= Solution
For the even periodic chain, every classical <Néel state> $\mathbf S_m=(-1)^mS\mathbf n$ has energy $-JNS^2$, with arbitrary unit vector $\mathbf n$. Thus there is a \b[continuous sphere of classical orientations], not merely two Ising-like states. Opposite staggered directions are connected by a global spin rotation.
A quantum Néel product is not an eigenstate: transverse exchange flips opposite neighbouring spins. The ground state therefore contains quantum superpositions and zero-point <spin waves>, reducing the classical staggered moment. A finite isotropic balanced chain has a symmetry-invariant ground state; a symmetry-broken reference is an approximation for analysing fluctuations, not its exact finite-system eigenstate.
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