Vertical-field magnetoconvection dispersion relation ($(\lambda +\sigma s)(\lambda +s)(\lambda +\zeta s)+\sigma Q\zeta {\pi }^2(\lambda +s)-\sigma R{k}^2(\lambda +\zeta s)/s=0$)

For a stress-free, fixed-temperature layer with vertical magnetic perturbation at its boundaries, take vertical velocity and temperature proportional to $\sin \pi z$, vertical magnetic perturbation proportional to $\cos \pi z$, and horizontal magnetic perturbations proportional to $\sin \pi z$. With $s=k^2+{\pi }^2$, the vertical velocity, temperature and magnetic amplitudes form a three-variable linear system whose determinant is the stated polynomial. The Prandtl number is $\sigma$, magnetic-to-thermal diffusivity ratio is $\zeta$, and field strength is the Chandrasekhar number $Q$. The steady marginal curve is $R^{(e)}=[s^3+Q{\pi }^{2}s]/k^2$.

For one stress-free planar roll mode, thermal-time rescaling reduces the linear dynamics to $a^{\prime }=\sigma (-a+rb-\zeta qc)$, $b^{\prime }=a-b$, $c^{\prime }=a-\zeta c$. Its characteristic polynomial is $(s+\sigma )(s+1)(s+\zeta )-\sigma r(s+\zeta )+\sigma \zeta q(s+1)$. For $\sigma ,\zeta >0$ and $q\ge 0$, steady marginality is $r_s=1+q$. Oscillatory marginality is $r_H=(\sigma +\zeta )(1+\zeta )/\sigma +\zeta (\sigma +\zeta )q/(\sigma +1)$, with ${\omega }_H^2=\zeta [\sigma (1-\zeta )q/(\sigma +1)-\zeta ]$. A genuine Hopf bifurcation onset requires ${\omega }_H^2>0$; a formal imaginary-root formula with negative frequency squared is not an oscillation. The Routh-Hurwitz stability criterion selects the first crossing, while nonlinear coefficients determine criticality.

At the minimum of the steady vertical-field magnetoconvection dispersion relation, stationarity gives $R_c=2(k_c^2+{\pi }^2{)}^3/{\pi }^2$. With nonmagnetic value $R_0=27{\pi }^4/4$, eliminating the critical wavenumber gives the stated relation and $k_c^2={\pi }^2[\frac{3}{2}(R_c/R_0{)}^{1/3}-1]$. These hold for every nonnegative Chandrasekhar number, including its zero-field limit.

Both steady and oscillatory neutral curves have the form $R=C(k^2+p{)}^3/k^2+DQp(k^2+p)/k^2$, with $p={\pi }^2$. Their exact minimizing condition is $x^2(2x+3p)=p^3+(D/C)Q{p}^2$, where $x=k^2$. For fixed positive coefficients and large Chandrasekhar number, $x\sim [D{p}^{2}Q/(2C){]}^{1/3}$ and $R_{\min }=DpQ+3C^{1/3}(D{p}^{2}Q/2{)}^{2/3}+O(Q^{1/3})$. The optimum balances transverse diffusion against the magnetic penalty, leading to increasingly narrow rolls.

A nonzero imaginary root of the vertical-field magnetoconvection dispersion relation requires $0<\zeta <1$ and $Q>\zeta (1+\sigma )s^2/[\sigma (1-\zeta ){\pi }^2]$. Its marginal curve is $R^{(o)}=C{s}^3/k^2+DQ{\pi }^{2}s/k^2$, with $C=(\sigma +\zeta )(1+\zeta )/\sigma$ and $D=\zeta (\sigma +\zeta )/(1+\sigma )$. A formal minimum of that algebraic curve is a physical oscillatory threshold only if the frequency squared there is positive.

For thermal-time magnetoconvection, a physical oscillatory neutral mode has ${\omega }^2=\sigma (1-\zeta )Q_{\kappa }{\pi }^2/(1+\sigma )-{\zeta }^2(k^2+{\pi }^2{)}^2>0$. The steady and oscillatory neutral curves meet when this frequency tends to zero. Thus $0<\zeta <1$ is necessary and the displayed relation locates the merger. A formal oscillatory-curve minimum with negative frequency squared is not a physical onset.

At a steady-curve minimum, differentiation and the steady-Hopf merger in vertical-field magnetoconvection give the first displayed wavenumber. At an oscillatory-curve minimum they give the second. For fixed positive Prandtl number $\sigma$ and large $Q_{\kappa }$, the corresponding merger diffusivity deficits are $1-{\zeta }_s\sim [(1+\sigma )/\sigma ]({\pi }^2/(4Q_{\kappa }){)}^{1/3}$ and $1-{\zeta }_o\sim [(1+\sigma ){\pi }^2/(16\sigma Q_{\kappa }){]}^{1/3}$. The two parameter surfaces are distinct: coincidence of neutral curves at one wavenumber does not imply coincidence of their minimizing wavenumbers.