  It is noted that the pair correlation matrix $\hat{\chi}$ of the nearest
neighbor Ising model on periodic three-dimensional ($d=3$) kagome-like lattices
of corner-sharing triangles can be calculated partially exactly. Specifically,
a macroscopic number $1/3 \, N+1$ out of $N$ eigenvalues of $\hat{\chi}$ are
degenerate at all temperatures $T$, and correspond to an eigenspace
$\mathbb{L}_{-}$ of $\hat{\chi}$, independent of $T$. Degeneracy of the
eigenvalues, and $\mathbb{L}_{-}$ are an exact result for a complex $d=3$
statistical physical model. It is further noted that the eigenvalue degeneracy
describing the same $\mathbb{L}_{-}$ is exact at all $T$ in an infinite spin
dimensionality $m$ limit of the isotropic $m$-vector approximation to the Ising
models. A peculiar match of the opposite $m=1$ and $m\rightarrow \infty$ limits
can be interpreted that the $m\rightarrow\infty$ considerations are exact for
$m=1$. It is not clear whether the match is coincidental. It is then speculated
that the exact eigenvalues degeneracy in $\mathbb{L}_{-}$ in the opposite
limits of $m$ can imply their quasi-degeneracy for intermediate $1 \leqslant m
< \infty$. For an anti-ferromagnetic nearest neighbor coupling, that renders
kagome-like models highly geometrically frustrated, these are spin states
largely from $\mathbb{L}_{-}$ that for $m\geqslant 2$ contribute to
$\hat{\chi}$ at low $T$. The $m\rightarrow\infty$ formulae can be thus
quantitatively correct in description of $\hat{\chi}$ and clarifying the role
of perturbations in kagome-like systems deep in the collective paramagnetic
regime. An exception may be an interval of $T$, where the order-by-disorder
mechanisms select sub-manifolds of $\mathbb{L}_{-}$.
