# Levels and designs

- 2026-09-09 [Proved] The product law: for 1-periodic `f, g` the mean of `f(mx) g(nx)` over the unit interval is `sum_j hat f(j n') hat g(-j m')` with `g = gcd(m,n)`, `m' = m/g`, `n' = n/g`, and the covariance is the same sum over `j != 0`; the parity specialisation returns `1/15, 1/3, 1/3` at `(3,5), (3,9), (5,15)` and `gcd^2/(mn)` at all 820 pairs to 40. Witness: lab/py/stack-levels `check_parity`.
- 2026-09-09 [Proved] The base-3 carpet stack correlates 3-adically, not by gcd: the shadow at level `level` has `hat f_level(k) = (-1)^k 2^level sin(pi k/3^level) prod_{i <= level} cos(2 pi k/3^i)/(pi k)`, zero exactly when `3^level | k`, so `Cov(f_level(mx), f_level(nx)) = G_level(m' mod 3^level, n' mod 3^level)/(m' n')` with `G_level(a, 3^level - b) = -G_level(a, b)` and `G_level(0, b) = 0`; at level 1 the closed form is `(2/9) chi(m') chi(n')/(m' n')` with `chi` the character mod 3, exact at all 1600 ordered pairs to 40; the covariance vanishes whenever `|v_3(m) - v_3(n)| >= level`, and the 2D field carries the same zero set through `Cov_2D = Cov_1D (M + (2/3)^(2 level))`. Witness: lab/py/stack-levels `carpet_law_level1`, `carpet_kernel`.
- 2026-09-09 [Proved] Levels are products, not layers: `f_level(x) = prod_{i < level} f_1(3^i x)`, so a layer at level `level` and scale `n` is the product of level-1 layers at `n, 3n, ..., 3^(level-1) n`, with `f_level(nx) <= f_{level-1}(3nx)` pointwise and gap measure `(2/3)^(level-1)/3`; level `level-1` at scale `3n` is not redundant against level `level` at scale `n`, `Cov(f_2(x), f_1(3x)) = 4/27` against `Cov(f_1(x), f_2(3x)) = 0`. Witness: lab/py/stack-levels `check_levels`.
- 2026-09-09 [Proved] Coprime independence is the half period's: the 1-periodic odd strip `p(x) = 1` iff `floor(2x)` odd carries covariance `g^2/(4mn)`, nonzero at 159 of the 490 coprime pairs to 40 and `1/60` at `(3, 5)`, while the tree's `chi_n` at odd `n` carries `(g^2 - 1)/(4mn)`, zero at every coprime pair; `p(nx) = chi_{2n}(x)`, so this is the landed law read at even scales, a sharpening and not a break, and the prime detector is a statement about odd scales under the half-period sampling. Witness: lab/py/stack-levels `check_parity`.
- 2026-09-09 [Proved] Design pairs at level 1: two base-3 one-digit-removed shadows correlate by `c/(27 m' n')` with `c` in `-6, -3, 3, 6`, zero exactly when 3 divides `m' n'`, so no two base-3 designs are coprime-independent; a base-2 strip against a base-3 shadow correlates by `c/(18 m' n')` with `c` in `-3, 0, 3`, and the odd strip against the Sierpinski shadow is exactly zero at all 1600 scale pairs, because the centred middle-thirds indicator is even and the centred strip odd under `x -> -x`. Witness: lab/py/stack-levels `design_law_33`, `design_law_23`.
- 2026-09-09 [Proved] The parity layers are linearly independent: the Gram matrix `gcd(m,n)^2/(mn)` over odd `m, n <= 2K + 1` has determinant `prod_{k odd <= 2K+1} J_2(k)/k^2 = prod_{k odd} prod_{p | k} (1 - p^-2)`, from `k^2 = sum_{d | k} J_2(d)` and a unitriangular incidence factorisation on the factor-closed odd set, exact at `K = 1..12`, `11399736556781568/21994507608198125` at `K = 12`, positive. Witness: lab/py/stack-levels `check_gram`.
- 2026-09-09 [Conjecture] The carpet zero law is exact at every level, `Cov(f_level(mx), f_level(nx)) = 0` iff `3^level | m' n'`, and the cross-level law `Cov(f_level(mx), f_{level'}(nx)) = 0` iff `3^level | n'` or `3^{level'} | m'` likewise: Proved at `level = 1` from the closed form, Verified only over `m, n <= 40` and `level <= 3`, 14400 ordered triples with zero breaches and zero unpredicted zeros. Witness: lab/py/stack-levels `carpet_kernel`, `check_levels`.
- 2026-09-09 [Refuted] The gcd law and coprime independence hold outside the parity family: at base 3 level 1 coprime layers carry `(2/9) chi(m') chi(n')/(m' n')`, witness `Cov(f_1(x), f_1(2x)) = -1/9`, a coprime pair, anticorrelated; the zero set is `v_3(m) != v_3(n)`, not coprimality. Witness: lab/py/stack-levels `carpet_law_level1`.
- 2026-09-09 [Refuted] The carpet kernel is separable above level 1: `G_2(1,1) G_2(4,4) - G_2(1,4)^2 = 76/729`, so no `theta(a) theta(b)` form exists at `level >= 2` and the level-1 character law does not lift. Witness: lab/py/stack-levels `check_carpet`.