= Block-spin coarse-graining
{title2=$S_B=\operatorname{sgn}\sum_{i\in B}s_i$}
Consider <Ising spins> $s_i=\pm1$ on a square lattice of spacing $a$. Divide it into disjoint $3\times3$ blocks $B$ and retain the majority sign $S_B$. Each block contains nine spins, so a tie is impossible. For example, five positive and four negative spins produce $S_B=+1$. All $2^9$ configurations are assigned a retained sign; the individual pattern within a block is lost. A <coarse-grained variable> has fewer possible values than the microscopic configuration it summarizes.
The spacing between block centres is $3a$. In coordinates $x'=x/3$ it is again $a$, so the retained pattern can be compared with the original lattice at the same numerical cutoff. This coordinate change does not physically shrink the sample. A majority picture alone does not specify the new interactions: the <blocking kernel> and original <Boltzmann weights> determine their probabilities. The figure is a constructed configuration, not a simulated equilibrium state or a claim that majority blocking closes the nearest-neighbour <Ising model>.
\Image[/renormalization-block-spin-coarse-graining.png]
{title=<Block-spin coarse-graining> retains a majority variable, then restores the coordinate convention}
{description=Original constructed $9\times9$ <Ising spin> pattern, its nine majority signs at spacing $3a$, and the same signs after $x'=x/3$. Red and blue mean $+1$ and $-1$. The last panel occupies one third of the original linear extent in the rescaled coordinates.}
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