coprime-terms

• Exact A(level), the points of a base-3 mask design at level level with coprime coordinates, without walking the fill^level points: the Menger sponge to level = 18, the Sierpinski carpet and the Vicsek plus to level = 20.
• Mask form: with mask(x) the set of base-3 positions where x has digit 1 (a digit away from 1 for the plus), a point lies in the level set iff its coordinate masks are pairwise disjoint.
• Mobius over the common divisor with the base prime peeled: A(level) = W(level) - W(level-1) where W(level) = Sum_{m < 3^level, gcd(m,3) = 1} mu(m) (N_level(m) - 1), since N_level(3m) = N_{level-1}(m) whenever the zero digit vector is filled; the plus has A(level) = W(level) and no -1.
• N_level(m) is a pairwise-disjoint count over the masks of the Y = ceil(3^level / m) multiples of m, on at most 2^level masks whatever the box; the sponge splits its moduli by Y over five counters, each cheapest in its band by measurement.
• tail, Y <= 3: closed forms N - 1 = 3 + 4 [mask(m) = 0] for Y = 2 and 6 + 7 [mask(m) = 0] + 7 [mask(2m) = 0] + 6 [mask(m), mask(2m) disjoint] for Y = 3, so the top two bands of W(level) are Mertens sums over automatic sets.
• bitset, Y <= (512 level 2^level)^(1/3): one Y-bit row per multiple marks its disjoint partners and N = 6 T + 3 z (Y - 1) + z, T the popcount sum of row pairs above the diagonal and z the zero-mask multiples; about Y^3 / 1000 cycles against Y^3 / 100 for the nested loops it replaces.
• rows, U <= 6 sqrt(level 2^level) distinct masks: one u16 zeta transform g of the mask histogram c, then 2 Sum_{a < b disjoint} c_a c_b g(complement(a | b)) plus the zero-mask diagonal, the disjoint pairs read off bitset rows built four masks per NEON lane inside buckets keyed by the top six bits.
• cube, the rest: u32 ranked zeta transforms truncated at rank K ~ level/2 with the rare heavier masks folded back by subset enumeration, then N = Sum_T <c_hat(complement T), Delta[sq(T)]> per set T, sq the square of the rank polynomial and Delta its binomial difference table, in wrapping u64 where Y^3 < 2^64 and u128 otherwise.
• residue: the automaton on (Z/m)^3 for m <= 16; the earlier nested-loop, zeta and ranked-convolution counters stay as references inside the gate.
• Cost: 3^level moduli a level; the bitset/rows crossover costs 3^level (level 2^level)^(2/3) / 8 and the rows/cube balance 3^level level 2^(level/2) cycles, so the level ratio in terms/menger.txt is 4.32 from level = 17 (28.317 s) to 18 (122.294 s) on eight threads, 3.7x and 3.8x the 104 s and 463 s of coprime.md:330, and tends to 4.76 only past level = 22.
• check gates the engine: direct enumeration to level = 6 for the sponge and 8 for the plane designs, every counter agreeing with every other on every modulus to level 8, each counter pinned alone agreeing with auto to level 8 (bitset), 10 or 11, 45 probed moduli at the sample level across counters, two pinned level-19 probes for the u64 cube gate (m = 440) and the u32 rows branch (m = 3^8 + 1), the 3m peel identity, and the fill N_level(1) = fill^level.
• terms/ holds the three ladders with seconds per level and their b-files.

RUN

• CARGO_BUILD_JOBS=4 cargo run --release -p coprime-terms -- terms menger 18 8
• Arguments: design (menger, carpet, vicsek), top level, threads.
• -- check menger 6 14 runs the gate battery; -- profile menger 16 8 prints seconds per log2 Y band and counter for one level; cargo test --release -p coprime-terms pins the stored terms.
• The sponge ladder to level = 14 takes 1 s, to level = 18 about 2.5 min (terms/menger.txt).

WITNESSES

• coprime.md:330: the sponge ladder A(10) .. A(18), ending 215134797774716879278017, with the 104 s and 463 s timings of the previous engine.
• research/claims/:161: the same ladder and the gate battery.
• A399364: DATA level = 0..17 and the b-file terms/menger_bfile.txt to level = 18.