= Exact zero-field Ising-chain decimation
{title2=$\tanh K'=(\tanh K)^b$}
Keep every $b$th spin in a ferromagnetic nearest-neighbour <Ising model> at zero field, with coupling $K=J/(k_BT)$. Multiplying its two-state <transfer matrix> $b$ times yields the displayed recursion. For $b=2$, direct summation of one intermediate spin gives $2\cosh[K(s+s')]=A e^{K'ss'}$, with $K'=\tfrac12\log\cosh2K$. The new spacing is $ba$ before coordinate rescaling. All finite positive couplings flow toward the disordered fixed point $K=0$; the ordered zero-temperature endpoint is unstable to a nonzero domain-wall number per bond.
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