Vajrayana

Abstract

Progenitors (DICTO DOI: 10.5281/zenodo.23413184) set the DICTO Ogdoecontahedron O into the Spheres of Dharma in the dodecahedron's place and found an exact nine-station golden cycle, I ⊂ ID ⊂ O ⊂ rTI ⊂ TI ⊂ H ⊂ jtI ⊂ RT ⊂ I·φ³, closing on φ⁹ in volume. This companion asks where the dodecahedron went and what fills the cycle's loose steps. Every solid here is exact in ℚ(√5), and every nesting and volume is checked with remainder 0 unless marked float. The dodecahedron returns inside the icosidodecahedron, as the hull of DICTO's Cleo star: the icosahedron's 20 Cleo-Icos with 20 more turned outward have their tips exactly at (2/3)·D, and 82 pieces of four kinds fill that dodecahedron exactly. The loose step from the rhombic triacontahedron to the next icosahedron is closed by the DICTO Metaxyhexecontahedron: the member of the dodecahedron–icosahedron hull family whose icosahedron edge is 9/φ⁴ times its dodecahedron edge. It touches both neighbours, both of its layers are a single kind of piece, and its dual closes the same step in the polar-reversed cycle. The full-height icosidodecahedron star gives a second member of the same family, the DICTO Palintriacontahedron, the rhombic triacontahedron turned inside out. The cycle grows to twelve stations; the loose step O ⊂ rTI stays open. Prior-art status is stated claim by claim in §5. Data, scripts and exact check outputs are included (see README.md); the full study is at https://github.com/DICTOR-Master/kaleidohedra (branch research: studies/2026-10-16-progenitors). Computation, exact arithmetic, searches and checking with Claude (Anthropic); scope and publishing order advised by Claude Fable; an independent check with Antiprism 0.29. Explore DICTO's worlds in the apps: https://dictospheres.com Version 2: the cut figures re-rendered with solid cut faces (each cut solid filled in its own colour; glass shells hollow), following DICTO's standing practice for cut views; text and results unchanged.