# The hexagonal slice

- 2026-08-28 [Verified] The carpet face-count law `V(i) = 2 * 20^i + 4 * 8^i`, visible faces `6, 72, 1056, 18048, 336384`, is A332705 verbatim, the surface area of the stage-`i` Menger sponge, with the same closed form on the entry. Witness: mrlymath::formulas::surface::surface, A332705.
- 2026-08-28 [Proved] The carpet slice census `6, 42, 306, 2250, 16578` is `A299916(level+1)`: sectioning the sponge at level `level` on `x + y + z = 1.5 * 3^level`, a surviving cube cuts a hexagon of 6 mesh triangles or a triangle of 1, refinement triples the plane offset, and the 20 surviving subcubes split by coordinate sum as `1, 3, 3, 6, 3, 3, 1`, so `H_(level+1) = 6 H_level + T_level` and `T_(level+1) = 6 H_level + 3 T_level`, the hexagon-triangle substitution proved by exhaustion; the ledger `54 = 6*6 + 6*1 + 12` punches exactly one 12-triangle hexagram per hexagon and none per triangle, so hexagram holes of the `n`-th size number `A299916(n)` and the mesh census is one index up; the recurrence `a(n) = 9 a(n-1) - 12 a(n-2)` gives the slice dimension `log((9 + sqrt(33))/2)/log(3) = 1.818410`; the empty area at `level = 0..5` resolves into `1, 6, 42` components of descending size, each with six radial maxima, sixfold symmetry to `0.001` and `max/min` radius `1.68` against the hexagram's `sqrt(3)`. Witness: slice-recurrence-order, `mrlymath::six::topology` test `the_carpet_slice_percolates_at_base_three`, A299916.
- 2026-08-28 [Verified] The slice vertex count `12k^2 - 6k + 1` is A154105 at `n = k - 1` and the centered hexagonal number A003215 at index `2k - 1` (`3m(m+1) + 1` at `m = 2k - 1`), so a prime vertex count is a cuban prime, A002407; at `k = 1..20` ten values are prime, `7, 37, 271, 397, 547, 919, 1657, 1951, 2269, 4219`, and ten composite, `91, 169, 721, 1141, 1387, 2611, 2977, 3367, 3781, 4681`. Witness: mrlymath::formulas::six::solid_slice_vertices, A154105, A003215, A002407.