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.