# Coprimality density

- 2026-08-28 [Proved] The base-local coprimality factor is exact at every finite level: `#{x in S_level : e | x_i for all i} = fill_e * fill^(level-1)` for every squarefree `e | rad(base)`, holding on all 763 census lines; at composite base it does not factor over primes, the base-6 sample giving `B(F) = 1/2` for the code's digits `F` (256 and 240 of 512) against the naive `0.46875`; the character contraction `c(base,fill) = 1 - (2/fill)(1 - cos(pi/(2 base)))` gives `0.804738, 0.966506, 0.946410, 0.986603, 0.991481` at `(base,fill) = (2,3), (3,8), (3,5), (3,20), (6,8)`. Witness: coprime-density-above-dimension-one, lab/rs/design-census.
- 2026-08-28 [Verified] Lemma B, the equidistribution estimate the coprimality programme needs, is a uniform equidistribution statement for digit-restricted sets across moduli growing with the level, the subject of Erdos, Mauduit and Sarkozy 1998, Konyagin 2001 and Maynard 2019, with Lemma A supplying the per-character input; it is proved for every design with `fill > base` by the dimension-above-one theorem and remains open only at `fill <= base`. Witness: coprime-density-above-dimension-one, REFS.md.
- 2026-08-28 [Proved] Every digital design of fractal dimension above one has the classical coprimality density: for `dim >= 2`, condition (E) and `fill > base`, `A(level)/fill^level -> B(F) prod_{p not dividing base} (1 - p^(-dim))`, by the box bound `N*_level(m) <= (base+1)^dim fill^level m^(-log(fill)/log(base))` driving a Chebyshev log-gcd sum and a fixed-`z` sieve; this settles the gasket `16/(3 pi^2)`, the or-triangle `8/pi^2`, the carpet `189/(32 pi^2)`, the Vicsek plus `27/(4 pi^2)` and the sponge `(513/520)/zeta(3)`, and the finite levels approach with oscillating signed error: carpet gap `-3.52e-07` at `level = 20`, sponge `-3.45e-05` at `level = 18` (lab/rs/dimension-one-ladder), gasket `0.539591` against `0.540380` at `level = 16` (lab/rs/oeis-terms, A396934). Witness: coprime-density-above-dimension-one, lab/rs/dimension-one-ladder, lab/rs/oeis-terms, A396934.
- 2026-08-28 [Proved] The 36 open dimension-one census lines are one problem: for any full-rank `base = 3`, `fill = 3` design, collecting the corner-choice classes into `E_j = sum_{c_l = j} 3^l` maps `S_level` bijectively through the gasket, and for every `m` coprime to the design's difference determinant the two divisibility conditions become one shifted-target congruence on the gasket pair, so Lemma B stands or falls for the whole family at once, with `T*` vanishing at `m >= 3^level`; the same argument reduces every full-rank `fill = dim + 1` design at any base to the simplex at that base. Witness: lab/rs/design-census.
- 2026-08-28 [Verified] Not every census collision is a shear: base-3 codes 11 and 161 have identical `A(level)` at every level (`2, 4, 12, 34, 108, 322, 992, 3006, 8924, ...` through `level = 12`, and equal `T_e` for every `e <= 40` at `level <= 7`) though 161 is alone in its `GL_2(Z)` orbit over the exhaustive entry range `[-8, 8]`: both have zero corner `v_0 = 0` hence gasket target 0, determinants with prime support `{3}`, and matching base-3 peel, so the finite Mobius sums agree termwise. Witness: lab/rs/design-census.
- 2026-08-28 [Proved] `A(level)` is not C-finite for any design meeting the dimension-above-one theorem with `B(F) > 0` and `dim` even or `dim = 3`: a rational C-finite sequence with `A(level)/fill^level` convergent has a rational limit (roots above `fill` have zero coefficient, oscillatory roots on `|z| = fill` die by mean-square averaging, the remaining constant is fixed by every Galois automorphism), while `delta` is an irrational multiple of `1/zeta(dim)`; all five eligible base-2 plane designs through `level = 12` admit no rational constant-coefficient recurrence of order at most 6 and approach their irrational limits (the gasket-type designs read `0.5378546` at level 12 against `0.5403796`); the theorem says nothing at odd `dim >= 5`, at `B(F) = 0` or `fill <= base`, or about polynomial-coefficient recurrences, which 2729 exact P-recursive fits with held-out terms exclude only empirically. Witness: coprime-density-above-dimension-one.
- 2026-08-28 [Proved] The Menger sponge's pairwise coprimality density is `(13/20) prod_{p != 3} (1 - 3/p^2 + 2/p^3) = 0.251620868451255 = (351/400) C_3` with `C_3 = 0.286747428434479`, so the sponge rule lowers the full-lattice benchmark by exactly `12.25%`: the three-modulus Mobius inversion over the three coordinate pairs does not collapse to one modulus, the base factor `13/20` is exact at every level (13 of the 20 legal digit vectors have at most one zero), each foreign prime contributes `(1 - 1/p)^2 (1 + 2/p)`, and the tail closes on the pair-fibred box bound with `kappa_I = 3` and `alpha = log_3(20/3) = 1.726833 > 1`; the exact census `0, 60, 1434, 32268, 721524, 15141288` at `level = 1..6` gives `0, 0.150000, 0.179250, 0.201675, 0.225476, 0.236583`; the local factor is not `1 - p^(-s)`, so no reciprocal zeta value is claimed and the coefficient in `0.4138997384/zeta(2)` carries no rationality claim. Witness: menger-pairwise-coprimality.
- 2026-08-28 [Proved] The density theorem's spanning hypothesis retires to condition (E): a finite abelian quotient of order `m` is killed by `m`, so `m Z^3` sits inside the difference lattice, and a character mod `d` vanishing on `F - F` forces `m t = 0 mod d` with `gcd(m, d) = 1`, hence `t = 0`, the only step of Lemma A that used spanning; non-parity index-4 and index-8 designs at bases 4 and 6 measure `0.105072, 0.035346, 0.936067` against the widened predictions `0.105639, 0.035261, 0.950751`, converging; those three non-parity measurements have no generator in lab/, which computes parity codes only. Witness: coprime-density-above-dimension-one.
- 2026-08-28 [Proved] Every parity code at every even `base >= 4` has `delta * zeta(3) = (8/7)(1 - W_0/|P|)`, nine values only and independent of the base, because multiples of an odd `m | base` split evenly by parity while multiples of `2m` are all even, so every odd base prime cancels its own Euler correction exactly; all 255 nonempty codes at every even `base <= 40` give exactly nine band values, every lattice index at `base = 4, 6, 8, 10` lies in `{1, 2, 4, 8}`, enumerations reproduce `0.712853, 0.951771, 0.709137`, and in `dim = 2` the parity carpet's band value `8/9` gives `8/9 / zeta(2) = 0.5403796460924681 = 16/(3 pi^2)`, an even-base band constant rather than the gasket's own. Witness: lab/py/mrlybang-density-classes.
- 2026-08-28 [Proved] Odd bases are self-similar across bases and the density trichotomy is exhaustive for odd `base >= 5`: `fill_e(base) = fill_P(base/e)` for squarefree `e | base`, so the bracket is a Mobius convolution of the code's corner-count cubic; the 149 spanning codes converge to `1/zeta(3)` along the odds while frozen on their even band, the 43 codes inside their difference span take the corrected factor `1 - 2^(-s2)` at 2, and the 63 codes whose affine span avoids the origin have no density at all, with zero all-even points at every odd level and two subsequential limits in ratio `1 - 2^(-s2)`; the even value equals the odd limit exactly on the 16 subgroup codes; checked for all codes at all odd `base <= 75`, twelve measured cases to the printed digit including exact zeros at the even levels of `{111}` at `base = 5` and the axes pair `0.987338 / 0.739563` against `0.987319 / 0.740489`. Witness: lab/py/mrlybang-density-classes.
- 2026-08-28 [Proved] Slice coprimality is finite arithmetic of the height: on `x + y + z = s` the gcd divides `s`, so `A_s = sum_{d | s} mu(d) N_s^(d)` exactly with no tail and no Lemma B; prime slices are fully visible up to the three axis points (coordinates forced into `{0, p}`), the base prime peels the slice to the previous level one step off-centre, a code without the origin corner owes nothing at its base on any slice, and each foreign prime costs the slice `1/p^2` where it costs the solid `1/p^3` (aggregated locals `0.040902` against `1/25`, `0.020446` against `1/49`); the parity-walk factor at 2 is `9121792/32002048` on the integer, the net's immunity at 3 holds on all `7^7` points, and the tree dichotomy holds: even slices hold zero visible points, odd slices zero even gcds. Witness: lab/py/slice-coprimality.
- 2026-08-28 [Proved] The central slice never converges and its bill is the repunit: `s* = (3(base-1)/2) R_level(base)` owes 3 always, 2 exactly when `base = 1 mod 4` or `level` is even, and a foreign odd prime exactly when `ord_p(base) | level`, so the centre's visible density is a quasiperiodic function of the divisors of `level`; at `base = 3` the two streams read `0.892, 0.898` against `0.571, 0.611, 0.652`, at `level = 7` the whole foreign bill is the Wieferich prime 1093 (`2^1092 = 1 mod 1093^2`), the central count is `A299916(level)` on the `(9, -12)` recurrence exact to `level = 14`, and the sixth peeled term is 83835 by a meet-in-the-middle count over all `20^7` level-7 points. Witness: lab/py/slice-coprimality, A299916.
- 2026-08-28 [Conjecture] The visible density inside a design's `base`-periodic pattern is exactly `delta`: `4/pi^2 = 0.405284734569` for the base-2 gasket pattern, `21/(4 pi^2) = 0.531936214122` for the carpet, `19/(26 zeta(3)) = 0.607932310732` for the sponge, by coprime tuples splitting evenly over the nonzero residue classes (`(2/3)(6/pi^2)`, `(7/8)(6/pi^2)`, 19 of 26 classes over `1/zeta(3)`), with worst Mobius-count error `6.05e-07` at `N = 10^6`; designs differing only in the all-zero corner have identical visible density on 8579 pairs, since that class holds no visible points.
- 2026-08-28 [Conjecture] Exhaustive endpoints at bases `2..6` and `dim = 2, 3` agree with the predicted `delta` with no inferable rate: base-6 code 34376528265 reads `0.454413` against `0.455945` at `level = 8`, and the base-5 `dim = 3` pair is the worst case at `8.3e-03` and `1.5e-02`.
- 2026-08-28 [Conjecture] The central-slice peel ratio tends to `(sqrt(33) - 5)/8 = 0.0930703308`, measured `0.093070331`, forced from the shared `(9, -12)` recurrence of the peeled streams, which itself stays Conjecture. Witness: lab/py/slice-coprimality.