Hexagon Moire

• The stack of diagonal cube slices, one hexagon per odd side n, for the carpet, net, tree and void designs.
• Cut ink of every layer as an exact rational against the four closed forms, and the void centre cell.
• Exact area-weighted Pearson correlations of layer pairs on the full hexagon: the doubling sign law, its Richardson limit, adjacent and gcd-echo pairs, the other families.
• The pair limit in closed form: the cut cell law s_y = s_x + s_z + w, the phase map n = a m + c, and the exact rational correlation of every family against every phase map, with the finite-layer gap beside it.
• The ideal-frame stack at 3601 samples per axis: quarter-line bands, the one-sided plateau at X = 1/4, the void star.
• The twisted triangle averages and the coarse crosshair law at 200003 samples.
• The ghost star of the rendered carpet stack on a 1200 by 2399 raster, and its log-corrected decay in the ideal and the lattice frame.
• The cell frame of the ghost star: the arm's rational ink law to n = 2001, the decay ladders at L = 0 mod 4, L = 2 mod 4 and odd L out to L = 6400, and the band half-widths with their held-out predictions.
• The width family in closed form: the derived tail weight E(K) against its 8-periodic run, the per-layer identity in exact rationals at 14 distinct half-widths broken out by n mod 8, and the summed ladders with the constant, both 1/L^2 branches and the sliding window's own L residue trap.
• The constants from the mrlynum series and the partial character sums at N = 53 and 55.
• The one-layer law: the six ink-law coefficients of each family, the summed identity against the counted hexagons in exact rationals, and the approach in both classes of the layer count.
• Every domain is the source's own; nothing was shrunk. Under a minute.

RUN

• bash scripts/cargo.sh cargo run --release -p hexagon-moire from mrlyprod/

WITNESSES

• hexagon.md the cut ink of one layer: the ink laws match 28/28 layers in exact rationals for all four designs at every odd n <= 55, and the void centre cell is ink in 28 of 28 against 14 for the other three.
• hexagon.md layer pairs and the doubling law: the sign law holds on all 18 pairs from (3,5) to (601,1201); the full-hexagon blind pairs read (5,9) = -0.14179450 and (5,7) = -0.08542646.
• hexagon.md layer pairs and the doubling law, the magnitude: Richardson r_inf from -0.11715991 to -0.11711630, branches 0.1171270 and 0.1171274, against 253/2160 = 0.11712963 and 19/162 = 0.11728395.
• hexagon.md the other pair families: tree and void doubling at (201,401) +0.00049938 and +0.00107260; tree adjacent -0.07016649, void adjacent +0.00786627; echo (67,201) carpet +0.21473346, tree +0.14984772, void +0.07721358.
• hexagon.md the quarter line and the crosshairs: plateau 0.12210, 0.12340, 0.12412, 0.12446 at N = 151, 301, 601, 1201; the X = 1/4 bands +0.066465 and -0.090881 at N = 55.
• hexagon.md the quarter line and the crosshairs, the coarse model: A = 1/3 excess -0.083726 against -0.083333 at N = 5555; A = 1/7 -0.058065 at N = 55 against -0.036036 at N = 5555.
• hexagon.md the void star: arms 0.513667, 0.486492, 0.491844, 0.491525 and the Z arm 0.399416 over a mean 0.278670.
• hexagon.md the twist that kills the ray family: the twisted average falls like 1/N at every x, the A = C excess -0.003580 at N = 55 and -0.000052 at N = 5555.
• hexagon.md the ghost star: star minus background -0.09385, -0.03102, -0.01756 at 5, 28, 56 layers; excess times L runs -0.7779 to -1.2519 in the ideal frame and -1.0212 to -1.3645 in the lattice frame from L = 28 to L = 400, ln L slopes -0.18 and -0.125.
• hexagon.md the ghost star, the cell frame: the arm ink law 1/2 + chi_8(n)/(2n) matches 1001/1001 layers at odd n <= 2001 and the three arms agree in 111/111 at odd n <= 221; excess times L + (ln L)/4 runs -0.2939128437 at L = 28 to -0.2937605886 at L = 6400 against the closed form -0.2937605857, ln L slope -0.250000.
• hexagon.md the ghost star, the parity split: the residual times L^2 runs -0.11975831 to -0.11979188 at L = 100 to 6400 with ratio 4.00 per doubling against -23/192 = -0.11979167; the L = 2 mod 4 ladder reads +0.13020825 at L = 3202 against +25/192 = +0.13020833; the odd ladder misses by +1/8 and what is left times L reads -0.250708, -0.250244, +0.249934, +0.249984, -0.250004 at L = 27, 99, 401, 1601, 6399.
• hexagon.md the ghost star, the width family: the twelve swept half-widths read -1/4, -1/6, -1/10, -3/28, -5/36, -3/22, -3/26, -9/68, -5/42, -13/100, -17/132 and -33/260 at L = 800, each the derived rational -(K + b)/(4(2K + 1)) and each found again by a blind best-rational search.
• hexagon.md the ghost star, the width family checked: W = 0 to 12 recomputed to L = 3200 land within 5.1e-8 with every gap falling by four per doubling; the held-out W = 14, 18, 28, 36 land within 4.0e-8 of -7/60, -5/38, -7/58 and -9/74 at L = 1600; W = 1 and W = 3 read -0.249999 and -0.166666, the same point set and the same rational as W = 0 and W = 2.
• hexagon.md the constants: pi/4 + pi^2/32 = 1.0938233009 with 1.0826656664 at N = 55; (pi + pi^2)/16 = 0.8131998159 against pi^2 ln2/(7 zeta3) = 0.8130217042; G/8 = 0.1144956993 with 0.1144757884 at N = 55 and G/8 - 1/8 with -0.0104828892 at N = 53.
• hexagon.md the constants: L(2, chi_-3) = 0.7813024129 printed from its series and matched by nothing above.
• hexagon.md the one-layer law: M(mean - A - c eps) = -B S - d s1 + e s3 - f s2 holds against the counted hexagons in exact rationals at every layer count to N = 55, all four families, 7 of 7 in each class n = 1, 3, 5, 7 mod 8.
• hexagon.md the one-layer law, the limits: carpet G/8 and G/8 - 1/8, net -G/8 and 1/8 - G/8, tree pi/48 + pi^2/48 + G/6 and void (pi + pi^2)/16 at both classes of the layer count.
• hexagon.md the one-layer law, the approach: at M = 3200 and 3201 the gap times M^2 reads -0.01562500 and +0.01562500 against -+1/64, and the gap times M reads -0.18750000 and -0.06250000 for the void against -3/16 and -1/16, -0.06250651 and -0.02082683 for the tree against -1/16 - 1/(48M) and -1/48 + 1/(48M), and -0.31249999 for the carpet split against -5/16.
• hexagon.md the width family: excess_W(n) = kappa chi + (m + q chi_8(n))/n + chi/(8 n^2) matches the counted band in exact rationals 1354 of 1354, at 14 distinct half-widths from W = 0 to 64 and every odd n from K to 201, with the four classes n = 1, 3, 5, 7 mod 8 counted apart; the nine odd rows repeat their even neighbours and are not counted twice.
• hexagon.md the width family summed: C_W = m(ln2 + gamma/2) + q L(1, chi_8) - G/8 + Delta_W holds to 1e-9 at L = 1600 for seven widths, W = 0 returning -0.2937605857 and Delta_W = 2/7 at W = 6.
• hexagon.md the width family, the two 1/L^2 branches: residual times L^2 reads -0.1197915 and +0.1302082 at W = 0 and +0.0027574 and +0.0174632 at W = 16, against -q/4 + 1/64 + m/48 and +q/4 + 1/64 + m/48.
• hexagon.md the width family, the L residue trap: the sliding window from L/2 to L reads -0.24999980 at L = 1600 and -0.43078703 at L = 1602, against m/2 = -1/4 and the limit m/2 + kappa/ln 2 = -0.43033688.
• hexagon.md the width family, the derived tail weight: E(K) = #(|j| <= K, j = 3,4,5 mod 8) + floor((K+2)/4) - K agrees with the 8-periodic run 0 -1 -1 0 1 2 2 1 at 201 of 201 values K = 0..200.
• hexagon.md the exact doubling constant: the limit integral returns covariance 253/9216 and variance 15/64 at all four branches, so r = 253/2160, with (301,601) reading -0.11745304 and (601,1201) -0.11729091, gap times m at -0.0974 and -0.0969.
• hexagon.md every pair limit: carpet and net read -+253/2160, -11/135 and +29/135, the tree 0, -61/864 and +4/27, the void 0, +7/864 and +2/27; the tree and void doubling covariances are exactly 0.
• hexagon.md every pair limit, the finite-layer gap: (249,251) reads -0.08150224 against -11/135 and (99,297) reads +0.21476417 against +29/135, gaps 2.1e-5 and 5.1e-5 with gap times m at -0.0052 and -0.0050.
• hexagon.md every pair limit, the second residue class: (103,205) reads +0.11914004 and (203,405) reads +0.11814528 against +253/2160, gap times m at +0.207 and +0.206, about twice the m = 1 mod 4 class's.