= Solution
The <Hellmann–Feynman theorem> applied to the source field normalization gives
$$
\frac{\partial E_0}{\partial g}=\left\langle\frac{\partial H}{\partial g}\right\rangle
=-2J\sum_n\langle S_n^z\rangle.
$$
Differentiating the <Ising-chain ground-state energy>, using $\partial_g\epsilon_k=4J^2(g-\cos k)/\epsilon_k$, therefore proves
$$
\boxed{\sum_n\langle S_n^z\rangle=\sum_k\frac{J(g-\cos k)}{\epsilon_k}.}
$$
This is the <Ising-chain field derivative and transverse polarization>, not its spontaneous x-order parameter. It tends to zero at g=0 and to N/2 at large positive field. The <critical transverse polarization of an Ising chain> is finite: in the thermodynamic critical limit this gives $\langle S^z\rangle=1/\pi$. A zero-energy isolated momentum is treated by the appropriate limiting state, rather than by assigning a value to a literal 0/0 summand.
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