Moire local limit

• 2026-10-02 [Proved] For every dim 2 code and integer gap h >= 1, the difference overlay of the level-1 designs at sides N and N + h on the unit square has local limit Phi(hu, hv), Phi(s, t) = (1 - w_0^2 - w_1^2 Lambda(s) - w_2^2 Lambda(t) - w_12^2 Lambda(s) Lambda(t))/2, Lambda(s) = 1 - 2 dist(s, 2Z), w the Walsh coefficients of the code's sign on the corners; for N >= 2h every box with sides at least l has box mean within (36/l + 20h)/N of the limit's. Witness: stack.md, The local moire limit.
• 2026-10-02 [Proved] The 16 codes of dim 2 give five local limits: 0 for codes 0 and 15, (1 - (1 + Lambda(s))(1 + Lambda(t))/4)/2 for codes 1, 2, 4, 7, 8, 11, 13, 14, (1 - Lambda(s))/2 for codes 3 and 12, (1 - Lambda(t))/2 for codes 5 and 10, (1 - Lambda(s) Lambda(t))/2 for codes 6 and 9, checked exactly at gaps 1, 2, 3, 4, 6 on 169 points and by an unfactorised raster at sides 21 and 23. Witness: stack.md, The local moire limit; lab/py/moire-local-limit.
• 2026-10-02 [Proved] At code 7 and gap 2 the local limit is H(u, v) = (1 - abs(1 - 2u) abs(1 - 2v))/2, mean 3/8, independent of N, so 2323 over 2321 looks like 23 over 21 only finer; at gap 2k it is H tiled k by k, and at gap 1 it is (1 - (1 - u)(1 - v))/2. Witness: stack.md, The local moire limit.
• 2026-10-02 [Verified] The worst 32 by 32 window error of code 7 at gap 2 against H reads 5.396e-01, 1.227e-01, 4.232e-03, 4.170e-05, 7.516e-06 at 21/23, 101/103, 2321/2323, 23231/23233, 232321/232323, and 4.845e-03, 5.080e-03, 4.117e-03, 4.091e-03 against the gap limit at 2320/2322, 2321/2322, 2321/2325, 2319/2325. Witness: lab/py/moire-local-limit.
• 2026-10-02 [Proved] For 0 < c < 1/2 the level set H = c is the four arcs abs(XY) = 1 - 2c in the centred coordinates X = 1 - 2u, Y = 1 - 2v, so no level set of H is a polygon, and the region H >= c is the unit square with its four corners cut along those arcs; its eight vertices form a regular octagon exactly at c = 1 - 1/sqrt 2 = 0.292893, where each arc bows inward from its chord by 0.089820, which is 0.108423 of the chord. Witness: stack.md, The local moire limit.
• 2026-10-02 [Verified] At 232321/232323 on windows of side 1/1024 on the diagonal, the window at the chord midpoint u = 0.146447 reads 0.249744, where a regular octagon at the level 0.292893 would put its side, and the window at the arc point u = 0.178203 reads 0.292906. Witness: lab/py/moire-local-limit.
• 2026-10-02 [Proved] The local correlation of the two parities tends to Lambda(hu) and covers [-1, 1], while int_0^1 Lambda(hu) du = 0 for every integer gap, so the limit's global mean is 2 fill (1 - fill) for every code and gap; at code 7 the local joint ink 1/2 + abs(1 - 2u) abs(1 - 2v)/4 averages to 9/16, the product of the fills, which is why the global mean 3/8 is the independent value. Witness: stack.md, The local moire limit.
• 2026-10-02 [Verified] At 2321/2323 the 32 window correlations run from -0.9367 to 0.9376, the exact global covariance of the two parities is 0 and the exact global overlay mean is 10895080998961/29070245572489, the independent value; the exact covariance is also 0 at 2320/2322, 2321/2322, 2321/2325 and 2/5391675 at 2319/2325. Witness: lab/py/moire-local-limit.
• 2026-10-02 [Refuted] The naive level-2 product formula 9/16 - (9/32) abs(1 - 2u) abs(1 - 2v) for code 7 at gap 2, which lets the second level decorrelate on its own: it misses the level-2 limit by at least 0.037617 at odd sides and 0.005552 at even sides, its largest misses on the 1/256 grid, and the direct window error against it reads 3.796e-02 at 1001/1003. Witness: lab/py/moire-local-limit.
• 2026-10-02 [Proved] For the level-2 design of code 7 at gap 2, with f = {Nx}, the second parities at sides N and N + 2 are p(Nf) and p(Nf + lambda), lambda = 4f + 4x + (N mod 2) floor(f + 2x), exactly, and the two-scale law is H2 = 5/8 - alpha(u) alpha(v) - gamma(u) gamma(v) - 2 beta(u) beta(v), alpha = abs(1 - 2u)/2, beta = (abs(1 - 2u) + w)/8, gamma = 1/4 at even N, (1 + 2w)/4 at odd N, w(u) = -2u(1 - 4u) on [0, 1/4], (4u - 1)(1 - 2u) on [1/4, 1/2], w(1 - u) = w(u). Witness: stack.md, The local moire limit.
• 2026-10-02 [Proved] For N >= 24 every box with sides at least l has level-2 overlay mean of code 7 at gap 2 within (146 + 16/l + 64/(N l))/N of the box mean of H2 of the same parity as N, by freezing the second lag on each second-level period and then x on each top period. Witness: stack.md, The local moire limit.
• 2026-10-02 [Verified] The direct level-2 window error against H2 reads 1.692e-02, 4.307e-03, 2.188e-03 at 201, 1001, 2001 and 1.621e-02, 4.505e-03, 2.064e-03 at 200, 1000, 2000, against 0.032 to 0.040 from the other parity's formula, and the closed form equals the two-scale law at 169 points at both parities. Witness: lab/py/moire-local-limit.
• 2026-10-02 [Proved] The level-2 overlay of code 7 at gap 2 has no single local limit: odd and even sides converge to two images differing by (w(u) + w(v) + 2 w(u) w(v))/8, at most 9/256 = 0.035156, attained at u = v = 3/8. Witness: stack.md, The local moire limit.
• 2026-10-02 [Proved] The level-2 limit kernel has rank exactly 3, spanned by 1, abs(1 - 2u) and w, its 3 by 3 minor at u, v in 0, 1/8, 1/4 being 99/524288 at odd and 9/524288 at even sides, so no formula a + F(u) G(v) exists, and it is not a function of abs(1 - 2u) abs(1 - 2v), reading 121/256 at (0, 3/8) and 63/128 at (1/4, 1/4) at odd sides. Witness: stack.md, The local moire limit; lab/py/moire-local-limit.
• 2026-10-02 [Verified] The global law of the four level-2 parities is uniform at both parities and the global mean of H2 is 63/128 = 2 (9/16)(7/16), the independent value. Witness: lab/py/moire-local-limit.
• 2026-10-02 [Verified] The indicial equations model of Amidror and Hersch puts the centrelines of the (1,-1)-moire of two parallel gratings of periods T_1, T_2 at x (T_2 - T_1) = T_1 T_2 p, a band period T_1 T_2/abs(T_2 - T_1), which is 2/h for the parities at sides N and N + h, read at source in section 2. Witness: Amidror Hersch 2010.