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.