Design Census

• The three design censuses in this tree, rerun in full.
• Coprime: every design with fill >= 2 at base 2 in dimensions 2 and 3 and at base 3 in dimension 2, the Menger sponge and two base-6 samples; the measured coprime density at the deepest level inside a budget of 200000 points against the predicted B(F) / zeta(dim) * prod (1 - p^-dim)^-1 for F the filled digits, the exact base-local identity at level 4, the spanning index, the split of the spanning lines by log_base(fill) against one, and how many lines widen their gap to the prediction at some level.
• Fill: one row per cube-group orbit at bases 2 and 3 in dimensions 2 and 3, base 3 dimension 3 by the four named designs only; level-1 fill and void at sides 1..12, the fill quasi-polynomial, orbit size, popcount and three algebraic degrees, every row rendered cell by cell and checked against the closed form and the level law.
• Fill classes: the parity-rule generator against the popcount-profile closed form over every base-2 design at dim 2, 3, the distinct fill sequences at dim 1..4 against Prod_k (1 + C(dim, k)), the 16 published terms of A129824, and the per-dimension counts table with A000616 by class sums to dim 6.
• The base-2 cube group counted three ways at dim 1..4: orbit walk, Burnside average, class sums.

RUN

• CARGO_BUILD_JOBS=4 cargo run --release -p design-census
• Half a minute on 8 cores; writes sequences.csv and counts.csv beside this file.

WITNESSES

• coprime.md:29 the exact identity at level 4 on all 763 lines, zero failures.
• coprime.md:35 code 34376528265 at base 6 with fill = 8, fill_2 = 3, fill_3 = 2, fill_6 = 1, bracket 1/2.
• coprime.md:101-110 the family table 11/5/3, 247/149/27, 502/365/175, 1/1/1, 2/2/2, total 763 designs, 522 spanning, zero flagged; the sponge at level 4 measuring 0.7719 against 0.8207; fill 3 reaching level 11.
• coprime.md:111-113 219 distinct term-vectors among the 502 base-3 designs and 175 spanning; 490 spanning lines above dimension one, 32 at exactly one, four more at index 3, none below; 426 lines whose gap to the prediction widens at some level.
• core.md:47-51 fill at side 3 of 8 of 9, 20 of 27 and 12 of 27, with 20^2 = 400 at level 2.
• core.md:81-86 and core.md:156-163 carpet and net share one orbit at base 2, rep code 7 in dimension 2 and 23 in dimension 3 so only the net label survives, and the six 2D classes with orbit sizes 1, 4, 4, 2, 4, 1 and degrees -1, 2, 1, 1, 2, 0.
• core.md:96-106 base-2 orbit counts 3, 6, 22, 402 agreeing three ways at dim 1..4.
• core.md:243-248 the census this study writes: representatives, orbit sizes, fill polynomials, three degrees, per-dimension counts.
• method.md:109-112 58 designs across bases 2 and 3 and dimensions 2 and 3, rendered cell by cell, orbit counts against Burnside, a 59-line csv.
• method.md:113-117 763 designs measured against the predicted density, 522 spanning, 241 degenerate, none flagged.
• method.md:322-330 bang dim 3, code 23 filling 4*k^3 - 3*k^2 at odd side 2k - 1 and reading 20 at side 3, bang dim 3, code 15 and bang dim 3, code 27 sharing one polynomial, 21 distinct polynomials over 22 classes and 64 over 256 designs.
• bijection.md:88-99 group orders 2, 8, 48, 384, orbit counts 22, 402, 1228158, 400507806843728, and 6 canonical designs in dimension 2, 22 in dimension 3; bijection.md:184 the base-3 dim 3 Burnside 111618.
• sequences.md:11 and sequences.md:105-108 fill classes counted by A129824 with zero profile collisions, A000616 3, 6, 22, 402, 1228158, 400507806843728, all 16 published A129824 terms, 144 and 2304 generator checks with zero mismatches, and the A129824 column to dim 8.