Even a Square Is Not Free — The Star Theorem

We study a one-parameter Euclidean square configuration obtained by extending the sides of a unit square according to a clockwise order-four rule and iterating the resulting direct similarity. The exterior square is generated by the multiplier w(t) = (1 + t) - it, which preserves the common centre and the exact C4 symmetry of all generations. Explicit coordinates for the second and third generations yield two non-degenerate axis-parallel alignment orbits. These occur precisely at t = phi and t = 1/phi, giving the Star Theorem. To distinguish genuine generational coincidences from fixed support-line contacts, we introduce a functional degeneracy criterion and perform an exact symbolic sweep of all 240 vertex-coordinate comparisons among generations 0 through 3. The sweep proves that the two golden orbits are the only non-degenerate golden alignments in the stated finite setting. At the reciprocal golden parameter, the area multiplier is exactly 3. The associated clockwise rotation has magnitude theta = arctan(1/phi^2); we prove that theta/pi is irrational and consequently that the orientation sequence is non-repeating and equidistributed modulo 2pi. The computational classification is finite, exact, and reproducible through the supplied symbolic script. A final section examines the higher-dimensional reach of the construction: the map is called isotropic when every point of a centred sphere is displaced by one common length, and we show that a nontrivial isotropic MJ map exists if and only if the ambient dimension is even, and we classify these maps, the rotations involved being exactly the isoclinic ones. This record contains the sealed preprint version 1.5, dated 30 July 2026. It is an author manuscript that has not undergone peer review. The PDF is deposited without modification. SHA-256: 0c2acebec3890688d8d4981c503735ccffacd0e238b284572afd6a0f3d85442e.