= Solution
For kinetic and field energies small compared with $mc^2$, the minimally coupled lower spinor is $\tilde\chi\simeq(\boldsymbol\sigma\cdot\boldsymbol\pi)\tilde\varphi/(2mc)$. Inserting it into the upper equation and using $[\pi_i,\pi_j]=i(e\hbar/c)\varepsilon_{ijk}B_k$ gives the <Pauli equation>
$$
\boxed{i\hbar\partial_t\tilde\varphi=
\left[\frac{\boldsymbol\pi^2}{2m}+e\Phi-
\frac{e\hbar}{2mc}\boldsymbol\sigma\cdot\mathbf B\right]\tilde\varphi.}
$$
The <spin> moment is $e\hbar/(2mc)$ multiplying $\boldsymbol\sigma$, or gyromagnetic factor two relative to physical <spin> $\hbar\boldsymbol\sigma/2$.
For the alternative equation, elimination gives
$$
\tilde\chi\simeq\frac1{2mc^2}
\left(c\boldsymbol\sigma\cdot\mathbf p-\frac{i\kappa}{c}\boldsymbol\sigma\cdot\mathbf E\right)\tilde\varphi.
$$
To leading kinetic order and leading magnetic interaction this yields
$$
\boxed{i\hbar\partial_t\tilde\varphi=
\left[\frac{\mathbf p^2}{2m}-\kappa\boldsymbol\sigma\cdot\mathbf B\right]\tilde\varphi
+\text{electric-gradient and spin-orbit terms suppressed by }c^{-2}.}
$$
Keeping the source electric coupling during the same elimination writes the kinetic part more explicitly as
$$
\frac1{2m}\left(\boldsymbol\sigma\cdot\mathbf p+\frac{i\kappa}{c^2}\boldsymbol\sigma\cdot\mathbf E\right)
\left(\boldsymbol\sigma\cdot\mathbf p-\frac{i\kappa}{c^2}\boldsymbol\sigma\cdot\mathbf E\right).
$$
This product includes <derivatives> acting on the field and must not be treated as an ordinary scalar product of commuting variables. The parameter $\kappa$ is the intrinsic magnetic-moment coefficient: the moment operator is $\kappa\boldsymbol\sigma$. As printed, the alternative equation has no minimal charge coupling, so it can describe a neutral particle carrying a <magnetic moment>. If a Pauli term were added to the charged equation instead, its coefficient would represent an additional, anomalous contribution to the charge-generated Dirac moment.
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