Core Cycles: when the cores of DICTO's star chains alternate

In The Truncated Icosahedron Star (#78) DICTO gave the truncated icosahedron (TI) its own face pyramids turned outwards and found that the core of the turn, the solid common to the icosahedron of the pentagon tips and the dodecahedron of the hexagon tips, alternates with TI: two turns give TI back. DICTO proposed that this holds for every solid whose faces lie in two families of planes, on the five-fold and three-fold axes or the four-fold and three-fold ones. We prove it. Let U be the corner directions of a regular solid and W those of its dual, and let c be the cosine of the angle from a direction of U to the nearest one of W. A solid cut by the planes u·x = a and w·x = b, with both kinds of plane as faces, has as the core of its turn the solid cut by the planes u·x = tcb and w·x = tca: the two distances swap, scaled by tc, where t is the spike factor. So two turns give the solid back, scaled by t²c², whatever a and b are; c² = (5 + 2√5)/15 for the icosahedron and dodecahedron, 1/3 for the octahedron and cube, and 1/9 for the tetrahedron. Every solid whose face planes are square to the symmetry axes of the icosahedral or octahedral group reaches such a cycle after at most one turn. The theorem accounts for 28 of the 29 convex shapes of DICTO Solids whose cores keep forming, and for all 88 solids of the two-family sweep, each matched exactly; the Belted Triacontahedron is the one exception, with an exact factor of its own. The other 19 shapes have no core, and we prove why: their families of tips lie in a plane. The hulls of the chains do not follow the cores: with one family on the two-fold axes, whether the chain closes depends on how deep the planes cut. This corrects the hull part of the originally proposed main statement. Turn a solid's face pyramids outwards and each kind of face sends its tips to a different shell. Wrap each shell's tips and keep what all the wrappings share. For the truncated icosahedron this inner solid swaps the depths of the pentagons and hexagons, and the next turn swaps them back. DICTO's idea: the core cycle for every solid with two families of face planes. Computation and proof with Claude (Anthropic). More at dictospheres.com. Polyhedraverse, Star step: View in the app.