walsh-fill

• Computes the Walsh coefficients hat f(S) = 2^-dim sum_(c in F) (-1)^(|S cap c|) of every base-2 design at dim 1..4 by one Hadamard product, and their level sums W_j = sum_(|S| = j) hat f(S), kept as the integers 2^dim W_j.
• Corner i is the binary digits of i, most significant first, as mrlyrs::math::bang::factory::code_to_corners reads a code.
• section_expansion counts filled cells on literal grids, every nonempty code at dim 1..3 and 2000 seeded codes at dim 4, sides 1..9, plus a literal corner histogram against all 65536 codes at dim 4, and checks 2^dim fill(N) = sum_j 2^dim W_j N^(dim-j) at odd N and w N^dim at even N.
• section_profile checks 2^dim W_j = sum_w a_w K_j(w), K_j the Krawtchouk polynomial, on every code at dim 1..4, and K^2 = 2^dim I.
• section_mirror checks fill_F(-N) = (-1)^dim fill_F'(N) on every code at dim 1..4, tests fill_F(-N) = +- void_F(N) against the profile criterion a_j + a_(dim-j) = C(dim, j), the count formula and the self-dual designs.
• section_roots maps every root r != 1/2 of P_F(n) to mu = 1/(2r - 1) and evaluates R(mu) = sum_j W_j mu^j there, on the 77 profiles at dim 1..3, and checks R > 0 on a grid of (-1, 1).
• section_drift checks dim/2 - mean = W_1/(2 W_0) on every nonempty code at dim 1..4, and 2^dim W_1 = U, the bichromatic edge count, exactly on the down-sets.
• section_stability checks 2^dim sum_(c in F) fill_(F+c)(N) = sum_j 2^(2 dim) W^j N^(dim-j), W^j the squared-coefficient weight, and sum_c fill_(F+c)(N) = w N^dim over all flips, by literal counts at odd sides 1..9, every code at dim 1..3.
• section_entropy prints -log2 of the fill ratio at sides 2, 3, 9, ..., 729 for dim 1 code 1, dim 1 code 2, dim 2 code 7, dim 3 code 23.
• section_threshold checks 2^dim W_0 = S(dim, t) and 2^dim W_1 = (t+1) C(dim, t+1) for the rule "at most t odd" at dim 1..8 by a Hadamard transform of each rule, prints the majority ratios, and groups S(dim, t)/2^dim over 0 <= t < dim <= 2048 by exact value.
• section_flat finds the codes with W_j = 0 for 0 < j < dim at dim 1..4, checks them against the level-parity criterion, and intersects them with the 896 bent codes at dim 4.

RUN

uv run python research/lab/py/walsh-fill/walsh.py

• From the repo root; one core, about 4 seconds: expansion 0.1s, mirror 0.6s, drift 0.6s, threshold 2.1s, the rest under 0.2s each. A verb name runs one section.

WITNESSES

• method.md The coin, Theorem: 339045 expansion checks at odd sides and 271236 even-side checks, 0 mismatches; W of dim 2 code 7, dim 2 code 11, dim 3 code 23, dim 3 code 232.
• method.md The coin, N -> -N swaps fill and void exactly at a balanced profile: swapping designs 2, 4, 40, 2800 at dim 1..4, equal to the profile criterion and the count formula, self-dual 2, 4, 16, 256 inside; dim 3 code 27, corners 000, 001, 011, 100, the least non-self-dual swap, dim 2 code 3 swaps, dim 3 code 1 does not.
• method.md The coin, The roots are zeros of the biased mean: 77 profiles, worst residual 2.7e-15.
• method.md The coin, The drift is half the slope of the log biased mean at the fair coin: drift identity on 65808 codes; 2^dim W_1 = U on the nonempty down-sets, 2, 5, 19, 167 at dim 1..4, and on no other code.
• method.md The coin, Noise stability is the fill summed over the flips by filled corners: 2730 literal checks; the four polynomials of the dim 3 code 23 flip orbit.
• method.md The coin, A level costs log2(2^dim/w) bits at infinite side: side-3 costs 0.584962501, 1.584962501, 0.169925001, 0.432959407 at dim 1 code 1, dim 1 code 2, dim 2 code 7, dim 3 code 23, and N times the gap at side 729 against (W_1/W_0)/ln 2.
• method.md The coin, Threshold rules and Shared limit ratios: majority 1/2, 3/4, 1/2, 11/16, 1/2, 21/32, 1/2, 163/256 at dim 1..8; 5/16 and 11/16 at dim 4; beyond 1/2, held by the 1024 odd majorities, 26 shared values to dim 2048, each held by exactly two rules.
• method.md The coin, Lower-order-free designs: 4, 8, 4, 140 at dim 1..4, none of the 896 bent codes at dim 4.