# 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.