= Odd antiferromagnetic Ising ring
{title2=$g_0=2N,\quad E_0=-(N-2)J$}
An odd periodic nearest-neighbour <Ising model> with antiferromagnetic coupling has unavoidable <geometrical frustration>. For $k$ parallel bonds, $H=-NJ+2Jk$, while $\prod_jS_jS_{j+1}=1$ forces $N-k$ to be even. Thus odd $N$ requires at least one parallel bond. A <ground state> has exactly one, with $N$ possible locations and two global <spin inversion symmetry> choices, proving the displayed <ground-state degeneracy> and <ground-state energy>. The zero-<temperature> <canonical ensemble> distributes that defect uniformly. A directed arc of $n$ bonds contains it with probability $n/N$, giving <two-point correlation function> $(-1)^n(1-2n/N)$. The <finite antiferromagnetic Ising-ring correlation> gives the corresponding positive-<temperature> result.
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