moire-local-limit

• The local limit of the overlay of two dim 2 designs at sides N and N + h, both on the unit square, under the difference filter (a point is on when exactly one design inks it), computed exactly from window integrals.
• The window is a box I x J; the box mean of a product picture is the product of the 1D means, so every box mean is a bilinear form in the joint law of the digit parities on I and on J.
• law1 computes that joint law exactly on W windows by merging the cut points i/N, j/(N + h) and k/W over the common denominator N (N + h) W; law2 does the same with the four level-2 parities on the denominator N^2 (N + 2)^2 W.
• check_walsh checks the Walsh form of the limit, (1 - sum_S w_S^2 Lambda(hu)^[1 in S] Lambda(hv)^[2 in S])/2 with Lambda(s) = 1 - 2 dist(s, 2Z), against the lag-law kernel in exact rationals for all 16 codes at gaps 1, 2, 3, 4, 6 on 169 points, and prints the five classes the codes fall into.
• check_direct rasters the overlay at sides 21 and 23 on the full 483 x 483 grid, with no factorisation, and matches the factorised box means exactly for all 16 codes on 7 by 7 windows.
• ladder runs code 7 at gap 2 on 32 by 32 windows of side 1/32 at the pairs 21/23, 101/103, 2321/2323, 23231/23233 and 232321/232323, printing the worst window error against H(u,v) = (1 - |1 - 2u| |1 - 2v|)/2, the error scaled by N l and the proved bound (36/l + 20h)/N.
• gaps runs the gap-h limit at 2320/2322 (gcd 2), 2321/2322 (gap 1), 2321/2325 (gap 4) and 2319/2325 (gap 6, gcd 3), with the exact global covariance of the two 1D parities.
• obstruction prints the window correlation of the two parities against the window mean of Lambda(2u), and the exact global overlay mean at 2321/2323.
• octagon prints the regular chord octagon level 1 - 1/sqrt 2, the sag of its curved sides, and two box means at 232321/232323 on windows of side 1/1024 on the diagonal, one at the chord midpoint and one on the arc.
• level2 computes the level-2 limit law in exact rationals by integrating the lag law over the top-level phase, checks it is quadratic on each quarter, runs the direct box means at 201, 1001, 2001 and 200, 1000, 2000, checks the closed form H2 = 5/8 - alpha alpha - gamma gamma - 2 beta beta at 169 points, prints a nonzero 3 by 3 minor, the parity gap 9/256, the distance to the naive formula and the kernel rank 3.

RUN

uv run python research/lab/py/moire-local-limit/moire_local_limit.py

Eleven seconds, prints only, writes nothing; the level-2 runs at 2001 and 2000 take six of them.

WITNESSES

• The local limit on the lag torus for all 16 codes and every gap: check_walsh, check_direct, law1, kernel1, limit_law1.
• The code 7 limit H, its mean 3/8 and the rate: ladder, gaps.
• Local correlation against global decorrelation: obstruction.
• The curved octagon: octagon.
• The level-2 limit, its closed form, its parity split and the death of the naive formula: level2, law2, limit_law2, closed_form.