Ising-chain decimation constant ($A=2\sqrt{\cosh 2K}$)

For a periodic even chain, eliminating alternate spins gives $Z_N(K)=A^{N/2}{Z}_{N/2}(K^{\prime })$. Thus dimensionless free energy per original spin obeys $f(K)=-\frac{1}{2}\log A+\frac{1}{2}f(K^{\prime })$. The exact solution $f(K)=-\log (2\cosh K)$ satisfies this relation. The multiplicative constant cancels from normalized retained-spin probabilities but records eliminated-spin entropy; discarding it loses thermodynamics even if the interaction recursion is correct.