# Primes on a design - 2026-09-06 [Proved] Primes on a design at `fill = base^dim - 1` are primes with one restricted digit at base `base^dim`: the Morton code `x -> Sum_j (Sum_c base^(c-1) x_(c,j)) base^(Dj)` maps `S_level` bijectively onto the integers at base `base^dim` whose digits lie in the image of the digit set `F`, the gasket base 4 missing 3 and the carpet base 9 missing 4; the gcd-prime reading is a positive-density count when `B(F) > 0` and empty otherwise, base 32 on `{0,4,...,28}^2` having `fill = 64 > 32`, (E), and every gcd divisible by 4; the `x_1`-prime reading is a sum-of-digits large deviation. Witness: coprime.md PRIMES ON A DESIGN. - 2026-09-06 [Proved] Lemma A' the window rate: for (E), `gcd(d,base) = 1` and nonzero `t` in `(Z/d)^dim`, `Prod_(l= m_d` so it is never weaker than Lemma A; with a base part `e` and `t` nonzero mod the coprime part `m`, and `|eta|_inf < base^(-2n/3)/(4 base dim (base-1))`, the rate is `c'(base,fill)^floor(2n/(3 m_d))`, which is Maynard Lemma 8.2 in every dimension with an explicit constant and no consecutive-digit hypothesis; the hypothesis on `t` is sharp, the gasket at `d = 6` and `t = (3,0)` sitting at `1/3` at every `level`. Witness: lab/py/digit-transform-norms lemma, worst per-digit rate `0.830915` at `d = 257` over `d <= 301` against Lemma A's `0.986514`. - 2026-09-06 [Proved] The 2D Type I saves a power when `alpha_1* < dim/2`, the dyadic block `d ~ Q_1` costing `fill^level (Q_1^(2 alpha_1* - dim) + Q_1^dim base^(level(alpha_1* - dim)))` and the small moduli going to Lemma A': the carpet certified at `alpha_1* < 0.8124` gives `Sum_(d <= Q, gcd(d,3) = 1) |#{x in S_level : d | x} - fill^level/d^2| <<_A fill^level level^(-A)` at `Q = 3^(0.5938 level) level^(-C)`, and the gasket certified at `alpha_1^- >= 1.0126`, `alpha_1^+ <= 1.1022` closes the route, `min_x Sigma_2 > 4.059204` against 4 and `min_x Sigma_3 > 8.213932` against 8 in interval arithmetic with directed rounding, `Sigma_2(0) = (8 + 2 sqrt(5))/3` exactly. Witness: lab/py/digit-transform-norms certify. - 2026-09-06 [Verified] The carpet misses the one-dimensional criterion at every order: base 9 missing 4 has `g(1) = 0.3437` below `27/77` but `g(3/2) = 0.1531`, `g(235/154) = 0.1457`, `g(1.6) = 0.1262`, `g(1.7) = 0.1031`, `g(1.8) = 0.0835` against `0.1473, 0.1397, 0.1179, 0.0884, 0.0589`, a gap of `0.0058` on the printed pair at `s = 3/2` and `0.005749` in full, and `g(3/2)` moves `0.154389, 0.153068, 0.152921` over four, five and six digit-vectors, so windows do not close it. Witness: lab/py/digit-transform-norms moments. - 2026-09-06 [Verified] The gasket is out of reach at both numbers: base 4 missing 3 has `g(1) = 0.4820` against `27/77` and `g(235/154) = 0.3170` against `59/433`, so no Type II range opens at any order computed. Witness: lab/py/digit-transform-norms moments. - 2026-09-06 [Verified] The componentwise route is closed at source: Chow, Varju and Yu Remark 6.1 puts the Fourier `l^1` dimension below `1/2` for `(b,a)` in `{(3,0),(3,1),(3,2),(4,1),(4,2)}` by interval arithmetic at `level = 2`, so the base-3 design's coordinate marginals fall on the wrong side, while Proposition 2.4 puts base 4 missing 3, the base-2 gasket's Morton code, above `1/2`. Witness: arXiv:2402.18395v2 pp.25-26. - 2026-09-07 [Verified] The missing-digit criterion is unreachable for the carpet at every order: dividing by `2 - s` the criterion is the single inequality `g(s)/(2 - s) < (1/5)*(1 + c/2)` on the transform's moment exponents, and for base 9 missing `4` the shift sandwich at a power certifies `g(3/2) > 0.149397` and `g(235/154) > 0.142274` against the required `0.147320` and `0.139667`, with a monotone chain of orders anchored at the exact `Sigma_N^(2)(x) = (9/8)^N` covering `[3/2, 2)` in 21 closed cells sharing endpoints and `[1, 2)` in 87; the pointwise deficit is at least `0.001268` over `[3/2, 2)` and the decisive cell re-derived independently at `N = 5` clears by `0.000840`. Witness: lab/py/digit-transform-norms criterion, with an independent recomputation by a digit-tree fold reproducing every printed digit. - 2026-09-07 [Proved] The two-missing-digit transform is `(base-2)^2 abs(hat F)^2 = K^2 + 2 + 2 cos(2 pi D t) - 4 K cos(pi S t) cos(pi D t)` with `K = sin(base pi t)/sin(pi t)`, `D = a - c`, `S = a + c - (base-1)`, so a pair enters only through `abs(D)` and `abs(S)`; that implication does not run backwards, `{0,2}` and `{0,8}` at `base = 10` reading `(2,7)` and `(8,1)` with equal transforms, and the collapse is generated instead by the reflection `d -> base-1-d`, which flips both signs, together with the integer translation of `F` available exactly when `0` or `base-1` is excluded and identifying `{0,c}` with `{0,base-c}`, so the edge family is the one-missing-digit sets of a `(base-1)`-digit interval read at base `base` and the number of distinct transforms is `(C(base-2,2) + floor((base-2)/2))/2 + floor(base/2)`. Witness: coprime.md PRIMES ON A DESIGN. - 2026-09-07 [Verified] That pair count reads `7, 16, 21, 31` of the `15, 36, 45, 66` excluded pairs at `base = 6, 9, 10, 12`, is reproduced by grouping every one of the `C(base,2)` pairs by its sampled transform at every base `4 <= base <= 41`, and sums to 2373 distinct sets over `4 <= base <= 31`. Witness: lab/py/digit-transform-norms pairs. - 2026-09-07 [Verified] The least base carrying a certified two-missing-digit set with `alpha_1 < 1/4` is `base = 32` at the interval class `{0,1}`, `alpha_1 in [0.2499087, 0.2499779]` at four window digits, the same class at `base = 31` reading `[0.2518967, 0.2519717]`; over `4 <= base <= 31` the machine certifies `alpha_1 > 1/4` at 2363 of the 2373 distinct sets, closest `base = 26` missing `{2,23}` at `> 0.2502919`, and the ten it cannot bracket from below all have `S = 0` or `D = base/2` with `base/2` odd, a shared shape and not a cause since the clearing headline `base = 32` missing `{0,1}` has a transform vanishing at all 29 points `t = j/30`, with certified upper bounds `0.2538899` to `0.2826357`, above `1/4`. Witness: lab/py/digit-transform-norms pairs and pairfail. - 2026-09-07 [Verified] Against the bar `1/3` a two-missing-digit set first clears at `base = 13`, the interval class at `alpha_1 < 0.3318819` on three window digits with `base = 12` above at all 31 of its sets to five, and the whole pair family clears from `base = 21` on through `base = 26`, worst `base = 23` missing `{4,5}` at `< 0.3333284`, every base `4 <= base <= 20` carrying a certified witness above `1/3`, `base = 20` by `{3,11}` at `[0.3356579, 0.3356674]`. Witness: lab/py/digit-transform-norms pairs pairclear pairsome. - 2026-09-07 [Proved] The digit-uniform bound holds at any excluded-digit count: `abs(hat F(t)) <= (abs(sin(base pi t)/sin(pi t)) + m)/(base - m)`, the level product expands with weight `m^(N - card E)` and telescopes to the same Dirichlet kernels, so `a_N = m a_(N-1) + m Sum_(l= 649` at `m = 2` with the chain failing at `648`, and `125` at `m = 1` and `1873` at `m = 3`, with `32`, `105`, `230` against `1/3`, certified at 120 bits; the coarser `c_0 = 0.97` form of the same chain needs `base^l >= 86` and gives `126` at `m = 1`. Witness: lab/py/digit-uniform-bound pairs. - 2026-09-07 [Proved] The threshold `1/4` is the Mertens bar: in the GRH chain steps 1, 2, 4 and 5 never name the digit set and only step 3 substitutes a digit-free kernel bound, so feeding the certified `l^1` exponent there gives `abs(M_F(x)) <<_(base,eps) A_F(x) x^(alpha_1 - 1/4 + eps)`, that is `A_F(x)^(1 - delta + eps)` with `delta = (1/4 - alpha_1)/alpha_base > 0`, and `1 - b(a)` in place of `1/4` under a zero-free half plane; steps 2 and 3 alone force `alpha_1 <= 1 - alpha_base + c_base` and `gap_base(1) > 0` is exactly `1 - alpha_base + c_base < 1/4`, so the old certificate implies the new condition and the wall can only fall. Witness: coprime.md PRIMES ON A DESIGN. - 2026-09-07 [Verified] That wall falls from `3690` to `34`, on the interval certificates behind the one-missing-digit clearance `alpha_1 < 1/4` at `base = 34`, at every `35 <= base <= 125` and by the uniform chain above. Witness: coprime.md PRIMES ON A DESIGN.