Block-spin coarse-graining ($S_B=\mathrm{sgn}{\sum }_{i\in B}{s}_i$)

Consider Ising spins $s_i=\pm 1$ on a square lattice of spacing $a$. Divide it into disjoint $3\times 3$ 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 $x^{\prime }=x/3$ 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.