= Solution
For each pair of adjacent <spins>, define the <transfer matrix for a classical spin chain> by $T_{ss'}=e^{-\beta E_0(s,s')}$. Multiplying its entries gives the <Boltzmann factor> of a whole configuration. The <periodic boundary condition> identifies the last index with the first, so
$$
Z_L=\sum_{\sigma_1,\ldots,\sigma_L}T_{\sigma_1\sigma_2}\cdots T_{\sigma_L\sigma_1}
=\boxed{\operatorname{Tr}T^L=\sum_\alpha\lambda_\alpha^L.}
$$
The assumed interchange symmetry of $E_0$ makes $T$ a real <symmetric matrix>, hence it has an orthonormal <eigenbasis>. At finite positive <temperature>, every matrix entry is strictly positive, so the <Perron–Frobenius theorem> supplies a unique positive largest <eigenvalue> $\lambda_+$, larger in modulus than every other one. Therefore $L^{-1}\log Z_L\to\log\lambda_+$.
Local observables become matrix insertions. For example, with $S_{ss'}=s\delta_{ss'}$,
$$
\langle\sigma_1\rangle=\frac{\operatorname{Tr}(ST^L)}{\operatorname{Tr}T^L},
\qquad
\langle\sigma_1\sigma_{1+r}\rangle=\frac{\operatorname{Tr}(ST^rST^{L-r})}{Z_L}.
$$
Thus <partition functions> and <correlation functions> reduce to a fixed-dimensional <eigenvalue> problem, rather than a sum over exponentially many <microstates>.
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