Claims · 40 dated claims on Primes on a design, each with its tag and its witness; newest 2026-09-23.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Proved The Lebesgue constant L_M = max_x sum_(j < M) abs(D_M(x + j/M)) of the shifted Dirichlet grid is attained at the half offset for every M >= 2, L_M = 2 sum_(m < ceil(M/2)) csc((2m+1) pi/(2M)) - (M mod 2), and L_M <= M((2/pi) log M + gamma') + 2/pi with gamma' = (2/pi)(gamma + log(8/pi)). The maximum: each pair contributes 2 csc(mu_m) X_m with X_m <= 1, reduced by monotonicity in mu_m to m = 0 and there, at u = pi/(2M), to sin^2 u cos(ru) sin^2(pi r/4) >= sin^2(ru/2)(cos^2 u + cos(ru)) for 0 <= r < 1, proved from x cot x decreasing and cos(ru) >= cos u; the odd leftover contributes cos(M delta)/cos delta <= 1. The bound: split csc = 1/x + g with g convex, the midpoint rule and the harmonic bounds H_N <= log N + gamma + 1/(2N), H_N >= log(N + 1/2) + gamma; even M directly, odd M >= 5 through the constant 1 - 2/pi + 2/(pi(M-1)) < 2/pi, M = 3 by hand. Hence lambda_l <= c_1 l log base + gamma' + c_1 base^(-l) and the exact-input digit-uniform chain is a theorem at every base >= 125. Witness: lab/py/digit-uniform-bound README, THE LEBESGUE CONSTANT, with the proof in full in paper first-base-below-a-quarter Lemma 4.4 and its use in Theorem 4.7; verbs threshold and lebesgue.
Verified Base 21 missing 0 is the least base at which a one-missing-digit set is certified below 1/4, alpha_1 in [0.2499765, 0.2499771] at six window digits; bases 20, 10, 4 and 3 are certified above at every digit, base 5 at two of three, the end and middle digits of 14 and 18; the 83 other sets of bases 5 to 19 have no certificate in the tree, and a one-line extension of verb least, base_family(q, 4, 8, False) for q in 3..19 with the restriction clause at the constant-phase digits, would close it. Witness: lab/py/digit-transform-norms verbs six and least; lab/py/mobius-region verbs threshold and criterion; paper Fact 6.2.
Verified Every base q >= 34 clears alpha_1 < 1/4 at every excluded digit and 34 is the least base from which every base does, on base 34 at all 17 sets, the 3663 sets of 35..125, the theorem from 125 and the witness 33 missing 15 at alpha_1 > 0.2506145; whether 34 is the least single base whose every digit clears is uncertified at nine of the bases 21..32. Witness: lab/py/digit-transform-norms verbs least and family; paper Corollary 6.5.
Verified Base 21 missing 0 is the least base carrying a one-missing-digit set with alpha_1 < 1/4: every one of the 108 distinct sets of every base 3 <= q <= 20 certifies alpha_1 > 1/4 at four window digits and sub-scan 8, restricted infimum matrix, closest q = 20 missing 0 at [0.2528608, 0.2531118], the only cells under 0.27 the digit 0 at q = 16..20, each matched by its mirror digit. Witness: lab/py/digit-transform-norms verb floor; paper Fact 6.2.
Verified 34 is the least single base at which every excluded digit clears alpha_1 < 1/4: every base 21 <= q <= 33 carries a digit certified alpha_1 > 1/4 at four window digits, closest q = 33 missing 15 at [0.2506145, 0.2506783], next q = 32 missing 10 at [0.2528988, 0.2529666], and every base below 21 is above at every digit; this closes the nine bases an earlier row left uncertified. Witness: lab/py/digit-transform-norms verbs floor and least; paper Corollary 6.5.
Verified The restriction clause of the window machine, the infimum matrix cut to the states whose float Perron entry exceeds 10^(-9) of its maximum, bounds the constant-phase digit from below where the plain test returns nothing: q = 5 missing 2 at [0.4366508, 0.4496785], q = 21 missing 10 at [0.2664296, 0.2666524], q = 33 missing 16 at [0.2401948, 0.2404222] at four window digits. Witness: lab/py/digit-transform-norms verbs floor and least; paper Table 1.
Verified The digit-blind minorant ell_q = abs(abs(D_q) - 1)/(q - 1) <= abs(hat F), valid at every digit since D_q is real, certifies alpha_1 > 1/4 at every digit at once for every 3 <= q <= 8, lowest 0.2522652 at q = 8, while its own l^1 exponent is certified under 1/4 at every 9 <= q <= 20, 0.2463803 at q = 9 and 0.2041911 at q = 20, so no bound through it collapses the sweep below 21 to one certificate per base. Witness: lab/py/digit-transform-norms verb blind; paper Fact 6.8.
Refuted The middle digit is the dearest one-missing-digit set of its base: q = 20 missing 6 certifies alpha_1 > 0.2831944 against alpha_1 < 0.2830950 at the middle digit 9, and q = 26 missing 8 certifies alpha_1 > 0.2652263 against alpha_1 < 0.2650507 at 13, both at four window digits. Witness: lab/py/digit-transform-norms verbs floor and least; paper Table 1.
Verified 32 is the least base carrying a two-missing-digit set with alpha_1 < 1/4: the ten sets of 4 <= q <= 31 left without a lower certificate at five window digits all have S = 0 or D = q/2 at q/2 odd, and under the restriction clause all 238 distinct sets of that zero class certify alpha_1 > 1/4 at three window digits, closest q = 30 missing {0,15} at alpha_1 > 0.2527281, so with the 2363 certified by pairfail all 2373 sets sit above; this upgrades an earlier row's certified floor to the exact floor. Witness: lab/py/digit-transform-norms verbs pairholes, pairfail and pairs.
Refutedmin_x Sigma_1 = 2 at every interior digit 1 <= a_0 <= q-2 of a one-missing-digit set: the constant-phase digit 2 a_0 = q-1 has C < 0 and dips below 2 at every odd base, the least counterexample q = 3 missing 1, where abs(hat F(t)) = abs(cos(2 pi t)) and min_x Sigma_1 = sqrt 3. Witness: coprime.md PRIMES ON A DESIGN, The one-digit sum at the grid; lab/py/digit-transform-norms verb grid1.
Conjecture At every interior digit of a one-missing-digit set other than the constant-phase digit, min_x Sigma_1(x) = 2, attained at the grid; a float scan of every such digit of 4 <= q <= 40 never reaches 2 on 1e-3 <= qx <= 1 - 1e-3. With the supermultiplicativity of min Sigma_N it would give alpha_1 >= log_q 2, above 1/4 exactly for q <= 15. Witness: lab/py/digit-transform-norms verb grid1.
Verified The shift floor alpha_1 >= (1/N) log_q min_x Sigma_N(x), certified in interval arithmetic over one period with a centred Lipschitz slack per factor, puts 106 of the 108 distinct one-missing-digit sets of 3 <= q <= 20 above 1/4 without a window matrix: every digit of q <= 15 at one digit, every digit of 16 <= q <= 18 and all but the end digit of 19 and 20 at two, the tightest q = 18 missing 0 at alpha_1 > 0.2501422; q = 19 and q = 20 missing 0 stay with the window census. Witness: lab/py/digit-transform-norms verb shift; paper Fact 6.9.
Proved The one-digit sum of a one-missing-digit set at the grid: at every base q >= 3, with mu = abs(a_0 - (q-1)/2), Sigma_1(x) = sum_(i<q) abs(hat F(x + i/q)) = 2 + (pi^2 C/(q-1)) (qx)^2 + O(x^4) with C = mu(q - 2 mu) + 2 mu^2/(q-1) - (q^2-1)/(3q) - 4 mu^2/q, and C > 0 exactly at the interior digits 1 <= a_0 <= q-2 other than the constant-phase digit 2 a_0 = q-1, where the grid is a strict local minimum of value 2; at the end digits and the constant-phase digit C < 0, so min_x Sigma_1 < 2 there at every q >= 3. Witness: coprime.md PRIMES ON A DESIGN, The one-digit sum at the grid; lab/py/digit-transform-norms verb grid1.
Proved Primes on a missing-digit design by the unconditional dissection: for F omitting m >= 1 digits at base base, keeping two consecutive digits and carrying a shifted-grid l^1 certificate sum_(a < base^i) abs(hat F_i(s + a/base^i)) <= C_F fill^i base^(i alpha_1) with alpha_1 < 1/5 at every i and every shift, abs(sum_(n <= x, n in S_F) Lambda(n) - kappa_F A_F(x)) <= C A_F(x) exp(-c sqrt(log x)) at every x >= 2, with kappa_F = (base/phi(base)) #{f in F : gcd(f, base) = 1}/fill and C, c > 0 effective and depending on base and F alone; the wall condition (W) is such a certificate with C_F = 1, so the count holds at m <= 176 excluded digits at base 10^7. The proof is the Mobius dissection with four changes: the main term as the principal characters of C2, the minor arcs from Maynard 2022 Lemma 4.2, read at source in arXiv:1510.07711v1, applied at a coarser fraction, the hybrid l^1 bound run on the certificate with constant C_F^2 (1 + pi base^2 C_F/fill), and the arithmetic input from Koukoulopoulos 2019 Theorem 12.4 with the exceptional zero confined to the real primitive characters of conductor dividing 8 rad(base) and bounded by its Theorem 12.8, the term u^(beta_1) as printed there or u^(beta_1)/beta_1 as its proof produces being at most 2 u^(1 - c_base) in either form. Witness: coprime.md PRIMES ON A DESIGN, The dissection for the von Mangoldt function.
Proved The digit-uniform chain puts alpha_1 < 1/5 at every excluded digit of every base >= 584 with the certificate constant C_F = z, certified at 120 bits on [584, 1272] (tightest margin 6.0170 * 10^-3 in the cubic's units at 584, where 1/5 - alpha_1 >= 1.79 * 10^-5, failing at 583) and by the cap 1 + sqrt(2 (2/pi) log base + 0.97) from 1272 with a slope test from base 100; so the prime count holds on every one-missing-digit set from base 584, and the Mobius bound of the unconditional dissection holds from the same base, below its wall 39363. Witness: lab/py/prime-dissection, verb wall.
Verified The digit-uniform window at two window digits certifies alpha_1 < 1/5 at every excluded digit of every base 301..583, the largest bound 0.199923 at 301, reading 0.200021 at 300; with the chain, the prime count and the Mobius bound hold on every one-missing-digit set from base 301. Witness: lab/py/prime-dissection, verb window.
Verified Maynard 2022 read at source in arXiv:1510.07711v1, its only arXiv version: Theorem 1.1 at q > 2 000 000 with an ineffective (log q^k)^(-A) saving at x = q^k, the remark that q > 2500 suffices by a more involved calculation, and its constant log((q/(q-1)) log q + 3q/(q-1))/log q below 0.198 from 2 000 000 and below 1/5 from 1520573. What the dissection adds is exactly an effective exp(-c sqrt(log x)) saving with a written proof from base 584, and from base 301 by certificate, against that written 2 * 10^6 and the remarked 2500, by the same route and the same 1/5 gate with the gain wholly in the l^1 input; the asymptotic itself is asserted by Maynard 2019 at one excluded digit for q >= 12. Witness: lab/py/prime-dissection, verb wall.
Verified At s excluded digits the constant C_(q,s) = 1 + (2 + s)/log q of Maynard 2022 Section 9 closes alpha_(q,s) < 1/5 at s <= 7 at base 10^7 and s <= 19 at 10^8, against m <= 176 and 703 under (W), a budget of order base^(2/5) against the q^(1/5 - eps) of its Theorem 1.3 and the q^(1/4 - delta) of the Maynard 2019 remark; the consecutive-digit cases of both reach further. Witness: lab/py/prime-dissection, verb wall.
Verified Maynard 2022 Theorem 1.3 as printed in arXiv:1510.07711v1 carries the main constant q(phi(q) - s')/((q - 1) phi(q)), a misprint that the paper's own Section 9 corrects, any occurrence of q - 1 must be replaced by q - s; the principal characters give q - s, as the Maynard 2019 remark prints its kappa_B; at base 10 missing {0, 5} the prime count to 10^8 reads 0.9999 of (5/4) A_F(x) against the printed 10/9. Witness: lab/py/prime-dissection, verbs series and count.
Verified The dissection for Lambda on the whole grid of base 10 missing 5 and missing 1 at level 6 and base 5 missing 2 at level 9: the four regions meet the exact sum, the principal characters return kappa_F fill^k, C2 on the grid carries 0.999546, 0.999021, 1.000427 of it; the prime count against kappa_F A_F(x) reads 0.9994..1.0001 at the largest power of the base at most 10^8 and 0.9984..1.0009 at twelve seeded x on six sets with two consecutive digits, and below 0.0006 on base 3 missing 1 and base 5 missing {1, 3}, which keep no consecutive pair. Witness: lab/py/prime-dissection, verbs regions and count.
Proved The singular series of a missing-digit design is the principal characters of region C2: at a C2 point a = l y/d + j the main part of S_P vanishes unless j = 0, the exact fractions with squarefree d dividing base carry sum_(d divides rad(base)) (mu(d)/phi(d)) sum_(gcd(l,d)=1) hat F_k(l/d) = (base/phi(base)) #{t in D_k : gcd(t, base) = 1} = kappa_F fill^k, and the prime count to 10^8 meets kappa_F A_F(x) on six sets, base 10 missing {0, 5} reading 0.9999 against kappa_F = 5/4; the exact check of the identity at 4956 sets is against its own closed form and not independent evidence. Witness: coprime.md, the main-term bullet; lab/py/prime-dissection, verbs count and series.
Proved At m excluded digits the digit-uniform chain is a shifted-grid l^1 certificate with constant C_F = z/m at every choice of the m digits, z the root of (z - m)(z - 1)^2 = m((2/pi)(log base) z + gamma'(z - 1) + (2/pi)(z - 1)^2/(base z - 1)), and alpha_1 < 1/5 holds from base 4692 at two excluded digits and from base 17596 at three, certified at 120 bits up to 5477 and 27761 and above by the cap m + sqrt(m((2/pi) log base + 0.97)) with a slope test, failing at 4691 and 17595; so the prime count sum_(n <= x, n in S_F) Lambda(n) = kappa_F A_F(x) + O(A_F(x) exp(-c sqrt(log x))) and the Mobius bound abs(M_F(x)) <= C A_F(x) exp(-c sqrt(log x)), both effective, hold at every choice of two excluded digits from base 4692 and of three from base 17596, against 124332 and 153738 for (W). Witness: coprime.md PRIMES ON A DESIGN, One wall per number of excluded digits; lab/py/prime-dissection, verb walls.
Proved One wall per number m of excluded digits, from the better of the chain and (W): the chain at m = 1..6, 584, 4692, 17596, 46798, 102133, 195891; (W) at m = 7..13, 265981, 294064, 322357, 350906, 379742, 408886, 438358; and (W) at every m >= 14, whose wall sits below m^5 while the chain's sits above it, the smooth (W) gap at base m^5 being at least 0.2947 at m = 14 and growing in m. Every wall exceeds 2m + 1, so the consecutive pair survives, and the prime count and the Mobius bound hold at every choice of m excluded digits from that wall. Witness: coprime.md, the same bullet; lab/py/prime-dissection, verb walls.
Verified The least base from which every one-missing-digit set carries the shifted-grid certificate alpha_1 < 1/5 of the von Mangoldt dissection is 115: the supremum matrix of the window machine, which bounds the shifted grid sum at every depth and every shift, certifies alpha_1 < 1/5 at every excluded digit of every base 115 <= q <= 300 at two or three window digits, the digit-uniform window and chain covering q >= 301; at q = 114 the digit 56 reads [0.2000730, 0.2001280] at three window digits, and every base 3 <= q <= 114 carries a digit certified alpha_1 > 1/5 by the restricted infimum matrix. Witness: lab/py/digit-transform-norms verbs fifth and fifthbelow.
Verified The least base carrying a one-missing-digit set with alpha_1 < 1/5 is 65, missing 0 at alpha_1 < 0.1996822 on three window digits; all 1054 distinct sets of 3 <= q <= 64 certify alpha_1 > 1/5 by the restricted infimum matrix, the closest q = 64 missing 0 at alpha_1 > 0.2000193 on four. Witness: lab/py/digit-transform-norms verb fifth.
Verified The prime count sum_(n <= x, n in S_F) Lambda(n) = kappa_F A_F(x) + O(A_F(x) exp(-c sqrt(log x))) and the Mobius bound abs(M_F(x)) <= C A_F(x) exp(-c sqrt(log x)), both effective, hold on every one-missing-digit set from base 115, by the per-digit shifted-grid certificates alpha_1 < 1/5 at every excluded digit of base 115..300, the digit-uniform window from 301 and the chain from 584; 115 is the floor of this route at one missing digit, base 114 missing 56 being certified above 1/5. Witness: coprime.md PRIMES ON A DESIGN, The per-digit windows reach 115; lab/py/digit-transform-norms, verbs fifth and fifthbelow.