slice-at-large-side.md

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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.