Held by Shape Alone: the Octet's Mixed Layer and the Admiral Layer

Proposition 1 (the octet's mixed layer). In a single (111) layer of the octet every octahedron is locked, against slides and twists alike, and every tetrahedron is free: it can be lifted straight out through its face on the layer's surface. The proof is exact. Proposition 2 (the Admiral layer, computer check). In the Admiral layer of #47, every interior Admiral Cap (each a rigid pair of tetrahedra) and every icosahedron is locked against all small slides and twists; of the face pyramids tested, 18 are locked and 4, those resting on the layer's surface, are free. The check uses exact contact polygons and linear programs for slides in ℝ³ and for slides and twists in ℝ⁶, and is calibrated on the tetrahedron layer of Dyskin et al. (all 21 interior blocks locked, as published) and on a layer of upright cubes (every cube free). So the Admiral layer holds itself together by shape alone, apart from a skin of face pyramids on its two surfaces. The locking of a layer of octahedra is known (Dyskin et al. 2001, 2003; Akpanya, Goertzen and Niemeyer 2024); the mixed slab and the Admiral layer were not found in these sources. DICTO proposed running the interlocking test among the unused tools of the Concordance, and then the rotation check on the Admiral slab. Computation, proofs and drafting with Claude (Anthropic).