# Odd-base slice grammar

- 2026-08-28 [Verified] The odd-base generalisation rests on a choice of solid: Cook's predicate "at most one coordinate in the middle third" and the `bang dim 3, code 23` rule "at most one odd coordinate" agree at `base = 3` (20 of 27) and nowhere else, at `base = 5` filling `4^3 + 3 * 4^2 = 112` of 125 against `3^3 + 3 * 2 * 3^2 = 81 = 4k^3 - 3k^2` at `k = 3`, and there is no canonical base-5 Menger sponge. Witness: lab/py/odd-base-slice-grammar.
- 2026-08-28 [Verified] In the comparison of the slice dimension against the dimension minus one, the dimension is the solid's own `log(fill)/log(base)` and never the ambient 3: at the ambient 3 the value `3 - 1 = 2` exceeds all four slice dimensions `1.8184, 1.6869, 1.8026, 1.7204` and the mod-4 split collapses. Witness: lab/py/odd-base-slice-grammar.
- 2026-08-28 [Verified] The four printed dimensions are consistent with the four printed rules and this is not evidence for either: the dominant roots `(9 + sqrt(33))/2 = 7.37228`, `(11 + sqrt(369))/2 = 15.1047`, `(42 + sqrt(612))/2 = 33.3693`, `(28 + sqrt(3556))/2 = 43.8161` give `log(root)/log(base)` of `1.8183, 1.6870, 1.8026, 1.7204` at `base = 3, 5, 7, 9`, while `4k^3 - 3k^2` at `k = 2..5` gives `20, 81, 208, 425` and dimension minus one `1.7268, 1.7304, 1.7430, 1.7544`; a rule and its own dimension cannot cross-check each other. Witness: lab/py/odd-base-slice-grammar.
- 2026-08-28 [Proved] The middle diagonal layer sits at coordinate sum `3(base-1)/2`, odd exactly when `base = 3 mod 4` (`3, 6, 9, 12` at `base = 3, 5, 7, 9`). Witness: lab/py/odd-base-slice-grammar.
- 2026-08-28 [Conjecture] No definition of a "blow-up of 4" for the slice exists in this tree, so the phrase carries no claim.
- 2026-08-28 [Conjecture] The two-tile grammar closes at ten odd bases with a `2 x 2` substitution matrix rational in `base` within each class of `base mod 4`.
- 2026-08-28 [Conjecture] That parity forces structurally different cells into the middle layer in each residue class, which is the mechanism of the mod-4 split.