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<level) f_l(t) <= c(base,fill)^floor(level/m_d) with m_d = max(1, floor(log(d/2)/log(base)) + 1), and ord_d(base) >= 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<N) lambda_l a_(N-1-l) + lambda_N and the growth root solves (z - m)(z - 1)^2 = m(c_1 (log base) z + gamma'(z - 1) + c_1 (z-1)^2/(base z - 1)) on the exact Lebesgue input lambda_l <= c_1 l log base + gamma' + c_1 base^(-l), gamma' = (2/pi)(gamma + log(8/pi)), giving alpha_1 < 1/4 for every base >= 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.