Design census

• 2026-08-28 [Proved] Designs up to cube symmetry are the NP-equivalence classes of Boolean functions: an equivariant bijection carries B_dim (order 2^dim * dim!) onto the NP group, so the class counts are 3, 6, 22, 402, 1228158, 400507806843728 at dim = 1..6, reproduced by orbit walk on designs, orbit walk on truth tables and Burnside, checked against the entry to dim = 7; the NPN sibling gives 2, 4, 14, 222 at dim = 1..4. Witness: mrlymath::bang::counting::sequence, A000616, A000370.
• 2026-08-28 [Verified] The design census at dim = 2 and any base is the toroidal binary array count: Burnside over one dihedral group per residue axis gives 2, 6, 26, 805, 172112, 239123150, 1436120190288, 36028817512382026 at base = 1..8, with brute-force orbit closure agreeing at base = 3, 4. Witness: mrlymath::bang::baseq::distinct_designs, A255016.
• 2026-08-28 [Proved] The isotropic-class count is A005418(dim+2) minus one at even dim: a level set's orbit is {F_S xor t} and the image depends on t only through |t|, so the count is subsets of {0..dim} up to reversal with the even-dim merge, 3, 5, 10, 19, 36, 71, 136, 271, 528, 1055, 2080, 4159, 8256, 16511, 32896, 65791 at dim = 1..16; nameable classes grow like 2^dim, half the 2^(dim+1) subsets. Witness: A005418.
• 2026-08-28 [Verified] Census multiplicity in the base-2 census is a bounded perfect-power representation count: over the 16 two-dimensional and 256 three-dimensional designs at side 3 and levels 1..5, the 1360 design-level pairs give 119 distinct fill counts, M(N) = sum_{level=1}^5 sum_{fill=0}^27 c_fill [fill^level = N] with sum_fill c_fill x^fill = prod_{w in {1,2,4}} (1 + x^w) + prod_{w in {1,2,4,8}} (1 + x^w), maximum M(4096) = 29 from 4096 = 8^4 = 16^3 (14 designs of base fill 8 plus 15 of base fill 16), support exactly {fill^level : 0 <= fill <= 27, 1 <= level <= 5}, so only 119 integers occur up to 27^5 = 14348907, coverage 8.29e-6, in 91 maximal missing runs, the longest 11881377..14348906; M is not multiplicative (M(2) = M(3) = 5, M(6) = 12), opens 10, 10, 5, 5, 13, 8, 12, 12, 19, 19, 15, 15, 16, 16, 16, 16, and no classical arithmetic function is behind it: the raw Pearson signals against sigma and phi (-0.332, -0.315) are shared size dependence (M against N is -0.338), and partial rank correlations controlling for log N fall in -0.035..0.096 for d, sigma, phi, omega, Omega. Witness: mrlymath::formulas::counting::fill.
• 2026-08-28 [Verified] The census covers only a short prefix of the integers: through Kronecker level 6 the 256 three-dimensional designs produce 37 positive fill values in 1..262144, coverage 0.01411%, contiguous only on 1..9, because fill(code, level) = f^level with f the tile popcount in 0..8; the nine named observables reach a union of 368 integers to 1633932 with 10 the first gap, the graph observables core_edges, tips, junctions extend it to 437 integers and the prefix to 1..24; edges is the broadest single observable (73 distinct positive values, 60 exclusive), then faces 70, vertices 67, surface 66; design 23 reads fill 20, voids 7, surface 72, vertices 64, edges 144, faces 96, Euler -4 at level 1 and fill 400, vertices 896, edges 2304, faces 1728, Euler -80 at level 2, and every record obeys surface = 6 fill - 2 core_edges, faces = 6 fill - core_edges, cycle_rank = core_edges - fill + components, euler = vertices - edges + faces - fill; voids means every empty lattice site, 3D edges means cubical-complex unit edges and core_edges the branch count of the face-adjacency graph; the 2D census gives 176 distinct positive integers to 1064340, contiguous on 1..10. Witness: mrlymath::formulas::counting::fill, mrlymath::three::census::census.
• 2026-08-28 [Verified] Unbounded census observables cannot be automatic sequences and are at best regular in some base: an integer-valued automatic sequence has finite range, a finite-range regular sequence is automatic, so d(n) and sigma(n) are automatic in no base and can enter only through unbounded regular representations, weighted substitutions, Dirichlet convolutions or a purpose-built geometric model; Cobham's theorem is the one general rigidity constraint, and substitution incidence matrices generate additive recurrences while d and sigma are multiplicative over primes, so self-similar census counts c_m = u^T A^m v are generically sparse in the integers. Witness: REFS.md.