= Solution
The <linear spin-wave approximation> expands about the rotated all-up state, with $\widehat n_n=a_n^\dagger a_n$:
$$
S_n^z=S-\widehat n_n,\qquad S_n^+=\sqrt{2S}\,a_n+O(S^{-1/2}),\qquad S_n^-=\sqrt{2S}\,a_n^\dagger+O(S^{-1/2}).
$$
In the longitudinal product, $S_n^zS_{n+1}^z=S^2-S(\widehat n_n+\widehat n_{n+1})+O(S^0)$. Each site belongs to two bonds, so the resulting quadratic <Hamiltonian> is
$$
\boxed{H_2=-NJS^2+JS\sum_n\left[2a_n^\dagger a_n+a_na_{n+1}+a_n^\dagger a_{n+1}^\dagger\right]}.
$$
The omitted terms are of order $S^0$ at fixed small <occupation number>. The pair terms describe the quantum fluctuations that were missing from the classical <Néel state>.
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