Flat carpet stack

• 2026-08-28 [Proved] The moire correlation law: for odd m, n the mean of s(mu) s(nu) with s(x) = (-1)^floor(x) is exactly gcd(m,n)^2/(mn) (for general integers it needs m/g and n/g both odd, else the integral is 0), and the Pearson correlation of the 1D parity indicators is exactly (gcd^2 - 1)/sqrt((m^2 - 1)(n^2 - 1)), so the correlation is exactly 0 if and only if the scales are coprime, and zero covariance is independence for Bernoulli pictures; exact rational integration over all odd pairs to 99 matches the closed forms to 5.6e-17. Witness: moire-correlation-laws.
• 2026-08-28 [Proved] The stack is an exact prime detector: an odd n >= 3 is prime exactly when its carpet is uncorrelated with every earlier carpet; over odd 3..199 all 45 primes sit at exactly 0 and all 54 composites strictly positive, minimum 0.0517383 at n = 169 = 13^2; the finite-window corollary "the zero-redundancy layers of the 1..55 stack are the primes above 55/3" is a window artifact and not the statement. Witness: moire-correlation-laws.
• 2026-08-28 [Proved] Pi cancels out of every visible brightness of the stack: ray strengths, crosshair steps, hot-spot values, layer correlations and per-layer means are all rational, because the square wave's (4/pi)^2 meets the odd Basel sum pi^2/8; the diagonal is exactly twice the background in paper coverage in the limit, and the anti-diagonal is its pixel-for-pixel copy by the palindrome symmetry. Witness: moire-correlation-laws.
• 2026-08-28 [Proved] The moire rays obey a 2-adic law, not a Farey law: the slope-one family at offset a/b carries (-1)^a/b^2 for odd b and exactly nothing for even b, and the slope q/p ray through the origin carries exactly 1/(pq), the same number as the correlation of grams p and q. Witness: moire-correlation-laws.
• 2026-08-28 [Verified] The stack fades at the random rate in L^2: RMS contrast falls as c/sqrt(L) in the layer count L with c^2 = lim L * Var, c = 0.522, a constant factor 1.2054 above independent layers, so "does not fade like random noise" is false in L^2; the exact variance is a finite rational sum at every L. Witness: moire-correlation-laws.
• 2026-08-28 [Proved] Coprime independence holds exactly in all four flat families: carpet, net, tree and void layer pairs have covariance identically 0 at every coprime odd pair, checked exhaustively to 201 and in exact rationals at (3,5), (5,7). Witness: moire-correlation-laws.
• 2026-08-28 [Proved] The void flat stack obeys a gcd-to-the-fourth law, Pearson_void(m,n) = (g^4 - 1)/sqrt((m^4 - 1)(n^4 - 1)) (void being the pure pair field (1 + s(mu) s(mv))/2), tree obeys (g^2 - 1)/sqrt((m^2 - 1)(n^2 - 1)), net's closed form is carpet's under m, n -> -m, -n with slightly larger correlations, and the gcd echo orders tree > net > carpet > void; all four match exact lcm-grid counting on (3,9), (5,15), (9,15) with zero error. Witness: moire-correlation-laws.
• 2026-08-28 [Verified] The flat variance constants reduce to two gcd sums, S2(N) = sum g^2/(mn) and S4(N) = sum g^4/(m^2 n^2) over odd pairs: lim L * Var is the S2/(2N) limit for tree, the S4/(2N) limit for void (0.2768062) and S2/(4N) + S4/(8N) for carpet and net (their difference dying like ln^2 N / N), with the identity Var_L(carpet) = Var_L(tree)/2 + Var_L(void)/4, measured to 5 digits. Witness: moire-correlation-laws.
• 2026-08-28 [Proved] The S4 limit is a theorem: lim S4(N)/N = (16/31) T/zeta(5) with T = sum_{k,l odd} 1/(k^2 l^2 max(k,l)) = 1.1122336970, value 0.5536124372, measured 0.5536124482 at N = 3 * 10^6, by a bounded coprime tail plus Mobius over odd moduli, with an independent Jordan-totient recomputation. Witness: moire-correlation-laws.
• 2026-08-28 [Conjecture] The rendered diagonal-to-background ratio reads 2.1189 at N = 55 walking to 2.0000252 at N = 10^6, and the rendered grey ratio is 16/9 because ink is 17.
• 2026-08-28 [Conjecture] A primitive integer line alpha u + beta v = gamma is a ray iff alpha and beta are both odd, and the crosshairs at u = a/b carry 1/(4b), a factor b stronger than the diagonal family.
• 2026-08-28 [Conjecture] Pi survives only in the census and the decay arithmetic: distinct rays are indexed by odd-denominator reduced fractions, counted by sum phi(b) ~ (2/pi^2) B^2 through the odd-prime Euler product 8/pi^2, and the pairwise gcd sum obeys S2(N)/N -> pi^2 ln 2/(7 zeta(3)) = 0.8130217 (6 digits at N = 10^6 two ways), whence the tree constant lim L * Var = pi^2 ln 2/(14 zeta(3)) = 0.4065108521 and the carpet and net constant 0.2724570.
• 2026-08-28 [Conjecture] The sup-norm never fades: diagonal, crosshairs and the four brightest points (paper 9/14 at the inner-thirds crossings, grey exactly 170) hold their values forever while their width shrinks like 2/(N + 1) on an exactly triangular profile.
• 2026-08-28 [Conjecture] Chaining 27 8-bit blends and truncating each step shifts the rendered stack by six grey levels, the saved image's mean 61.9 against the true 67.8 and its brightest pixel 163 against 170; quote paper-coverage fractions, never absolute greys.
• 2026-08-28 [Conjecture] The 2D ray law transfers to the void flat stack at double contrast: void has no crosshairs (no single-wave terms) but carries the full odd/odd 2-adic diagonal web at twice the carpet's strength (offset a/b: (-1)^a/(2b^2); slope q/p through the origin: 1/(2pq)), with u = v and u + v = 1 solid ink, a black X on mid-grey, the negative of the carpet's paper X.