DICTO Janus c/h Pyrochlore

In short. The pyrochlore network, small tetrahedra touching only at their corners, hides inside every diamond-like crystal: the middles of the bonds round each atom make one tetrahedron. Diamond-like crystals stack their layers in many orders, so the pyrochlore network can too. Sharing out all of space round it by one weight, as in #57 (The Rule of Halves), we prove the same rule: in every close-packed stacking and for every weight from −11/36 to 1/4, every piece is borrowed from the pure cubic or the pure hexagonal stacking — balls by their own layer, diamond atoms by their nearest layer, the two empty holes of each gap as two halves by the layers on their sides. Exact weight walks for the cubic and hexagonal stackings (the cubic one passes through the tetrahedral–truncated tetrahedral honeycomb). Proof with #57's exact engine: locality over sixteen windows and every stretch of the walk (384 of 384), with an independent exact check at twelve weights (768 of 768).

The idea is DICTO's. Computation and proof with Claude (Anthropic); a first draft by a local open model (gpt-oss-120b), checked line by line. Sources checked are listed in the paper. Licence CC BY-SA 4.0. More at dictospheres.com.