# 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.