parity-fill

• Computes the inked fraction of the unit square under the parity blend of the odd carpet stack: the XOR of the layers C_n(u, v) = chi_n(u) chi_n(v), n = 1, 3, ..., N, with chi_n(u) = 1 iff floor(n u) is odd.
• The parity blend is the sum of the layers folded to its parity, mrlynum::spin::Blend::Parity, which on 0/1 layers is exactly their XOR.
• Layer n = 1 is blank on [0, 1)^2, so the XOR is carried by the scales 3, 5, ..., N alone and the expansion runs over subsets of those.
• The exact route is the subset expansion. With s_n = 1 - 2 C_n in {+1, -1}, prod_n s_n = (-1)^(sum_n C_n) = 1 - 2 XOR, so fill = (1 - E[prod_n s_n])/2; expanding prod_n (1 - 2 C_n) = sum_S (-2)^|S| prod_(n in S) C_n and splitting prod_(n in S) C_n(u, v) = [prod_(n in S) chi_n(u)][prod_(n in S) chi_n(v)] factorises each mean, so E[prod_(n in S) C_n] = m_S^2 with m_S the measure of the set of u in [0, 1) where every floor(n u), n in S, is odd. Hence fill(N) = (1 - sum_S (-2)^|S| m_S^2)/2.
• m_S is exact and rational: every chi_n, n in S, is constant on each cell of the grid of side 1/lcm(S), so m_S is a cell count over lcm(S). All 2^L masses, L the number of layers, come from one pass over the grid of lcm(3, 5, ..., 21) = 14549535, which records the scale set of each cell and then sums over supersets (mass_table).
• The independent comparison gives layer n the mean p_n = ((n-1)/(2n))^2 and pretends the layers are independent Bernoulli, which would make the XOR fill (1 - prod_n (1 - 2 p_n))/2 (fill_independent).

RUN

• uv run python research/lab/py/parity-fill/parity_fill.py
• From the repo root. One core, about three seconds, numpy and fractions only.
• Domain is the exact fill at N = 3, 5, ..., 21, the literal 2D cross-check at N <= 9, a 4096 x 4096 raster check at every N, the spun readings at N = 55 on rasters R = 256, 512, 1024, 2048, and the whole-degree increment sweep 0..90 at R = 512.

WHAT IT PRINTS

• section_exact prints fill(N) as an exact fraction and a decimal for each N, beside the 4096 x 4096 raster mean and the gap, and beside the literal 2D XOR count at N <= 9.
• fill_exact sums the 2^L subset terms in integers over the common denominator lcm^2, so no rounding enters; fill_literal XORs the actual layer products on the lcm x lcm cell grid and counts inked cells, sharing no line with the expansion.
• section_independent prints the exact fill, the independent-Bernoulli fill, their difference, the two deviations from 1/2 and their ratio, then the successive ratios of each deviation.
• section_spun prints the disc fill at N = 55 for eight schedules, the resolution table for three of them, and fill against layer count with the exact and square-raster columns beside the disc readings.
• section_sweep prints the disc fill of the fixed-increment stack at every whole increment from 0 to 90, its extremes, the eyes, and two symmetry checks.

WHAT IT FINDS

• The expansion is exact and the literal count confirms it: fill(N) reads 1/9, 53/225, 3524/11025, 36284/99225, 19619/51975, 117419647/289864575, 109067744/289864575, 17006699344/45107387325, 6812188030619/19244451701475 and 1114185811873/2749207385925 at N = 3, 5, ..., 21, matching the literal 2D XOR count at all four N <= 9 with zero mismatches and the 4096 x 4096 raster mean at every N within 2.42e-04 (fill_exact, fill_literal, raster_row).
• The fill is not monotone in N: it climbs to 0.405084502 at N = 13, falls to 0.376271381 at N = 15, falls again to 0.353981924 at N = 19, and returns to 0.405275287 at N = 21 (section_exact).
• Most subsets contribute nothing: 587 of the 1024 subsets of {3, 5, ..., 21} have m_S = 0, the smallest being {3, 5, 7}, since floor(3u), floor(5u) and floor(7u) are never all odd at the same u (mass_table).
• That zero is the whole mechanism. At two coprime scales the layers are independent and the exact and independent fills agree to the digit, 1/9 at N = 3 and 53/225 at N = 5; the triple {3, 5, 7} has joint mass 0 against the independent 2/35, so the two fills part company at N = 7 and the exact fill is below the independent one at every N from 7 to 21, by -1.31e-02 up to -1.40e-01 (section_independent).
• The independent approximation decays as advertised and the exact quantity does not. The independent deviation 1/2 - fill has successive ratios 0.680000, 0.632653, ..., 0.546485 falling toward 1/2, while the exact deviation's ratios run 0.680000, 0.682044, 0.744755, 0.912184, 0.774630, 1.303566, 0.993894, 1.187399, 0.648719 and rise above 1 twice; the ratio of the two deviations grows from 1.000000 to 29.337331 over L = 2 to L = 11 (section_independent).
• On the square the raster fill of the unspun stack still climbs toward 1/2: 0.320427, 0.375175, 0.424397, 0.444069 at L = 4, 8, 14, 28 at R = 1024, the first two against the exact 0.319637188 and 0.376271381, the raster reading high by 7.9e-04 and low by 1.1e-03 (section_spun).
• Spinning pushes the fill to 1/2. On the disc at N = 55 and R = 1024 the fill reads 0.472574 unspun, 0.490510 at one degree per layer index, 0.505291 prime degrees, 0.503331 golden, and 0.500821, 0.491967, 0.500658, 0.502868 at the eyes 90/j, j = 2, 3, 4, 5; the unspun stack is the outlier at 2.74e-02 from 1/2 and no spun schedule exceeds 9.49e-03 (section_spun).
• The raster reading is good to about 3e-03: the same schedule over R = 256, 512, 1024, 2048 reads 0.478511, 0.475647, 0.472574, 0.473205 unspun, 0.485506, 0.492253, 0.490510, 0.489691 at one degree, and 0.492967, 0.494657, 0.491967, 0.492404 at the eye 90/3. The unspun offset survives that band by a factor of nine; the individual spun readings mostly do not (section_spun).
• The eyes do not stand out. Over the whole-degree sweep at R = 512 the 89 nonzero increments span 0.490165 to 0.511161, the eyes at 18, 30 and 45 reading 0.500185, 0.494657 and 0.500282, inside that band and not at either end; the extremes are 0 and 90 at 0.475647 on one side and 23 and 67 at 0.511161 on the other (section_sweep).
• Two symmetries close the sweep. Every layer is invariant under the quarter turn about the centre, since chi_n(1 - u) = chi_n(u) at odd n and C_n is symmetric in its two coordinates, so increment 90 reproduces the unspun stack exactly and increments d and 90 - d are mirror images: the sweep reads fill(90) = fill(0) = 0.475647 and matches 45 of the 46 mirror pairs to the bit, the one exception differing by 9.71e-06 in float32 (section_sweep).

WITNESSES

• stack.md THE SPUN PICTURE: the same field this study folds by parity is the one that study averages, scales n = 1, 3, ..., 55, 28 layers, disc-masked raster; the mean-blend paper coverage 0.7682 there and the parity fill here are two blends of one stack.
• stack.md SELECTING THE SCALES: the moire correlation law's coprime independence is what makes the exact and independent fills agree at N = 3 and N = 5, and the joint mass m_{3,5,7} = 0 is what makes them disagree at N = 7.
• mrlynum::spin::Blend::Parity: the blend this study measures, the sum of the copies folded to its parity.