= Solution
In <mean-field theory> all sites have the same normalized single-site distribution $e^{x\cos\theta}/Z_1(x)$. Consequently
$$
\langle S_{kz}\rangle=\frac{d\log Z_1}{dx}
=\coth x-\frac1x\equiv\mathcal L(x),
$$
where $\mathcal L$ is the <Langevin function>. Requiring the average used to construct the field to equal the average in that field gives
$$
\boxed{S=\coth(\beta z_cJS)-\frac1{\beta z_cJS}.}
$$
Set
$$
\boxed{x=\beta z_cJS=\frac{4JS}{k_BT},\qquad
\tau=\frac1{\beta z_cJ}=\frac{k_BT}{4J}.}
$$
Since $S=\tau x$, the <self-consistency equation> is exactly \b[$\tau x=\coth x-1/x$]. It also follows by stationarity of $F_{\rm MF}/N=z_cJS^2/2-k_BT\log Z_1(\beta z_cJS)$. This independent route checks the constant term and all neighbor-counting factors in the <partition function>.
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