= Finite-field Ising-chain decimation
{title2=$K'=\tfrac14\log[W_{++}W_{--}/W_{+-}^2],\quad h'=\tfrac12\log[W_{++}/W_{--}]$}
Use the symmetric <transfer matrix> $T_{ss'}=\exp[Kss'+h(s+s')/2]$ with dimensionless field $h$. For blocking <length> two set $W=T^2$, then write $W_{ss'}=A\exp[K'ss'+h'(s+s')/2]$. Matching its three independent entries proves the title relations and gives $A=(W_{++}W_{--}W_{+-}^2)^{1/4}$. Explicitly $W_{\pm\pm}=e^{2K\pm2h}+e^{-2K}$ and $W_{+-}=2\cosh h$. At zero field the exact chain recursion is recovered; at $K=0$ the independent-spin field stays unchanged. Splitting the field equally between neighboring bonds is essential for this normalization.
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