The parity fill

• 2026-09-09 [Proved] The parity blend of the odd carpet stack has an exact rational fill: with s_n = 1 - 2 C_n the signed layer, the XOR of the layers is (1 - prod_n s_n)/2, and expanding the product over subsets S of the scales and splitting prod_{n in S} C_n(u, v) into its two coordinates gives fill(N) = (1 - sum_S (-2)^|S| m_S^2)/2 with m_S the measure of the set of u where every floor(nu), n in S, is odd, a cell count on the grid of lcm(S); the fill reads 1/9, 53/225, 3524/11025, 36284/99225, 19619/51975, 117419647/289864575, 109067744/289864575, 17006699344/45107387325, 6812188030619/19244451701475, 1114185811873/2749207385925 at N = 3, 5, ..., 21, matching a literal 2D XOR count on the lcm cell grid at every N <= 9 and a 4096 raster at every N within 2.42e-04; coprime layers being independent, the exact fill equals the independent-Bernoulli fill (1 - prod_n (1 - 2 p_n))/2, p_n = ((n-1)/(2n))^2, at N = 3 and 5, and the vanishing triple mass m_{3,5,7} = 0 against the independent 2/35 breaks the agreement at N = 7. Witness: lab/py/parity-fill fill_exact, fill_literal, fill_independent, mass_table.
• 2026-09-09 [Verified] The parity fill is not monotone in the layer count, falling from 0.405084502 at N = 13 to 0.376271381 at N = 15 and 0.353981924 at N = 19, because 587 of the 1024 subsets of the odd scales 3..21 carry joint ink measure zero, the smallest being {3, 5, 7}, whose floors are never all odd at one point; the exact fill sits below the independent approximation at every N from 7 to 21 with the deviation ratio growing to 29.337331; every odd layer is invariant under the quarter turn about the centre (chi_n(1 - u) = chi_n(u) at odd n), so the fixed-increment parity fill obeys fill(d) = fill(90 - d) (Proved), 45 of 46 mirror pairs bit-equal on the raster; the square raster climbs 0.320427, 0.375175, 0.424397, 0.444069 at L = 4, 8, 14, 28. Witness: lab/py/parity-fill mass_table, section_independent, section_sweep, fill_raster.
• 2026-09-09 [Conjecture] The parity fill of the odd carpet stack tends to 1/2: the raster climbs toward it while the exact deviation at L = 11 is still 9.47e-02 and non-monotone, no rate derived; the moire law controls pairs while the expansion needs every subset mass, and which subsets of odd scales have m_S = 0 is a covering question about the intervals [k/n, (k+1)/n) with k odd, open. Witness: lab/py/parity-fill fill_raster, mass_table.
• 2026-09-09 [Refuted] The rational-increment eyes stand out in the parity fill: at N = 55 and R = 512 the 89 nonzero whole-degree increments span 0.490165 to 0.511161 and the eyes at 18, 30 and 45 degrees read 0.500185, 0.494657 and 0.500282, inside the band at neither end, the one outlier being the unspun stack at 0.475647; coincident-class layers carry different scales and share a sublattice, not a picture, so nothing cancels; the independent approximation's O(2^-L) decay does not predict the exact deviation either, whose ratios run 0.680 to 1.304 and exceed 1 twice. Witness: lab/py/parity-fill section_sweep, section_independent.