R = √2−1 = tan(π/8) = 1/δₛcube side 2. δₛ = 1+√2, the silver ratio. insphere of every pyramid, touching all 5 facesk = 2−√2 = 1−Rdot sits on the axis, k from the cube centre|rot(cube)| = 24 ↔ 24 tetra/cell6 pyramids × 4 triangle faces. tetra = dot + one face. 3D COINCIDENCE: pieces 4n(n−1) vs rotations 2^(n−1)·n!, equal only at n=3. in 2D pieces match all 8 symmetries, from 4D they fall short. see MIRRORS|Oₕ| = 48 = 24 × 2mirror plane through dot, centre, base-edge midpoint halves each tetradots → octahedron (dual)six dots on the axes, one per face|dot−dot| = 2R : spheres kiss12 contacts per cell, the octahedron's edgeskissing number in lattice = 54 in its own cell + 1 through the face. neighbours meet at the face centrecentre void r = 3−2√2 = R²the gap left among the six spherespacking = π(√2−1)³ ≈ 0.2233sphere volume / cube volume. loose, and that is the pointred lines = 6 × 5 = 30 / celldot → 4 base corners + cube centrefold: ±θ counter-rotation, 4.8.8 → square packing2D lock at θ = 45° exactly (incentre dots; first contact checked on a 0.01° grid). strain −(√2−1)² = −0.1716, density 0.539 → π/4, ν = −1 all along. among the scanned dot positions, only the incentre ends on the regular square lattice. see MOTIONContinuous cab → fcc, one octahedron per cellExplicit smooth motion; exact non-overlap certificate; strictly increasing density to π/√18. Internal proof review only; priority open. Explore the motion and review status3D fold [111], cubic path: 6 contacts/ball at 60°spacing 8t/3, density 0.4686. Not a jam: a sheared branch keeps folding. [001] locks at 45°, [011] at 54.7356°axes: 3×C4 · 6×C2 · 4×C3keys 1 2 3. the body diagonal outlines a hexagon