Density theorem boundary

• 2026-08-28 [Verified] The base-2 dim = 4 design census is complete to level 4: 65536 designs close into 402 orbits under the 384 signed coordinate permutations, 400 with fill >= 2, 336 both spanning Z^4 and carrying fill > 2 so the dimension-above-one density theorem applies to them on its stated sufficient condition, and the 400 eligible canonical representatives realize 189 distinct coprimality sequences (A(1), A(2), A(3), A(4)) with 87 collisions covering 298 classes, the largest being 10 classes on (4, 16, 88, 436) and 10 on (5, 25, 165, 985), by exhaustive enumeration with exact integer minors for spanning and an exact four-coordinate gcd at every point; four terms cannot separate an infinite collision from a short coincidence, and the distribution is over minimum-bitmask representatives rather than unoriented orbits, since coordinate complement moves the arithmetic origin and preserves neither B(F) nor delta nor A(level). Witness: coprime-density-above-dimension-one.
• 2026-08-28 [Verified] Exact sponge visible census without enumeration, three levels past the feasible: the hybrid A(level) = Sum_(d <= G) mu(d) T*_d(level) - Sum_(gcd > G) S(gcd) with transfer matrix T_d and the big-gcd tail enumerated as multiples in a base^level/g box costs about base^(level(dim+1)/2) against enumeration's fill^level and gives A(7) = 1038074187, A(8) = 20860210527, A(9) = 418429711224 (22.6 seconds against half a trillion points, the whole ladder in 84 seconds), anchored by the four census terms, direct enumeration at level = 5, 6, and cutoff independence (G = 100 and G = 150 split the work differently and agree on A(9) to the integer). Witness: lab/py/sponge-visible-census.
• 2026-08-28 [Conjecture] Spanning is the wrong hypothesis for the density theorem and the sharp condition (E) is that F - F has full rank and rad(m(F)) | rad(base): six census degenerate lines satisfying it obey the formula unchanged, a prime p | m(F) not dividing base replaces the Euler factor at p by a coset-corrected one, and A(level)/fill^level can fail to converge at all (base 7, F = {v : v_1+v_2 = 1 mod 3}, fill = 16, index 3, period-3 subsequential limits matched to 3e-04), numerics at 1e-04 on five corrected designs plus seven null cases, provable-looking by the base-peel argument; the same failure shows at base 3, F = {0,2}^2, fill = 4 > 3 with A(level) = 0 at every level against a positive predicted density. Witness: coprime-density-above-dimension-one; A396934.
• 2026-08-28 [Conjecture] The b-visible local factor of a design at a base prime is exact at every finite level: on the gasket #{x in S_n : 2 | x_1 and 2^b | x_2} = (2^(b-1)/3^b) 3^n for n >= b, by the last-digit argument, exact to the integer at b = 1, 2, 3 on every level 3..12, thirty matches with ratios exactly 1/3, 2/9 and 4/27; the Euler-product assembly it feeds is unproved for b >= 2.
• 2026-08-28 [Conjecture] Directional coprime profiles separate designs that the scalar density cannot: two base-3, dim = 2, fill = 5 designs with identical B(F) = 4/5 and identical predicted density 0.5471344 differ by up to 0.0327 in an eight-bin angular coprime profile at level 7 while their scalar pairwise densities differ by 0.0001001809, by exact enumeration at levels 2 to 7 with displacement multiplicities validated against N(N-1); the bin gap decays by a factor near 0.6 a level (0.375, 0.1461, 0.0946, 0.0492, 0.0327 at levels 3 to 7), so a nonzero limit is open.
• 2026-08-28 [Conjecture] The digit-restricted coprime density over F_3[t] with S = {0,1} is 9/16: measured 0.564176, 0.563471, 0.562833 at level = 10, 12, 14 by exact enumeration against (2/3)(3/4)/(8/9) = 0.5625, the correction sitting entirely at the exceptional prime t where pi(t) = 1/2 exactly against the unrestricted 1/3, the other two linear primes measuring 0.333984 and 0.333008; the finite Euler product is neither exact nor monotone, crossing 9/16 between degrees 3 and 4 and landing at 0.560193 through degree 5 while the exact no-shared-prime-below-degree-6 probability is 592189/1048576 = 0.564755, a dependence gap of -0.004563, so the density is a theorem only under three hypotheses: existence of pi_S(p), asymptotic independence over finite prime sets, and a vanishing high-degree tail. Witness: lab/py/function-field-density.
• 2026-08-28 [Conjecture] Mixed-radix coprime density depends on the schedule through a mod-3 effect rather than parity: at level = 12 over 8386560 ordered pairs per schedule (4096 points each), alternating base-2/base-3 gives 0.511135 with digits {0,1}/{0,1} and 0.672615 with {0,1}/{0,2}, against pure base-2 0.607874 (near 1/zeta(2) = 0.607927) and pure base-3 0.514692; both alternating schedules are exactly half even, and the 0.161480 gap comes from the divisible-by-3 fraction falling from 1/2 to 1/4, local factor 3/4 to 15/16, log advantage 0.223144, with prime 5 opposing at -0.014253 and primes 2, 7, 11, 13 identical between them; the residue mechanism for {0,1}/{0,1} is exact (every place value from position 2 on is a multiple of 6, so a mod 6 = d_0 + 2 d_1 and residues 0..5 are hit 1024, 1024, 1024, 1024, 0, 0), but no limit is established and the aperiodic staircase is untouched.
• 2026-08-28 [Conjecture] The sponge census gaps delta 20^level - A(level) measure 0.347, 0.349, 0.344 in units of 12^level, the subdominant parity-walk scale, so A(level) = delta 20^level - c 12^level + smaller. Witness: lab/py/sponge-visible-census.
• 2026-09-02 [Verified] The sponge visible census to level = 18 by two engines: admissibility is pairwise disjointness of the digit-1 masks, so A(level) = W(level) - W(level-1) with W(level) = Sum_(m < 3^level, gcd(m,3) = 1) mu(m) (N_level(m) - 1) and N_level(m) a disjoint-triple count over at most 2^level masks of the multiples of m; the engine splits moduli by their multiples count into a closed-form tail, bitset rows, u16 zeta rows and a rank-truncated ranked cube, about 3^level (level 2^level)^(2/3) work, 122.3 s at level = 18 on eight threads with level ratio 4.32, 3.8x its previous form, which it reproduces term for term from A(10) = 8382927031902 to A(18) = 215134797774716879278017, both matching the hybrid census through A(9) and enumeration through A(6); nine counters agree on every modulus to level 8, pinned counters with auto to level 8-11, 38 probed moduli at level 14 and two pinned probes at level 19 cover the u64 cube gate and the u32 rows branch; the new engine alone gives A(19) = 4302768326366633733102921 in 515 s, unwitnessed. Witness: lab/rs/coprime-terms.
• 2026-09-02 [Proved] The tail of the sponge Mobius sum is closed and admits no hyperbola grouping: for 3^level/2 < m < 3^level, 3 not dividing m, N_level(m) - 1 = 3 + 4 [m has no base-3 digit 1], so the top band of W(level) is three times the Mertens sum over the band's moduli coprime to 3 plus four times a Mertens sum over the base-3 Cantor set; the Y = 3 band is 6 + 7 [mask(m) = 0] + 7 [mask(2m) = 0] + 6 [mask(m), mask(2m) disjoint], every band a Mobius sum over a digit-automatic condition on m, 2m, ..., (Y-1) m; N_level(m) depends on the digits of m, not on floor(3^level/m) (level = 2: m = 5, 8 share the floor with N - 1 = 3, 7; at level = 6 every floor band holding two admissible moduli is non-constant), verified exhaustively at level = 6, 7 against a brute triple loop. Witness: lab/rs/coprime-terms.