The HexOct Family: every close-packed stacking from two single solids, at every weight

In HexOct (#75 [1]) every close-packed stacking is a string of two layer tiles, the c-tile and the h-tile. Each tile is four pieces cut by DICTO's Rule of Halves (#57 [2]) at a weight λ, −11/24 ≤ λ ≤ 3/8. The h-tile is one solid, but the c-tile is two towers that touch only at a point. We show that moving one tower by one lattice step along its own layer joins the towers along a face. The layer is filled exactly as before, so every stacking is still a string of the two tiles, and any tile still sits on any tile. The move works at all 402 weights we checked exactly, a grid of step 1/480 over the whole walk that includes every break point. An exact check of all 42 stackings of period 2 to 6 finds no gap and no overlap. At the end of the walk, λ = 3/8, both tiles are convex: the c-tile is a brick of two cubes and the h-tile is a 12-faced solid we call the twisted cube (placeholder), with 6 squares and 6 triangles. Five single-solid pairs, at λ = 3/8, −1/8 (two ways), −11/24 and 1/10, make up the HexOct family. View in the app: the HexOct world in Kaleidohedra. DICTO proposed making the two HexOct tiles of #75 into two true simple pieces and keeping all the single-solid pairs found as one family; the family and member names by weight are placeholders until DICTO names them. Computation, exact checks and drafting with Claude (Anthropic).