= Solution
Differentiate the uniform <effective action>:
$$
0=\frac1\beta\frac{\partial S_{\rm eff}}{\partial M}=\frac{NM}{U}-\sum_{k,\sigma}\sigma n_F(\epsilon_k-\sigma M).
$$
To reproduce the printed coefficient, define the \b[spin-summed density of states per site] by
$$
\nu(\epsilon)=\frac2N\sum_k\delta(\epsilon-\epsilon_k),\qquad
\frac1N\sum_k f(\epsilon_k)=\frac12\int d\epsilon\,\nu(\epsilon)f(\epsilon).
$$
The <saddle-point approximation> then gives
$$
\boxed{M=\frac U2\sum_{\sigma=\pm1}\int d\epsilon\,\nu(\epsilon)\,\sigma n_F(\epsilon-\sigma M).}
$$
The <uniform spin field and magnetization> obey $M=U(n_\uparrow-n_\downarrow)=2U\langle S_z\rangle$ for occupations per site. If a one-spin <density of states> were used instead, the factor $1/2$ would be absent. The spin field is an energy, rather than directly a magnetic moment.
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