The coin

• 2026-10-03 [Proved] At every odd side N a base-2 design F fills sum_(j=0..dim) W_j N^(dim-j), W_j = sum_(abs(S)=j) hat f(S) the level sums of its Walsh coefficients hat f(S) = 2^-dim sum_(c in F) (-1)^(sum_(i in S) c_i), so the fill ratio is the mean of the design under coins of bias mu = 1/N, the noise operator T_(1/N) at the all-even corner. Witness: method.md The coin, Theorem; lab/py/walsh-fill verb expansion, 339045 checks at odd sides 1..9 and 271236 of w N^dim at even sides 2..8.
• 2026-10-03 [Proved] The level sums are the Krawtchouk transform of the odd-count profile, 2^dim W_j = sum_w a_w K_j(w), so they see a design only through its profile. Witness: method.md The coin, The level sums are the Krawtchouk transform of the profile; lab/py/walsh-fill verb profile, every code at dim 1..4.
• 2026-10-03 [Proved] fill_F(-N) = +- void_F(N) as polynomials iff the sign is (-1)^dim and a_j + a_(dim-j) = C(dim, j) for every j, equivalently W_0 = 1/2 and W_j = 0 at every even j >= 2; such designs number 2, 4, 40, 2800 at dim 1..4, 46 of the 273 nonempty codes at dim <= 3, against 2, 4, 16, 256 self-dual ones. Witness: method.md The coin, N -> -N swaps fill and void exactly at a balanced profile; lab/py/walsh-fill verb mirror.
• 2026-10-03 [Refuted] N -> -N swaps fill and void exactly in odd dim, or exactly on the self-dual designs. Witness: dim 2 code 3 swaps, dim 3 code 1 does not, and dim 3 code 27, corners 000, 001, 011, 100, swaps without being self-dual; lab/py/walsh-fill verb mirror.
• 2026-10-03 [Proved] A root r of P_F(n) is a zero of the biased mean R(mu) = sum_j W_j mu^j at mu = 1/(2r - 1), the drift is W_1/(2 W_0), and W_1 <= I/2, I = 2^-dim sum_x s(f, x) the sensitivity total influence, with equality exactly on the down-sets, 2, 5, 19, 167 nonempty at dim 1..4. Witness: method.md The coin, The roots are zeros of the biased mean, and The drift is half the slope of the log biased mean at the fair coin; lab/py/walsh-fill verbs roots and drift.
• 2026-10-03 [Proved] The fill summed over the flips F + c by the filled corners c is 2^dim sum_j W^j N^(dim-j) with W^j = sum_(abs(S)=j) hat f(S)^2, so Stab_(1/N) is 2^-dim sum_(c in F) fill_(F+c)(N)/N^dim, and the fill averaged over all 2^dim flips is w (N/2)^dim at every side. Witness: method.md The coin, Noise stability is the fill summed over the flips by filled corners; lab/py/walsh-fill verb stability, 2730 literal checks.
• 2026-10-03 [Proved] On a nonempty design at odd side N >= 3, -log2 R(1/N) = log2(2^dim/w) - (W_1/W_0)/(N ln 2) + O(N^-2) bits per level, log2(2^dim/w) exactly at every even side: 1 bit at dim 1 code 1 and dim 3 code 23, 0.415037499 at dim 2 code 7, and dim 1 code 2, drift -1/2, pays 1.584962501 at side 3. Witness: method.md The coin, A level costs log2(2^dim/w) bits at infinite side; lab/py/walsh-fill verb entropy.
• 2026-10-03 [Proved] The rule at most t odd has 2^dim W_0 = S(dim, t), 2^dim W_1 = (t+1) C(dim, t+1) and drift (t+1) C(dim, t+1)/(2 S(dim, t)); the majority ratio is 1/2 in odd dim and 1/2 + C(dim, dim/2)/2^(dim+1) in even dim; at dim 4 at most one odd is 5/16 and at least two odd is 11/16. Witness: method.md The coin, Threshold rules; lab/py/walsh-fill verb threshold.
• 2026-10-03 [Verified] Over 0 <= t < dim <= 2048 the ratios S(dim, t)/2^dim are shared only at 1/2, by exactly the 1024 odd majorities, and at 26 values held by exactly two rules each: 2^-m at radius 1 and dim = 2^r - 1, r = 3..11, at (23, 3) and at (90, 2), their complements, and S(274, 52) = 8 S(271, 51) and S(1871, 357) = 8 S(1868, 356) with their complements. Witness: method.md The coin, Shared limit ratios; lab/py/walsh-fill verb threshold.
• 2026-10-03 [Proved] A design fills W_0 N^dim + W_dim with no term between exactly when it holds a fraction alpha of every even-weight level and beta of every odd-weight level, then (alpha (N^dim + 1) + beta (N^dim - 1))/2; in odd dim only the empty, full and parity designs, in even dim more, for example dim 2 code 11, (3N^2 + 1)/4; 4, 8, 4, 140 at dim 1..4. Witness: method.md The coin, Lower-order-free designs; lab/py/walsh-fill verb flat.
• 2026-10-03 [Refuted] The designs with no lower-order fill terms are the bent-like ones. Witness: none of the 896 bent designs at dim 4 qualifies, x_1 x_2 + x_3 x_4 having profile (0, 0, 2, 4, 0); method.md The coin, Lower-order-free designs; lab/py/walsh-fill verb flat.