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.