DICTO Kyushindo Cores

The main theorem: every convex polyhedron S with icosahedral rotational symmetry lies inside (15 − 6√5)·P(S), and so inside φ·P(S), where P(S) is the hull of the foot points of S's face planes (Theorem 2, sharp; from a general containment bound, Theorem 1). Hence DICTO's OG turned out, which flips the 80 face pyramids (Aktis) of the DICTO Ogdoecontahedron (OG) outward at 1/φ of their height and takes the hull, beat after beat, is exactly φk times the k-th solid of the OG's downward family [2] at beat k and is nested for every k (Corollary 1), although the downward family itself is not: its first break, TQH → TQP, is caused exactly by the 30 ribbons of the TQH, trapezoid pairs whose foot points fall outside their faces (Proposition 1). Reciprocation exchanges the two steps, polar(P(S)) = T(polar S) (Lemma 3), so the polar of the OG turned out is the Universal PiTet sequence of the DICTO Hecatodihedron, shrinking inward and nested for every k. Reciprocation about the sphere of radius² 15 − 5√5 maps beat k of the Universal PiTet rule [1] to beat 5 − k (Theorem 3). With a lemma that every shell splits into towers, congruent within each symmetry orbit, these give, exactly in ℚ(√5): the DICTO Cleo Towers (shells of 1, 1, 2, 3 and 5 beats round the CLEO sphere, with 32, 30, 32, 32 and 60 towers), the DICTO Polar Cleo Towers (5, 3, 2, 1 and 1 beats from the centre; 30, 32, 30, 60 and 32 towers), the towers of the OG turned out (80, 132 and 140 in 2, 3 and 3 shapes) and of its polar (80, 132 and 162 in 2, 3 and 4 shapes). We also give the OG core of 80 Aktis (20 DICTO Hex Aktis and 60 DICTO Deltoid Aktis, taking (65 − 3√5)/110 and (45 + 3√5)/110 of the OG) and the OTT core of 38 OTT rays (8 DICTO Oktaktis, each exactly a Cleo-Icos lying in a hexagon plane of the OG, 24 DICTO Triaktis and 6 DICTO Tetraktis). The boundary solids change their surface pattern from shell to shell; on the TQH the trapezoids pair into 30 ribbons on the two-fold axes, which DICTO sees as a possible new shell structure; they return exactly at down 8 and down 9; on the hulls the two-fold axis word, of period 4 from the OG to down 7, breaks at down 8 (the axis point is a listed foot point but not a hull corner), and the listed points themselves leave the period at down 10. Each result is marked as proved in general, proved for the cases built, or observed. Finding #43 on dictospheres.com. Scripts, exact checks and data: https://github.com/DICTOR-Master/kaleidohedra/tree/c3c8ff7/studies (research branch; folders 2026-10-20-cleo-towers, 2026-10-20-og-spires, 2026-10-20-ott-rays, 2026-10-21-og-turned-out, 2026-10-21-nesting-proof), also attached here (kyushindo-cores-study-c3c8ff7.zip).