Bond moving ($K^{\prime }=\mathrm{artanh}[{\tanh }^b(b^{d-1}K)]$)

In the bond-moving-first hypercubic construction with blocking length $b$, combine $b^{d-1}$ transverse-equivalent bonds into each link of a surviving $b$-link chain. Summing that chain gives the displayed Migdal-Kadanoff approximation. For $b=2$, it becomes $K^{\prime }=\frac{1}{2}\log \cosh (2^{d}K)$. The $d=1$ specialization is exact; the higher-dimensional formula is a convention-dependent approximation, not an exact closure of the original hypercubic model.