Four integers name an element of ℤ[ζ₅]. Reading ζ₅ as a 72° rotation gives the physical point. The star map ζ₅→ζ₅² gives its internal coordinate, while the coefficient sum modulo five supplies a discrete class.
The four nonzero classes select the pentagons P, −φP, φP, and −P. A point becomes a tiling vertex exactly when its internal reading lies in its class's window. Connecting accepted points one cyclotomic unit apart produces the two Penrose rhombs and all seven geometric vertex-star types.
A phason translates every internal coordinate while leaving the projection direction fixed. The regular presets use exact seventh-unit shifts and have no window-boundary hits. Nearby choices change finite patches through local rearrangements while preserving the same quasiperiodic scheme. The centered choice restores fivefold symmetry but is singular, so its boundary inclusions are counted explicitly.