stack-levels

• The correlation law of two stacked layers when the layers are different designs, or the same design at different levels, computed exactly.
• The general law it runs on: for 1-periodic f, g with Fourier coefficients hat f(k), hat g(k), 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.
• check_parity recovers the landed parity law from that sum by exact rational integration on the lcm grid: the master integral gcd(m,n)^2/(mn) at all 820 pairs to 40, reading 1/15, 1/3, 1/3 at (3,5), (3,9), (5,15), and the layer covariance (g^2-1)/(4mn), zero at all 159 coprime odd pairs.
• The same block prints the price of the tree's half-period convention: the 1-periodic odd strip p(x) = 1 iff floor(2x) is odd carries g^2/(4mn), nonzero at 159 of the 490 coprime pairs to 40, so coprime independence belongs to the odd sub-stack and not to the design.
• check_carpet computes the covariance of f_level(mx) and f_level(nx) for the base-3 Sierpinski shadow f_level(x) = 1 iff none of the first level base-3 digits is 1, exactly, at level 1, 2, 3 and all 820 pairs to 40, by cell counting on the lcm(m,n) 3^level grid.
• It checks three laws there: the covariance depends only on m/g, n/g; the product Cov * m' n' depends only on (m' mod 3^level, n' mod 3^level); and the covariance is zero exactly when 3^level divides m' or n'.
• At level 1 it checks the closed form Cov = (2/9) chi(m') chi(n')/(m'n') with chi the nontrivial character mod 3, reading -1/9 at (1,2), 1/45 at (2,5) and 0 at (1,3), on all 1600 ordered pairs.
• It prints the kernels G_2 on the units mod 9 and the row G_3(1,b) on the units mod 27, and the non-separability witness G(1,1)G(4,4) - G(1,4)^2 = 76/729.
• carpet_2d cell-counts the 2D field f_level(mu) f_level(mv) against f_level(nu) f_level(nv) on the squared grid and confirms Cov_2D = Cov_1D (M + (2/3)^(2 level)), so the two zero sets coincide.
• check_levels verifies the level split f_level(x) = prod_(i < level) f_1(3^i x) as masks at level 1..4, the nesting f_level(nx) <= f_(level-1)(3nx), and the cross-level zero law Cov(f_level(mx), f_level'(nx)) = 0 exactly when 3^level | n' or 3^level' | m', on all 14400 ordered triples with level, level' <= 3 and m, n <= 40.
• check_designs verifies two closed forms at level 1: between two base-3 one-digit-removed shadows, Cov = c/(27 m'n') with c in -6, -3, 3, 6 and zero exactly when 3 divides m'n'; between the base-2 strip and a base-3 shadow, Cov = c/(18 m'n') with c in -3, 0, 3. The strip against the Sierpinski shadow is exactly zero at all 1600 pairs.
• check_gram computes det[gcd(m,n)^2/(mn)] over odd m, n <= 2K+1 by exact elimination and matches prod J_2(k)/k^2 over the same odds at K = 1..12, J_2 the Jordan totient; the value at K = 12 is 11399736556781568/21994507608198125.

RUN

uv run python research/lab/py/stack-levels/stack_levels.py

Two seconds, prints only, writes nothing.

WITNESSES

• The general law and its parity specialisation, against the landed gcd(m,n)^2/(mn): check_parity.
• The gcd law does not survive to base 3: the carpet covariance is a character-twisted gcd law at level 1 and a mod-3^level kernel above it, and coprime layers are correlated: check_carpet, carpet_law_level1, carpet_kernel.
• Levels are not new layers: check_levels, level_product_mask.
• Which design pairs are coprime-independent: check_designs, design_law_33, design_law_23.
• The layers of the parity stack are linearly independent: check_gram, jordan2, exact_det.