# Diagonal cuts of the parity solid

- 2026-08-28 [Proved] Every diagonal slice of `bang dim 3, code 126` holds exactly `3^level` points at every admissible height, by uniqueness of the binary expansion of the height offset; checked at `level = 1..8` by two enumerations sharing no code and to `level = 14` by a height recursion; the constancy separates it from the digit-scheduled slices of Nakajima and Watanabe, whose non-autonomous IFS uses `A_c^(j) = {0}` or `{1,2,3}` by the height digit, so their digit changes the number of maps and this one only the orientation. Witness: mrlymath::three::diagonal.
- 2026-08-28 [Verified] The two central cuts of `bang dim 3, code 126` decompose into six congruent Sierpinski gaskets of `3^(level-1)` points each tiling a hexagon with an order-12 symmetry group, checked at `level = 2..8` by rebuilding the pieces from the previous level (union, pairwise disjointness, sizes); the combined totals are `6 * 3^(level-1) = 2 * 3^level`: `18, 54, 162, 486, 1458, 4374, 13122`; the ambient object is the octahedron flake of dimension `log(6)/log(2)`. Witness: mrlymath::three::diagonal.