# 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.