# 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.