The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field

We address a stability threshold problem of the Couette flow $(y,0,0)$ in a uniform magnetic fleld $\alpha (\sigma ,0,1)$ with $\sigma \in \mathbb{Q}$ for the 3D MHD equations on $\mathbb{T}\times \mathbb{R}\times \mathbb{T}$. Previously, the authors in \cite{L20,RZZ25} obtained the threshold $\gamma =1$ for $\sigma \in \mathbb{R}\mathrm{\backslash }\mathbb{Q}$ satisfying a generic Diophantine condition, where they also proved $\gamma =4\mathrm{/}3$ for a general $\sigma \in \mathbb{R}$. In the present paper, we obtain the threshold $\gamma =1$ in $H^N(N>13\mathrm{/}2)$, hence improving the above results when $\sigma$ is a rational number. The nonlinear inviscid damping for velocity $u_{\mathrm{\ne }}^2$ is also established. Moreover, our result shows that the nonzero modes of magnetic field has an amplification of order ${\nu }^{-1\mathrm{/}3}$ even on low regularity, which is very different from the case considered in \cite{L20,RZZ25}.