= Solution
The <Hellmann–Feynman theorem> gives $\partial_gE_0=\langle\partial_gH\rangle=-2J\sum_n\langle S_n^z\rangle$. Differentiating the <Ising-chain ground-state energy> yields $\partial_gE_0=-J\sum_k(g-\cos k)/r_k$. Hence
$$
\boxed{\sum_n\langle S_n^z\rangle=\frac12\sum_k\frac{g-\cos k}{r_k}=\sum_k\frac{J(g-\cos k)}{\epsilon_k}.}
$$
This is also obtained from the occupations of the <Bogoliubov quasiparticle> vacuum: $\langle c_k^\dagger c_k\rangle=[1-(g-\cos k)/r_k]/2$ and $S_n^z=1/2-n_n$. At an exact zero mode the finite-sector limiting <ground state> specifies the occupation. In the bulk limits, the <magnetization> per site is zero at $g=0$, tends to $1/2$ as $g\to\infty$, and equals $1/\pi$ at $g=1$: there the integrand $(1-\cos k)/r_k=|\sin(k/2)|$.
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