Universal PiTet: The DICTO PiTet Proposition

DICTO specified a five-fold version of the octet: spherical shells grown outward from the Cleo star, the same pattern each time with different solids, triangles and pyramids staying constant (hence the name PiTet: Pi for the spherical progression, Tet for the triangles). We give a rule that does this and prove, in exact arithmetic in ℚ(√5), what it produces. Let T take a convex solid with corners v to the solid {x : v·x ≤ v·v for every corner v}: each new face passes through a corner of the last solid at right angles to it. Starting from the icosidodecahedron (ID), T gives the rhombic triacontahedron (RT), then the DICTO GTI (Golden Truncated Icosahedron: hexagon sides 1 : φ), then the DICTO FPD (Flat Pentakis Dodecahedron), and then the ID again, exactly 10/φ⁴ = 35 − 15√5 times larger; the cycle therefore repeats without end. Each solid lies in the next, each shell is filled by congruent pieces of one or two kinds (exact volumes), and the whole chain is its own polar about the GTI's circumsphere. Below the cycle, the Cleo star (82 pieces) and 12 twisted frusta (Monkey Staff B) close the ID; the pieces on one five-fold axis join into a convex five-sided Vajra, and the DICTO Ogdoecontahedron fits between the ID and the RT with 20 pure hexagonal Cleos. Every claim is checked exactly.