# The staircase law

- 2026-10-02 [Proved] On the `row` word of odd sides `3, 5, ..., 2L+1` a base-2 design with `w` corners in `dim`, `a_j` of them with `j` odd coordinates, has fill ratio exactly `(w/2^dim)^L prod_i Gamma(L+2-r_i)/Gamma(2-r_i) / (Gamma(L+3/2)/Gamma(3/2))^dim` over the roots `r_i` of `P_F(n) = sum_j a_j n^(dim-j) (n-1)^j`, all real roots in `[0, 1]`, hence `(w/2^dim)^L L^drift C (1 + c_1/L + O(L^-2))` with `drift = dim/2 - mean`, `C = Gamma(3/2)^dim / prod_i Gamma(2 - r_i)` real and positive, and `c_1 = dim/8 + drift - var/2` over the odd count of the corners. Witness: `magic.md` The staircase, the fill law on the row word; `lab/py/staircase-constants`, 77 profiles covering all 273 nonempty codes at `dim 1..3`, exact form at `L = 1, 2, 3, 10, 100, 1000` to `4.4e-47`, direct sum at `L = 10^6` to `1.13e-12`.
- 2026-10-02 [Proved] The row-word drift is 0 exactly when the corners hold as many odd coordinates as even ones, and the design of corners with at most one odd coordinate has ratio per level `(dim+1)/2^dim`, drift `dim/2 - dim/(dim+1)` and constant `(sqrt(pi)/2)^dim (dim+1) / Gamma(1/(dim+1))`, `3 pi/(4 Gamma(1/3)) = 0.879525401` at `dim 2 code 7` and `pi^(3/2)/(2 Gamma(1/4)) = 0.767916039` at `dim 3 code 23`. Witness: `magic.md` The staircase, the drift counts the corners; `lab/py/staircase-constants`, roots from the fill polynomial at `dim 1..8`, direct sum to `L = 10^6` at ten digits.
- 2026-10-02 [Proved] At every even side a base-2 design's fill ratio is `w/2^dim` exactly, so even letters carry no drift and no constant in any schedule. Witness: `magic.md` The staircase, even sides are silent; `lab/py/staircase-constants`, every code at `dim 1..3`, sides `2..8`.
- 2026-10-02 [Proved] The stacked-prefix staircase of any base-2 design has log fill ratio `(n(n+1)/2) log(w/2^dim) + sum_i [log G(n+3-r_i) - log G(3-r_i) - n log Gamma(2-r_i)] - dim [log G(n+5/2) - log G(5/2) - n log Gamma(3/2)]`, `G` the Barnes function, and `dim - dimension(n) = log(2^dim/w) / (log(2n) - 3/2) + O(1/n)`, while the row word has `dim - dimension_L = log(2^dim/w) / (log(2L) - 1) + O(1/L)`. Witness: `magic.md` The staircase, the rate of approach; `lab/py/staircase-constants`, the five `dim 2 code 7` staircase dimensions `1.892789261` to `1.895495742` from the Barnes form, gap `2.1e-50` against the direct sum on 77 profiles, `n` times the rate error `-0.75` to `-0.69` at `dim 2 code 7` over `n = 10^2 .. 2 x 10^5`.
- 2026-10-02 [Proved] Along distinct odd sides `N_k >= 3` the renormalised measure `prod_k (2^dim/w) P_F((N_k+1)/2)/N_k^dim` of a base-2 design is finite and nonzero exactly when its drift is 0 or `sum 1/N_k` converges, and no infinite schedule of sides `>= 2` of a design short of full keeps positive measure; along `N_k = 3^k` it is `1.56493401857`, `1.31484053105`, `1.87429848245` at `dim 1 code 1`, `dim 2 code 7`, `dim 3 code 23`. Witness: `magic.md` The staircase, the constant belongs to the schedule; `lab/py/staircase-constants`.
- 2026-10-02 [Proved] At odd side `N` the parity designs fill `(N^dim -+ 1)/2`, since `sum (-1)^(x_1+...+x_dim)` over the box is 1, so on the row word `2^L` times the fill ratio tends to `prod_(N odd >= 3) (1 - N^-dim) = Gamma(3/2)^dim / prod_(u^dim = 1) Gamma((3-u)/2)` for odd parity and `prod (1 + N^-dim)` for even, at every `dim >= 2`: `pi/4` and `cosh(pi/2)/2` at `dim 2`, `0.948815486` and `1.052420668` at `dim 3`, `pi cosh(pi/2)/8` and `(cosh(pi/sqrt2) + cos(pi/sqrt2))/4` at `dim 4`, `pi cosh(pi sqrt3/2)/24` and `cosh(pi/2)(cosh(pi/2) + cos(pi sqrt3/2))/4` at `dim 6`. Witness: `pi.md` Pi on the staircase, parity designs are the Wallis sieve at half rate; `lab/py/staircase-constants`, the zeta log series and the closed forms at 50 digits, the direct product at twelve.
- 2026-10-02 [Proved] The roots of a base-2 design's fill polynomial are closed under `r -> 1 - r` with multiplicity exactly when its odd-count profile is a palindrome, and wherever the roots outside `0, 1/2, 1` so pair the row-word constant is `(sqrt(pi)/2)^(m_0 + m_1) prod_pairs sin(pi r)/(4 r (1-r))`, `m_0` and `m_1` the multiplicities of the roots 0 and 1. Witness: `pi.md` Pi on the staircase, the roots pair exactly at a palindromic profile; `lab/py/staircase-constants`, 776 profiles at `dim 1..4`.
- 2026-10-02 [Proved] A design's row-word constant times its mirror's is `prod_i s(r_i)` over its roots, with `s(r) = sin(pi r)/(4 r (1-r))` and `s(0) = s(1) = pi/4`, the constant of `F x F'`: `9 sqrt(3) pi/64` for `dim 2` codes 7 and 14, `sqrt(2) pi^2/24` for `dim 3` codes 23 and 232, `pi/4` for `dim 1` codes 1 and 2. Witness: `pi.md` Pi on the staircase, a design times its mirror reduces by reflection; `lab/py/staircase-constants`, 77 profiles, gap `1.1e-50`.
- 2026-10-02 [Refuted] A row-word constant reduces by reflection exactly when the design is closed under flipping every coordinate. Witness: `dim 2 code 11`, corners `00, 01, 11`, not flip-closed, constant `3 cosh(pi/(2 sqrt3))/4 = 1.080152394`, and `dim 2 code 1`, one corner, non-palindromic, constant `pi/4`; `lab/py/staircase-constants` counts 18 flip-closed against 46 palindromic codes at `dim 2, 3`, and 37 of 77 profiles reducing by reflection, 13 palindromic.