# 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`.