= Solution
Put $r=\tanh K$, so $r\to1^-$ as the <temperature> tends to zero. For even $N$, the <finite antiferromagnetic Ising-ring correlation> becomes
$$
C_N(n)=(-1)^n\frac{r^n+r^{N-n}}{1+r^N}
\quad\Longrightarrow\quad
\boxed{\lim_{\beta J\to\infty}C_N(n)=(-1)^n\qquad(N\text{ even}).}
$$
The two perfectly alternating <ground states> give exactly the same product $S_1S_{n+1}=(-1)^n$; averaging them preserves its magnitude even though the <one-point correlation function> remains zero.
For odd $N$ the signs instead give
$$
C_N(n)=(-1)^n\frac{r^n-r^{N-n}}{1-r^N}.
$$
Both numerator and denominator vanish. Set $r=1-\delta$ and expand at fixed $N,n$:
$$
r^n-r^{N-n}=(N-2n)\delta+O(\delta^2),
\qquad1-r^N=N\delta+O(\delta^2).
$$
Hence
$$
\boxed{\lim_{\beta J\to\infty}C_N(n)
=(-1)^n\left(1-\frac{2n}{N}\right)\qquad(N\text{ odd}).}
$$
There is a direct <ground state> interpretation. The mandatory parallel bond of an <odd antiferromagnetic Ising ring> is uniformly distributed among its $N$ bonds in the limiting <canonical ensemble>. Along the directed $n$-bond arc from site 1 to site $n+1$, the product is $(-1)^n$ if the defect lies outside that arc, and $-(-1)^n$ if it lies inside. The respective probabilities are $1-n/N$ and $n/N$, giving the displayed <two-point correlation function>. Global <spin inversion symmetry> doubles the number of <ground states> but leaves each product unchanged.
\Image[/past-exam-of-the-field-theories-and-collective-phenomena-in-condensed-matter-course-of-the-university-of-oxford/2025-ising-ring-parity.png]
{title=One unavoidable parallel bond changes the zero-temperature correlations of an odd antiferromagnetic ring}
The reduction on an odd ring survives at separations comparable with $N$, despite the vanishing defect density $1/N$. For fixed $n$, however, taking $N\to\infty$ after the zero-<temperature> limit recovers $(-1)^n$ for either parity, consistently with the diverging <correlation length>. The linear envelope at finite odd $N$ comes from one freely located frustration defect; it is distinct from the exponential decay caused by thermal defects in the <thermodynamic limit>.
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