slice-ladder-controls

• Checks the anti-diagonal profile identity P_{A (x) B}(t) = P_A(t^side_B) P_B(t) on every base-2 word of length 2 and 3, counting mismatches.
• Checks that the level-1 central diagonal slice of the base-3 Menger analog equals the vertex count of the cube's central cross-section, dim 2..14.
• The slice side is enumerated over all 3^dim digit vectors and the vertex side is binomial, so the two sides share no formula.
• Prints the dim 4 central diagonal census at levels 1..6, the order-2 recurrence fitted on its first four terms and checked on all six, its dominant root and its slice dimension.
• Prints the 3, 5, 7, 9, 11 staircase dimensions at rung n = 1..5, assuming fill base^2 - ((base-1)/2)^2 for each base.
• Domains run are the page's own: words of length 2 and 3 over the 15 nonempty 2x2 tiles, dim 2..14, levels 1..6, rungs n = 1..5. Whole run is about three seconds.

RUN

uv run python research/lab/py/slice-ladder-controls/controls.py

WITNESSES

• slices.md:118 - the profile identity, zero mismatches over all words of length 2 and 3.
• the claims line (was DISCOVERIES.md:146) - the same identity, zero mismatches.
• the claims line (was DISCOVERIES.md:147) - level-1 slice counts 2, 6, 6, 30, 20, 140, 70 at dim 2..8.
• cuts.md:324 - the same counts, and the level-1 slice identified with the vertex set.
• cuts.md:330 - the identity at dim 2..8 and the dim 2..14 ladder.
• sequences.md:96 - 6, 132, 1848, 29040, 441408, 6772128, a(n) = 11a(n-1) + 66a(n-2), root (11 + sqrt(385))/2, dimension 2.483635500.
• the claims line (was DISCOVERIES.md:136) - the dim 4 characteristic polynomial x^2 - 11x - 66 and rho_4 = 15.310708.
• sequences.md:97 - 2, 6, 6, 30, 20, 140, 70, 630, 252, 2772, 924, 12012, 3432.
• dimensions.md:296-297 - 1.892789261, 1.892315261, 1.893034267, 1.894190425, 1.895495742.
• The run states its carpet fill assumption before any staircase number.