Spin Census

• Runs the four unrun questions of spin plus the Gaussian Farey, and prints only.
• The spin mass M(r) is read twice: exactly, as the integer shell histogram of filled cells about a fixed point, and through mrlynum::spin::profile with mass_within, the two agreeing to 1.7e-6 on the total and to 0.5% on partial radii. The hole about the raster centre is measured on cell rectangles in exact integer arithmetic, never on cell centres, which would return hole + 1/2 whatever the hole.
• Every fixed point is the attractor of one filled digit d, the point d/2 of the unit square, about which the design is exactly self-similar; the corner digit gives the widest window, r <= side, the centre digit only r <= side/2.
• The ripple is the residual ln M(r) - (log(fill)/log 3) ln r taken exactly, never fitted, folded into 24 bins of log_3 r over a whole number of periods so every bin carries the same sample count. Its drift bar is the same fold on the first half of the window against the second, computed from the code itself.
• Level 6 and level 7 for the census over all 256 codes with a filled corner digit, de-duplicated to transpose classes because the corner ripple is a class function; the transpose control asks the same code and its mirror for the same ripple and gets it to the last bit. The window's whole periods are counted by repeated multiplication, never by a logarithm, which floors to the wrong period at level 8.
• The powder is the arithmetic ring average of |F(k)|^2 over 240 logarithmic bins, band 3 pad/side to pad/8 in frequency index, at level 7 with pad 4096 and again with pad 8192 on three codes; the instrument spread is a three-period fit window slid a quarter period at a time across the band, not a split of the band in two, which understates it by an order.
• The spin spectrum is mrlynum::spin::harmonics at levels 1 and 2 over all 511 nonempty codes, orders m = 0..12, compared against the 101 orbits of the square group.
• The shape reading reduces the spin spectrum to exact integers. At level every P_m is a quadratic form in the indicators of the render's cells whose Gram matrix commutes with the raster's symmetry group, so P_m is a linear functional of the pair census Phi_level: the number of filled cell pairs in each orbit of that group on pairs. Equal pair censuses force equal P_m at every order, every ring count and every truncation, in exact arithmetic and with no transform run.
• The pair census is the whole instrument at level 1: base 3 plane has 11 classes, base 5 plane 55, base 3 dim 3 24 under the order-48 cube group; at level 2 the render is the Kronecker square and the counts are 461 and 24805.
• The control is the full 511. Codes de-duplicate to orbits by canonical form and the census reports the number of distinct Phi_1 and Phi_2 values against the Burnside orbit count, which the same pass computes from the cycle index and never from the sweep.
• The reduction is checked against the instrument rather than assumed: the level-1 coefficient matrix is solved from independent censuses by elimination, then predicts mrlynum::spin::harmonics at 1024 rings and m = 0..12 on all 511 codes, printing the worst residual and the rank of the 13 orders against the 11 classes. It holds for a constant-valued 0/1 render on a raster of side base^level; the spectrum count it is compared against is a greedy first-match bucketing at tolerance 1e-9.
• Base 5 and dim 3 run Phi_1 in full over all 2^25 and 2^27 codes, storing canonical representatives only, and run Phi_2 over the colliding groups inside a budget-capped weight window the generator prints; the cap did not bind and the window covered every group.
• The sponge shadow counts lattice lines in direction (a,b,c) meeting the sponge at level, as classes of filled cells under x -> x cross v, with the solid cube in the same direction as the exact ceiling.
• The Gaussian Farey counts radii new at scale n three ways: the direct union over reduced squared radii, the square-free rule, and the Mobius identity over the radical.

RUN

• CARGO_BUILD_JOBS=4 cargo run --release -p spin-census
• About thirty seconds; prints only, writes nothing. Peak memory is the pad-8192 transform, about 1.1 GB.
• CARGO_BUILD_JOBS=4 cargo run --release -p spin-census -- shape runs the shape reading, the one verb group the default pass leaves out.
• About two minutes and 0.4 GB, the peak being the base-5 sweep holding 4.2 million packed censuses; the dim 3 sweep visits all 2^27 codes and keeps 2.85 million.

WITNESSES

• spin.md the spin dimension rows: slopes 1.465054, 1.649432, 1.783588, 1.761814, 1.897854, 1.879522, 2.000100 about the corner at level 6 against the exact log(fill)/log 3.
• spin.md the centre hole: codes 239 and 495 have first occupied radius 41.000 against side/6 = 40.5 at level 5, mass zero inside.
• spin.md the acid test: 127 against 239 ripple gap 0.11984 on drift bar 0.04126; 255 against 495 gap 0.12042 on bar 0.01461; the solid 511 ripple flat at 0.00277 under its own bar 0.00578.
• spin.md the ripple census: 256 codes read at levels 6 and 7, transpose control 0.00e0, 13 equal-fill transpose class pairs inside their own bar at level 6 and 6 at level 7 with ratios 0.71 to 0.95, closest 7 and 273 at 0.01371 with swings 0.01114 and 0.01217.
• spin.md the powder: pad 4096 slopes -1.37986, -1.51762, -1.73012, -1.83071, -1.97886, -2.00433, slide spreads 0.1652 to 0.4405, pad 8192 moving 127, 255, 495 to -1.81607, -2.03225, -2.12289; the fractal slide floor -2.27308 against the solid's -2.75781.
• spin.md the centre hole: four times the squared distance to the nearest filled cell 6561 = (side/3)^2 for 239 and 495 at level 5, 12802 for 79.
• spin.md the spectrum: 0 isospectral pairs, 101 distinct spectra against 101 nonempty square classes.
• spin.md the shadow: (0,0,1) sponge 8, 64, 512, 4096, 32768 against cube 9, 81, 729, 6561; (1,1,1) sponge and cube both 19, 217, 2107, 19441, 176419.
• spin.md the Gaussian Farey: 2, 3, 9, 11, 22, 18, 40, 38, 55, 52, 91, 64, 123, 97, 128, 126, 199, 136, 243, 180, the Mobius identity to n = 64, the radical-six ratios 0.56250 to 0.63801.
• spin.md the shape reading at base 3: 11 pair classes at level 1 and 461 at level 2, 97 level-1 censuses against 101 orbits, the homometric pairs 45-105, 61-121, 78-102, 94-118 at level-1 spectrum gaps 1.30e-16, 1.03e-17, 6.51e-17, 1.64e-16 and level-2 gaps 0.151, 0.0689, 0.253, 0.105; P_m read at level 1 alone, bucketed greedily at 1e-9, gives 97 spectra, the count the census predicts.
• spin.md the reduction: the level-1 coefficients solved from 11 independent censuses reproduce mrlynum::spin::harmonics on all 511 codes at worst relative residual 1.14e-14, coefficient rank 9 of 11, the six odd orders at the cap 3 that half-turning one member of a pair forces and the seven even orders at 6 of a possible 8.
• spin.md the completeness away from base 3: 3993511 level-1 censuses on 4211743 base-5 orbits with 204856 ties over 423088 orbits, and 1461693 on 2852287 cube orbits with 757066 ties over 2147660; every tie broken at level 2, the window covering every group out to weights 21 and 24, with no pair surviving, and the canonical counts matching the Burnside averages 4211744 and 2852288.
• mrlynum::spin::mass_within and the crate test the_spin_mass_scales_by_the_fill_about_a_filled_corner; the window regression test the_window_counts_whole_periods_at_every_level.