staircase-constants

• Computes the fill ratio of every base-2 design along the row word of odd sides 3, 5, ..., 2L+1, against the closed form R_L = (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 the fill polynomial P_F(n) = sum_j a_j n^(dim-j) (n-1)^j, a_j the corners with j odd coordinates.
• Corner i is the binary digits of i, most significant first, as mrlyrs::math::bang::factory::code_to_corners reads a code; the fill depends on the profile a_j alone, so 77 profiles cover all 273 nonempty codes at dim 1, 2, 3, 3 + 15 + 255.
• section_render counts filled cells of every code at dim 1..3 on odd sides 3..9 and even sides 2..8, and the row word by np.kron at L = 2..5, 2..3, 2 in dim 1, 2, 3, against P_F and w/2^dim.
• section_exact checks the Gamma identity against the exact integer product at L = 1, 2, 3, 10, 100, 1000 at 50 digits.
• section_asymptotic sums log1p of the exact per-letter excess in float to L = 10^6 and compares with drift log L + log C + c1/L, drift = dim/2 - mean, c1 = dim/8 + drift - var/2, mean and variance of the odd count over the corners.
• constant asserts C real and positive before taking its real part.
• section_named compares C with its closed form on named codes; section_parity compares the Gamma form at roots of unity for the two parity designs at dim 2..6 with the independent zeta log series exp(-+ sum_m (+-1)^m (lambda(dim m) - 1)/m) at 50 digits, with the closed form where one is printed, and with the direct product; section_family takes the roots of the at-most-one-odd design from its fill polynomial at dim 1..8, so the 50-digit match tests only the Gamma recurrence and the direct sum at ten digits is the independent check.
• Values print at 12 significant digits and rounded to nine decimals, the form the notes copy.
• section_pairing checks P(1-n) = (-1)^dim P(n) iff the profile is a palindrome on 776 profiles at dim 1..4, the mirror identity C_F C_F' = prod sin(pi r)/(4 r (1-r)) on all 77, and counts the profiles whose roots outside 0, 1/2, 1 pair under r -> 1-r.
• section_fence multiplies the renormalised factors along sides 3^k and along the odd primes; section_staircase evaluates the stacked-prefix staircase dimension by Barnes G at the roots, against the direct fill sum, and the gap to dim against log(2^dim/w)/(log 2n - 3/2) to n = 2 x 10^5.

RUN

uv run python research/lab/py/staircase-constants/staircase.py

• From the repo root; one core, about 7 seconds: render 0.1s, roots 0.1s, exact 0.3s, asymptotic 1.0s, pairing 0.3s, staircase 2.8s, the rest under 0.2s each.

WITNESSES

• magic.md The staircase: the Gamma identity on 77 profiles, worst log gap 4.4e-47; the direct sum to L = 10^6 against drift log L + log C + c1/L, worst residual 1.13e-12, residual times L^2 flat at 1.12 from L = 10^4.
• magic.md The staircase: renormalised limits along 3^k, 1.56493401857, 1.31484053105, 1.87429848245 at dim 1 code 1, dim 2 code 7, dim 3 code 23; the five staircase dimensions 1.892789261, 1.892315261, 1.893034267, 1.894190425, 1.895495742 from Barnes G; Barnes G against the direct sum on 77 profiles, gap 2.1e-50; n times the rate error -0.75, -0.71, -0.70, -0.69, -0.69 at n = 10^2 .. 2 x 10^5, dim 2 code 7, and -0.41 .. -0.36 on the row word.
• pi.md Pi on the staircase: the parity constants 0.785398163, 1.254589239, 0.948815486, 1.052420668, 0.985352084, 1.014708181, 0.995477949, 1.004525474, 0.998553027, 1.001447181 at dim 2..6, each against the zeta log series at 50 digits, gap below 1e-50, and its closed form where one is printed, gap below 1e-50.
• pi.md Pi on the staircase: the named constants and the mirror products 9 sqrt3 pi/64, sqrt2 pi^2/24, pi/4; 18 flip-closed codes against 46 palindromic codes at dim 2, 3; 37 of 77 profiles reduce by reflection, 13 of them palindromic.