= Solution
At $B=0$ a global reversal $\sigma_j\mapsto-\sigma_j$ leaves the energy and statistical weight unchanged but changes the sign of the one-spin observable. Pairing reversed configurations gives
$$
\boxed{\langle\sigma_j\rangle_\beta=0.}
$$
Equivalently, $\langle\pm|\sigma^z|\pm\rangle=0$ in the transfer <eigenbasis>, so $\operatorname{Tr}(\sigma^zT^N)=0$. This holds for every finite even ring and all finite temperatures, and for the zero-field Gibbs limit. A symmetry-selected zero-temperature alternating state can have nonzero \b[staggered] <magnetization>; its uniform <magnetization> is still zero. The classical spin variable in this model has no separate quantum spin-operator dynamics despite the conventional matrix notation $\sigma^z$ used for the insertion.
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