A lattice coefficient vector names one cyclotomic integer. Reading ζ₈ as a 45° rotation places it in physical space; the star map ζ₈→ζ₈³ gives its internal-space coordinate. A point survives exactly when the internal reading lies in the regular octagon.
The two panels carry the same lattice provenance. Color is internal angle, so rotating by 45° permutes both panels coherently. The window is the projection of a unit four-cube and has eight exact halfspaces over ℤ[√2]. This centered cut is nonsingular: the boundary-hit count is zero.
The phason presets translate internal coordinates by exact quarter units. They produce distinct, locally related patterns without changing the projection direction or using floating membership decisions.
Gold rings mark the visible part of the exact lattice automorphism. It expands physical positions by 1+√2 while its conjugate 1−√2 contracts internal positions back into the octagon. Self-similarity is therefore an integer transformation of provenance, not a visual rescale.