m^2{\pi }^2$ and strictly increases the threshold at fixed horizontal wavenumber, so $m=1$ is selected. The displayed formula describes the prescribed fundamental roll; part (b) optimizes its width.