# Slice at large side

- 2026-10-02 [Proved] For a word of odd sides listed coarsest first, the central hexagon count of `bang dim 3, code 23` is `(S_(b_L) .. S_(b_1))[0, 0]`, with `S_b` the spectra block and the finest letter on the left. Witness: spectra.md THE SLICE AT LARGE SIDE, The word product.
- 2026-10-02 [Verified] Brute-force central hexagon counts `60, 72, 2412, 2688, 300` on the words `(3,5), (5,3), (3,5,7), (7,5,3), (5,7)` equal the block product, as do seven more words and four words for each of seven other codes. Witness: lab/py/slice-at-large-side verb words.
- 2026-10-02 [Proved] `S_b/b^2 = A + B/b + B_2/b^2` exactly on each class of `b mod 4`, with `A = [[3u/4, v/8], [3v/2, u/4]]`, `u = 3/4` and `v = 1/4` at `3 mod 4`, the reverse at `1 mod 4`, and `B = (3/8)[[1, 1/2], [2, 1]]` in both; the Perron roots are `(3+sqrt(3))/8`, `(1+sqrt(7))/8` and `3(1+sqrt(2))/32` for the pair. Witness: spectra.md THE SLICE AT LARGE SIDE, The letter at infinite side.
- 2026-10-02 [Proved] At `b = 1 mod 4` the dominant root of `S_b` is `(b^2 + 3b + 4 + (b-1) sqrt(7b^2 + 32b + 34))/8`, and in both classes `(log b)(log_b rho_b - log_b fill + 1) = log(2 lambda) + (mu - 3/2)/b + O(b^-2)`, with `mu = 1` at `3 mod 4` and `1 + 2/sqrt(7)` at `1 mod 4`. Witness: spectra.md THE SLICE AT LARGE SIDE, The letter at infinite side.
- 2026-10-02 [Proved] The central hexagon ink is `lambda^L L^gamma (C + o(1))` with `C > 0`: `lambda = (3+sqrt(3))/8`, `gamma = 1/4` on sides `3, 7, 11, ..`; `lambda = (1+sqrt(7))/8`, `gamma = 1/4 + 1/(2 sqrt(7))` on `5, 9, 13, ..`; `lambda = sqrt(3(1+sqrt(2))/32)`, `gamma = (2+sqrt(2))/4` on `3, 5, 7, ..` in either order, with `C` depending on the parity of `L`. Witness: spectra.md THE SLICE AT LARGE SIDE, The drift.
- 2026-10-02 [Proved] On sides `3, 7, .., 4L-1` coarsest first the central hexagon count is `(L!)^2 2^-L [z^L] (1 - 6z)(1 - 24z + 96z^2)^(-3/4)`, and the ink constant is `Gamma(3/4)(1 + sqrt(3)) / (3 (sqrt(3) - 1)^(3/4)) = 1.410085329792638597969`, rounded. Witness: spectra.md THE SLICE AT LARGE SIDE, The constants.
- 2026-10-02 [Proved] On sides `3, 5, 7, ..` in either order, the ink constants at odd and at even length have ratio `sqrt((5 sqrt(2) - 1)/3) = 1.4225643291682`. Witness: spectra.md THE SLICE AT LARGE SIDE, The constants.
- 2026-10-02 [Conjecture] The ink constants are `0.72001825738796` on sides `5, 9, 13, ..`; on `3, 5, 7, ..` they are `0.53693769481512` and `0.76382841162981` at even and odd length coarsest first, and `0.66052225470496` and `0.93963539816504` finest first. Witness: lab/py/slice-at-large-side verb constants.
- 2026-10-02 [Proved] For every dim 3 parity design the limit letter is `[[3u/4, v/8], [3v/2, u/4]]`, with `u = o/4`, `v = e/4` at `b = 3 mod 4` and `u = e/4`, `v = o/4` at `b = 1 mod 4`; at large side the census exponent minus `log_b fill - 1` therefore has the sign of `u - v`. Witness: spectra.md THE SLICE AT LARGE SIDE, Every parity design.
- 2026-10-02 [Proved] A dim 3 parity design with as many even-weight as odd-weight patterns has automaton Perron root exactly `fill/b` at every odd base, since its digit polynomial vanishes at every nontrivial `b`-th root of unity. Witness: spectra.md THE SLICE AT LARGE SIDE, Every parity design.
- 2026-10-02 [Proved] Computer-assisted: for all 255 nonempty dim 3 parity designs and every odd base, the census exponent of the central hexagon, a limsup over levels, minus `log_b fill - 1` has the sign of `o - e` at `b = 3 mod 4` and of `e - o` at `b = 1 mod 4`, with an empty slice counted below; that is 93, 93 and 69 codes. Witness: spectra.md THE SLICE AT LARGE SIDE, Every parity design; lab/py/slice-at-large-side verb designs.