A Miller family indexes a set of lattice planes, and the sum of squared indices fixes the spacing through the lattice constant. For h=2, k=2, l=1 the squared sum is 4+4+1 = 9, so with lattice constant 12 m the spacing is 12/sqrt(9) = 4 m exactly for the stated schematic indexing ledger; real instruments have indexing ambiguity, peak width, calibration, crystal disorder, and measurement uncertainty.
A plane-family spacing is a checked geometry ledger. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.
highlighted = computed this step
Miller indices define the training family
The family is indexed by 2, 2, and 1. The squared denominator is 9.
h2+k2+l2=9
The plane spacing is forced by geometry
With lattice constant 12 m, the exact spacing is 4 m. This is schematic indexing, not structure refinement.