= Solution
Partition the lattice into blocks of side $b$ sites, with $b>1$, and retain one coarse spin $\sigma'_R$ per block. A <blocking kernel> is a nonnegative conditional weight $K_b(\sigma',\sigma)$ normalized by $\sum_{\sigma'}K_b(\sigma',\sigma)=1$ for every microscopic configuration. A deterministic majority rule, with fair resolution of ties, is one example.
Define the blocked <statistical Hamiltonian> and its field-independent constant by
$$
e^{-\beta_{\rm th}[H(u',\sigma')+N'C']}=
\sum_\sigma K_b(\sigma',\sigma)e^{-\beta_{\rm th}[H(u,\sigma)+NC]},\qquad N'=N/b^D.
$$
Summing over $\sigma'$ proves exact equality of the old and new <partition functions>. The lattice spacing becomes $ba$; expressing lengths again in the original units completes the <real-space renormalization group> step. Blocking generally generates many-body and longer-range interactions, so exactness requires retaining the full operator space and the additive constant. A finite-coupling truncation is a calculational approximation.
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