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.