Progenitors

The DICTO Hecatodihedron H (102 faces) and its polar, the DICTO Ogdoecontahedron O (80 faces), came out of the Spheres of Dharma, an exact cycle of icosahedral shells that repeats every φ³. DICTO asked whether the twin pair could seed a cycle of its own. Every solid here is exact in ℚ(√5), and every nesting and volume is checked with remainder 0. On their own the twins do not close golden. H, O and their classical relatives, nested tightly in every order (8,742 cycles), never return on a power of φ, and the tightest 3-cycle gains exactly 9/φ⁴ = (3/φ²)² per round: a cycle governed by 3, as in Double Vision. Set into the Spheres of Dharma with O in the dodecahedron's place, they close it again on φ⁹ in volume: I ⊂ ID ⊂ O ⊂ rTI ⊂ TI ⊂ H ⊂ jtI ⊂ RT ⊂ I·φ³. The cycle then holds the core and hull of two dual compounds, the icosahedron–dodecahedron compound (ID, RT) and the twins' own compound, the DICTO Hecatogdoecontahedron (rTI, jtI). Prior-art status is stated claim by claim in §4.

Data, scripts and exact check outputs are included (see README.md); the full studies are at https://github.com/DICTOR-Master/kaleidohedra (branch research: studies/2026-10-16-progenitors, studies/2026-10-16-polar-dharma, studies/_lib). 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