The Mobius meter across digit designs

Fix a base base >= 3 and a digit set F inside {0..base-1} with fill = |F| >= 2. The digit-restricted set S_F holds the positive integers n whose digits at that base all lie in F, with no leading zero: a one-dimensional digit design, the same restriction rule that carves every fractal in this tree, read on the integer line instead of the square. This page measures how much the Mobius function cancels along each design, against the design's own size - and proves that the columns are not independent: digit sets that are scalar multiples of each other carry exactly transferred meters, including one family whose meter vanishes identically and one base-4 pair locked in exact anti-symmetry. Every number is printed by lab/rs/mobius-designs, the divisor section's census numbers by lab/rs/rho-decoupling, the GRH section's by lab/rs/mertens-numerology and the pair route's by lab/py/mobius-region.

The zeros are the other face and they are a different page. The design's own Dirichlet series zeta_F(s) = sum_(n in S_F) n^(-s) has zeros inside its own half-plane of absolute convergence, a comb of them along the pole lattice, three products where the integers have one Euler product, and a Mertens function of its own that runs the wrong way; none of it reaches the meter measured here, and the decoupling is why this page's question is about mu restricted to S_F and nothing else: zeta.

The meter and its yardstick

• A_F(x) counts S_F up to x. The count is exact at every checkpoint: A_F(base^level) = fill^level - 1 when 0 in F (plus 1 when 1 in F too, for the boundary element base^level itself), and A_F(base^level) = (fill^(level+1) - fill)/(fill - 1) when 0 is not in F, by counting digit strings of each length. Proved; the lane's tests pin it against direct enumeration. Between checkpoints A_F(x)/x^(log(fill) / log(base)) carries the log-periodic ripple every design in this tree carries - the classical fluctuation of digital sums (Flajolet, Grabner, Kirschenhofer, Prodinger and Tichy 1994) - so a checkpoint value is a grid value, never a constant.
• The meter is M_F(x) = sum of mu(n) over n in S_F, n <= x, and the exponent is theta(F) = limsup of log|M_F(x)| / log A_F(x). A single cut of |M_F| is a bad estimator - the meter crosses zero freely - so the census prints two readings per level: M_F(base^level) itself, and the running maximum max of |M_F(x)| over x <= base^level, whose exponent thetamax is monotone in the numerator and is the estimator the slope tables use.
• The yardstick matters. A_F(x) grows like x^(log(fill) / log(base)), so S_F is sparse, and a bound of shape o(x) is weaker than the trivial |M_F(x)| <= A_F(x). The indicator of S_F is automatic in that base, so Mullner 2017 (automatic sequences fulfill the Sarnak conjecture) gives M_F(x) = o(x) for every F: orthogonality holds and the question is well-posed, but against the set's own mass that bound says nothing at all. The same shape repeats in base 2 through circuits: the indicator is computable in bounded depth from the binary digits, so Green 2012 also gives o(x), again below the trivial bound. Verified against the literature. The honest question is theta, and it is open at every 2 <= fill <= base - 1.
• The Dirichlet series over S_F is built territory, and this page claims nothing about it: the abscissa is log(fill) / log(base) (Kohler and Spilker 2009, with position-varying digit rules in Nathanson 2021); the series continues meromorphically to C with simple poles among s = log(fill) / log(base) - m + 2 pi i j / log base (the automatic-series mechanism of Allouche, Mendes France and Peyriere 2000, carried out for missing digits in Burnol 2026 and unified in Allouche, Shallit and Stipulanti 2025); a pole lattice of period 2 pi i / log base reads as log-periodic oscillation through the Mellin dictionary of Flajolet, Gourdon and Dumas 1994, and the oscillation is visible in the series' own numerical moments (Burnol 2026 oscillations); the Mobius function itself is automatic in no base, so the Mobius-weighted series inherits none of that continuation (Coons 2010); and no Mobius or Mertens sum appears anywhere in that literature. Verified against the sources in REFS.md. The series does not carry the meter the way zeta carries Mertens: S_F is not multiplicatively closed - at base 3, F = {0,1}, both 4 = 11 and 13 = 111 lie in S_F while 4 x 13 = 52 = 1221 does not - so there is no Euler product and M_F is not the coefficient sum of an inverse series. Proved by that witness.
• The full digit set is the classical boundary. S_F is then every integer, M_F is the Mertens function of Mertens 1897, and M(x) = O(x^(1/2 + eps)) for every eps > 0 is equivalent to the Riemann hypothesis (Titchmarsh 1986, Theorem 14.25 (C)), while limsup |M(x)|/sqrt(x) >= 1.06 unconditionally by Odlyzko and te Riele 1985, so the exponent over all x equals 1/2 exactly when RH holds. Verified against the literature. This page claims nothing about RH: the full-set column below is a control rendered for scale, and the tree's own claims live in the restricted columns.
• The even moments of the digit transform are additive energies. Proved. sum_{a mod base^level} |hat F_level(a/base^level)|^(2r) = base^level E_r(level) with E_r(level) the number of 2r-tuples of digit strings of length level with n_1 + ... + n_r = n_(r+1) + ... + n_(2r) mod base^level, by orthogonality, and E_r(level) is counted by a carry DP on the carry pairs of the two sides, so each moment is C-finite in level of order at most r(r+1)/2 and its growth constant Lambda(2r) = base rho is an algebraic number, rho the Perron root of the transfer matrix, certified in exact rationals (lab/rs/rho-decoupling, the riesz module).
• The fourth-moment constants. Verified. Lambda(4) = 18 at {0,1}, {0,2} and {1,2} in base 3 (rho = 6), 2(23 + sqrt 353) = 83.5766 at {0,1,2} in base 4 (x^2 - 23x + 44), (275 + 5 sqrt 2369)/2 = 259.1809 at {0,1,2,3} in base 5 (x^2 - 55x + 164), 95 at {0,2,4} in base 5, and 6566.412 to 6567.410 over the reflection classes of one excluded digit in base 10; every value sits strictly inside [max(fill^4, base fill^2), base fill^3] and a hair above fill^4 at the dense families (log_base(Lambda(4)/fill^4) is 0.107 at base 3, 0.0004 at base 10), and the sixth, eighth and tenth moments at {0,1} base 3 are 39 + 3 sqrt 79, 3(99 + sqrt 5265)/2 and a cubic (lab/rs/rho-decoupling).
• What a moment buys the bilinear sum. Proved. Holder with the 2r-th moment on the digit side and Parseval on the bilinear side bounds the Type II sum over m up to M and l up to N, 4MN <= x, by x^(theta_p/p + 1/2 - 1/p) with theta_p = log Lambda(p)/log base, which is at least x^(alpha + 1/4) for every even p >= 4 and every digit set, above the trivial x^alpha; so no moment of the digit transform alone beats the trivial bound, and the route needs the bilinear sum on the minor arcs below its own root mean square, which random-sign coefficients defeat on the census (Verified, lab/rs/rho-decoupling the arcs lines).
• The multiplicative energy of a digit column has no exponent of its own. Proved. With E_x(level) = #{(n_1, n_2, n_3, n_4) in D_level^4 : n_1 n_2 = n_3 n_4} and K = fill^level, the two diagonals give 2K^2 - K <= E_x(level), and E_x(level) = sum_m r(m)^2 <= K^2 max_m r(m) with r(m) <= d(m) gives E_x(level) = fill^(2 level) x^(o(1)) for every base and digit set (the census reads 58760487 at {0,1} base 3, level 12, the exponent 1.356938 falling toward 2 alpha = 1.261860); so a Type II sum estimated through the energy obeys |Sigma| <= (2MN)^(1/2) x^(alpha/2 + o(1)) and misses the trivial bound by (1 - alpha)/2; the excess over the diagonal is structure, not arithmetic: the shift family (base^i u, base^j v, base^(i') u, base^(j') v) with i + j = i' + j', counted in closed form when 0 is a digit, is 0.44 of it at {0,1} base 3, level 12 (lab/rs/rho-decoupling, the menergy module).
• Above the sup the L^p norms of the transform buy nothing. Proved. The sandwich max(fill^p, base fill^(p/2)) <= Lambda(p) <= base fill^(p-1) forces Lambda(p)^(1/p)/fill down to 1, and it reads 1.224744, 1.029883, 1.004288, 1.000653, 1.000107 at p = 2, 4, 6, 8, 10 for {0,1} at base 3, so every higher norm is the supremum up to a factor tending to 1 and no ladder of moments reaches past the bullet above (lab/rs/rho-decoupling, riesz higher moments).
• The unbalanced kernel carries no Type II estimate uniform over bounded coefficients at any digit set containing 0. Proved. At a_m = b_l = 1 the Type II sum is the box representation count and some admissible box carries R >= x^(alpha - o(1)), so the trivial bound is attained and the only target left is the balanced sum; there the route returns the box's own trivial bound, the ratio of bound to trivial rising through 1 (1.0134 at level 12 and 1.0730 at level 14 at {0,1} base 3) while the margin of alpha over the achieved exponent falls from 0.035675 to 0.027009, and the digit column is worse for the method than a random column of the same density at every cell of the arc regime (lab/rs/rho-decoupling, menergy type II).
• Two box witnesses floor every coefficient-free route at x^alpha. Verified at the dense cells. At a_m = b_l = 1 the Type II sum reads 0.19 to 0.41 of fill^level over the nine dense cells, and at a_m = 1_(base | m), b_l = 1 the balanced sum still reads 0.0024 to 0.104 of fill^level there, carried by the frequencies a'/base^j at bounded j, which are exactly the major arcs the pair route below removes and computes; the statement for every digit set rests on R >= fill^level (log x)^(-C) and stays Conjecture, and the witness is void at alpha = 0.15, where the box is empty (lab/rs/rho-decoupling).
• The Mobius signs cancel the column's correlation no better than random signs. Verified. The digit column carries a real off-diagonal multiplicative correlation, zero in the mean for a random column of the same density, and the Mobius and Liouville signs cancel it no better than an unstructured sign vector on the same support does, |Sigma_mu| sitting at 0.0913 to 0.7178 of the random-sign root mean square against 0.0359 to 1.5048 for the support-matched controls over sixteen boxes, with the split against those controls 3, 9, 4 at chi-square 0.375 against the uniform-rank null; a sign vector built by greedy flips against a known column drives the same Cauchy-Schwarz bound to 0.0265 of its diagonal floor, so the census measures the arithmetic of the coefficients and not a limit of the method (lab/rs/rho-decoupling, menergy signed and menergy signed engineered).
• The coefficient the method is given is not the coefficient it would need. Verified. At seven of the eight swept boxes with both sides above x^(2/5), the boxes a Vaughan decomposition actually produces, the coefficient sequence it hands the bilinear sum takes values in {-1, 0, 1} and needs no normalisation, and its full quadratic form sits between 0.69 and 1.21 of its own diagonal, where a sign vector engineered against the column reads 0.13 to 0.21 on the same boxes, so the sequence the method is given and the sequence the method would need are different objects (lab/rs/rho-decoupling, menergy signed vaughan).
• The large-values refinement is the moment route itself. Verified. The large-values refinement of the moment route is costed out and is the l^2 route itself, exponent (1 + alpha)/2 at every threshold (lab/rs/rho-decoupling, riesz large values chain, 66 cells over six families); the large frequencies are adjacent grid points (407 in 331 runs at {0,1} base 3, level 12, eta = eta_4), so the grid offers no spacing gain, and at the dense families the bilinear sum at a_m = b_l = 1 equals the box representation count, 0.38 fill^level there, so no bound uniform over bounded coefficients holds at those families.

The exact transfer between designs

The census columns are tied together by one carry-free mechanism. Proved:

• Scaling. If every digit of F is a times a digit of F', so F = aF' inside {0..base-1}, then m -> am maps S_F' bijectively onto S_F preserving digit length: am = sum (a d_j) base^j and each a d_j <= base - 1, so no carry occurs and the digit string scales digitwise. Hence A_F(base^level) equals the string count of F' at the same depth, and M_F(base^level) = sum of mu(am) over m in S_F' with at most level digits.
• Vanishing. If a has a square factor then mu(am) = 0 for every m, so M_F is identically zero: at base 5, F = {0,4} = 4 x {0,1}, the meter reads 0 at all 21 levels. A census that reads cancellation without factoring out the digit gcd reads this as infinite cancellation; the digit gcd must be squarefree before theta means anything.
• Prime twist. If a = p is prime then mu(pm) is -mu(m) on p-free m and 0 otherwise, so M_(pF')(base^level) = -sum of mu(m) over the m in S_F' not divisible by p. At base 3, F' = {0,1}: an element m = sum of 3^j is odd exactly when its count of 1-digits is odd, and reading the digit string as a binary index that parity is the Thue-Morse sign, so the {0,2} column is the Thue-Morse-twisted {0,1} column.
• Base-4 anti-symmetry. At base 4, M_{0,2}(4^level) = -M_{0,1}(4^level) exactly: since 4 | base, an element of S_{0,1} is 0 or 1 mod 4 by its unit digit, so every even element is divisible by 4 and carries mu = 0, and the odd-part twist above is minus the whole meter. Stronger, M_{0,2}(x) = -M_{0,1}(x/2) at every real x, and since S_{0,1} has no element strictly between (4^level - 1)/3 and 4^level the running maxima agree level by level as well. The census confirms both at all 22 levels, e.g. meters -110/110 at level 15, -342/342 at level 17, 34/-34 at level 22, and Mmax = 1553 for both at level 22.

Verified: the generator recomputes all eight scaled census families ({0,2} at base 3; {0,2}, {0,3} at base 4; {0,2}, {0,3}, {0,4}, {2,4}, {0,2,4} at base 5) from their primitive families through mu(am) and asserts equality at every level. The mechanism needs a common digit factor, so it partitions the census into primitive columns and their twists and says nothing across primitive columns.

The census

Every M_F(base^level) below is an exact integer: restricted families enumerated in ascending order with mu from deterministic factorization (trial division, Miller-Rabin on the twelve witnesses 2..37, Pollard rho), controls by a linear Mobius sieve; one family (base 3, F = {1,2}, level 16) is computed by both methods and asserted equal at every level. The base-10 control reproduces A084237 (-1, 1, 2, -23, -48, 212, 1037, 1928 at 10^1..10^8). Every table below is extracted by script from the generator's printed rows, never assembled by hand. Verified.

The three base-3 columns, checkpoint meter and running maximum Mmax = max of |M_F(x)| over x <= 3^level per row:

The final checkpoint of every family, with thetamax = log(Mmax)/log A and its drift (max minus min) over the last five levels:

Base 10 with one digit excluded, the Kempner designs (fill = 9, the sets behind the convergent harmonic series of Kempner 1914, revisited at s = 1 in Allouche, Hu and Morin 2024), at x = 10^8:

The full-set controls at comparable depth: M(3^17) = -1423 with Mmax = 4610 (thetamax 0.4517), M(4^13) = 329 with 2845 (0.4413), M(5^11) = 617 with 2573 (0.4436), M(10^8) = 1928 with 3448 (0.4422). The Mertens function itself - limiting exponent exactly 1/2 if and only if RH, and at least 1/2 unconditionally - reads 0.4413..0.4517 at these depths, which calibrates every reading above: at census mass even the classical meter sits a few hundredths under 1/2.

The distribution of the apparent exponent across designs at fixed base, sorted by the generator: at base 3 the three columns read 0.4802, 0.4869, 0.4962; at base 4 the ten run 0.4527 to 0.5197; at base 5 the twenty-four with nonzero meter run 0.4471 to 0.5358; at base 10 the ten Kempner columns run 0.4465 to 0.5273. All 47 readings sit within 0.054 of 1/2, against cut readings (theta at the checkpoint alone) that scatter over 0.22..0.53 for the same data - the single-cut estimator is noise, the running maximum is the meter.

Digit strings across divisors

The meter weighs mu along a design; this section weighs the design itself against a divisor, the arithmetic input any multiplicative estimate over S_F has to have. Write N_F(level; d, r) for the number of digit strings of length level over F whose value sum_j f_j base^j is r mod d, and N_F(level; d) = N_F(level; d, 0). The value map is injective on strings of one length, so with 0 in F this counts the multiples of d below base^level whose padded digits lie in F, and with 0 outside F it is the block of S_F at length level, the blocks l <= level partitioning S_F below base^level. Write fill = |F|, e(x) = exp(2 pi i x), g_F(t) = sum_{f in F} e(f t) (the digit transform hat F itself, written g_F on this page), Delta_F for the gcd of the digit differences, gamma_F(d) = max over a not 0 mod d of |g_F(a/d)|/fill, and normalized error for d |N_F(level; d) - fill^level/d| / fill^level. Residue distribution of digit-restricted sets is the subject of Erdos, Mauduit and Sarkozy 1998; what follows is derived here from the transform, each statement carrying its own hypotheses, and alpha = log(fill) / log(base) is the design's dimension throughout. Every census number is printed by lab/rs/rho-decoupling; the rest is exact arithmetic carried out in the sentence that prints it.

• Orthogonality. Proved. N_F(level; d, r) = (1/d) sum_{a mod d} e(-a r/d) prod_{j < level} g_F(a base^j/d): expand the divisibility indicator in additive characters mod d; the digits are independent, so the character sum factors over positions. The a = 0 term is fill^level/d and every bound below is a bound on the rest.
• The uniform geometric bound. Proved. For fill >= 2, d >= 2, (d, base) = 1 and gcd(d, Delta_F) = 1, every r and every level >= 1: |N_F(level; d, r) - fill^level/d| <= ((d-1)/d) fill^level (1 - 8/(fill^2 d^2))^level <= fill^level exp(-8 level/(fill^2 d^2)). Coprimality to base keeps a base^j nonzero mod d at every position, |g_F(a/d)|^2 = fill^2 - 4 sum_{f < f'} sin^2(pi a (f' - f)/d), and if d divided a (f' - f) for every pair then d/gcd(a, d) would divide Delta_F and force d | a, so one pair sits at distance >= 1/d from an integer and gamma_F(d)^2 <= 1 - 16/(fill^2 d^2). The census asserts the weaker form as an exact integer inequality at every cell where the hypotheses hold; the largest observed-to-bound ratio is 0.187, at base 100, F = {0,1}, level 16. The exponent d^(-2) is not slack: at d | base - 1 with F an arithmetic progression of common difference m', taking a m' = 1 mod d gives |g_F(a/d)|/fill = sin(pi fill/d)/(fill sin(pi/d)) = 1 - Theta(fill^2/d^2). Summed over a range it is microscopic and never a route on its own: D exp(-8 level/(fill^2 D^2)) < 1 fails past D ~ sqrt(level)/fill, so this bound alone certifies a level of distribution of that size and nothing like a power of x.
• The dense-digit bound. Proved. For F = {0..base-1} minus E with m = |E|, fill = base - m and (d, base) = 1: gamma_F(d) <= (d/2 + m)/fill, since g_F is the full Dirichlet kernel less g_E, |D_base(a/d)| <= 1/(2||a/d||) <= d/2 and |g_E| <= m; hence for d/2 + m < fill the error is at most fill^level ((d/2 + m)/fill)^level, uniform in r.
• A power saving at level base^(1-eps). Proved. Fix eps in (0,1) and take base >= 4^(1/eps), m <= base^(1-eps)/2, level >= 4/eps. Every 2 <= d <= base^(1-eps) coprime to base then has per-digit factor (d/2 + m)/fill <= base^(-eps/2), so sum over those d of |N_F(level; d) - fill^level/d| <= fill^level base^(1 - eps level/2) <= fill^level x^(-eps/4) at x = base^level: a power saving over the whole block, not one divisor at a time. The saving is carried by the digit count and not by the base. At the fixed divisor d = 7 the per-digit error rate reads 0.4869, 0.3312, 0.2484, 0.1104, 0.0167 along base 3, 4, 5, 10, 100 at the designs {0,1}, {0,1,2}, {0,1,2,3}, {0..9} less 7 and {0..99} less 37, as fill runs 2, 3, 4, 9, 99, against per-factor ceilings gamma_F(7) = max_a |sum_{f in F} e(a f/7)|/fill reading 0.9010, 0.7490, 0.5617, 0.2002, 0.0221, the ceiling of the design and not of fill, so the dense row's 0.0221 is attained at a = 2 while the other four are attained at a = 1; for F = {0,1} at base 100 it stays 0.4992 with ceiling 0.9010, the same ceiling F = {0,1} has at base 3.
• The split across the base's own divisors. Proved. For d = d1 d2 with d1 | base^m for some m <= level and (d2, base) = 1, the low m digits fix the value mod d1 and reach the rest only through the invertible multiplier base^m mod d2, so N_F(level; d) = sum over w in F^m with d1 | val(w) of N_F(level - m; d2, r_w), r_w = -val(w) (base^m)^(-1) mod d2. The density splits exactly, rho_F(d1 d2) = (N_F(m; d1)/fill^m) (1/d2), and the base part is a digit-string count rather than 1/d1: a divisor sharing a factor with base is read off the digits, never off a density. Checked against direct enumeration at base 6, d = 10.
• The digit-gcd hypothesis is a wall, not a convenience. Proved. If gcd(d, Delta_F) > 1 there is no equidistribution at all: at base 3, F = {0,2}, d = 2 every value is even, N_F(level; 2) = fill^level, and the normalized error is exactly 1 at every level. Over d <= 200 the unrestricted worst error for that family reads 1.0483 at level 32, pinned at d = 164, against 0.019166 once d is required coprime to Delta_F. Such families reduce to a primitive one through S_(aF') = a S_(F'), the scaling map of the transfer above.
• The slow column at fixed digit count. Verified. The bound decays in level only, at a rate the digit count controls, and the census sees nothing better: F = {0,1} at base 100 has worst normalized error 28.593, 14.590, 9.0340, 7.2034 at level 16, 32, 64, 96 over d <= 500, per-digit factor 0.9929, with argmax d = 481 | base^3 - 1 at the first two depths and d = 303 | base^2 - 1 at the last. Sparse digit sets are outside the reach of every per-divisor estimate here, exactly as they are outside the reach of the exponent census above.
• The worst divisor is pinned. Verified. The obstruction is small multiplicative order: the orbit a base^j mod d visits only ord_d(base) points, so no averaging happens across positions, and at every family's deepest level the sweep argmax has ord_d(base) <= 8, hence divides base^t - 1 with t <= 8 (d = 164 at base 3, d = 143 at base 10, d = 101, 303, 481 at base 100); shallow depths can stray, d = 199 with ord = 99 at base 10, level 6.
• The signed pinned sum does not cancel. Verified. Weight each squarefree pinned modulus e = (base^t - 1)/g, g | base - 1, e >= 2, t <= level, by mu(e), with T_level(e) = N_F(level; e) - fill^level/e, and set the signed sum Sigma_level = sum mu(e) T_level(e) against the absolute sum Abs_level = sum |T_level(e)| over the same moduli. Printed at every level 3..40 by the generator, the ratio Sigma_level/Abs_level swings across [-1, 1] (-1.00 at level 5, 6 in the first family) with no decay: -0.211, -0.123, +0.069, -0.498 at level 10, 20, 30, 40 for F = {0,1} at base 3 and +0.812, -0.495, -0.127, -0.192 for one excluded digit at base 10, while Abs_level/fill^level reads 2.1 * 10^-4 and 3.9 * 10^-12 at level 40; the sum rests on 6 of 29 terms in the first family and 4 of 60 in the second, the four largest at t = 7, 9 and at t = 5, 7, 8, 10. The signed weight is itself a Mertens-type sum and that is why signing buys nothing here: with T_level(d) = N_F(level; d) - fill^level/d and P_level(e) = sum_{(a,e) = 1, 0 < a < e} hat F_level(a/e), real because a and e - a conjugate, grouping each frequency by its reduced denominator gives T_level(d) = (1/d) sum_{e | d, e >= 2} P_level(e) and hence sum_{d <= U} mu(d) T_level(d) = sum_{e >= 2} (mu(e)/e) M_e(U/e) P_level(e) with M_e(y) = sum_{f <= y, (f,e) = 1} mu(f)/f, Proved; a pinned primitive modulus enters weighted by M_e, so mu(e) fixes the sign and a Mertens-type sum fixes the size, and the signed route restates the wall one layer down rather than escaping it. No bound on M_e(y) uniform in e is available to lean on: at the primorial e of all primes up to P and y = P the only f <= y coprime to e is f = 1, so M_e(P) = 1 exactly, Refuted for any unrestricted uniformity. Every N_F(level; e) is an exact integer of the carry count, which sums the digits in each residue class of positions mod t and counts the targets j e by carries, polynomial in level at every t; mu(e) is read off a complete factorisation of every base^t - 1 to t = 40, cyclotomic factors first, then Pollard-Brent, every prime certified by deterministic Miller-Rabin below 3.317 * 10^24. The sign of mu across the family does not organise the errors: a signed Type I sum over these moduli buys only a bounded factor over the absolute one, |Sigma_level|/Abs_level reading 0.498 and 0.192 at level 40, and that factor does not grow with depth at any depth computed.
• The worst orbit at a pinned divisor, two-sided. Proved. The full kernel's orbit product telescopes: for t >= 1, d | base^t - 1 with d >= 2 and a nonzero mod d, prod_{j < t} |D_base(a base^j/d)| = 1 exactly, since a base^t = a mod d and d | base^t - 1 forces (d, base) = 1, so no factor degenerates and the full digit set sees no closed shift orbit at all: every damping comes from the excluded digits. With |g_F| <= |D_base| + m and |D_base(a base^j/d)| <= B = min(base, d/2), convexity of log(e^y + m) puts the maximum of sum_j log(D_j + m) on {sum_j log D_j = 0, log D_j <= log B} at a vertex and gives prod_{j < t} |g_F(a base^j/d)| <= (B + m)^(t-1) (m + B^(1-t)). At one excluded digit that is sharp both ways: for d = base^t - 1, t >= 2, base >= 10 and any single excluded digit, fill^(-1/t) (1 - 9/base) <= max_{a not 0 mod d} (prod_{j < t} |g_F(a base^j/d)|/fill^t)^(1/t) <= fill^(-1/t) (1 + 3/(base-1)) uniformly in t, the lower bound witnessed by a = 1 through |e(f y) - 1| <= 2 pi f y at the first t - 1 positions and sin(pi y) >= 2 y at the last. So the orbit carries t - 1 undamped positions and one damped by ~ 1/fill, and the 1 + o(1) is a two-sided O(1/base) that does not grow with t; the folded constants 9/base and 3/(base-1) are stated at base >= 10 and are recomputed before any smaller base quotes them. The census's pinned probes read a different quantity, the finite-depth error rate (|N_F(level; d) - fill^level/d|/fill^level)^(1/level) d^(1/level) at level 12 and base 100: 0.1059 at d = base^2 - 1 and 0.2369 at d = base^3 - 1 against fill^(-1/2) and fill^(-1/3), with the proper divisor d = 3367 | base^3 - 1 at 0.0549 and d = 101 | base + 1 pinned but harmless at 0.0261. The d^(1/level) of that normalisation is why a depth-12 rate sits beside the band and not inside it, and the probes corroborate the size rather than test the bound. The a-average at the same modulus is exact, at every digit set and every m: sum_{a mod d} prod_{j < t} |g_F(a base^j/d)|^2 = d (fill^t + 2w) for d = base^t - 1, with w = 1 when both 0 and base - 1 lie in F and w = 0 otherwise, since congruent pairs of length-t strings are the diagonal plus the one wraparound pair {0...0, (base-1)...(base-1)} when both endpoints are strings over F. Under the band's own hypotheses, one excluded digit and base >= 10, the worst orbit therefore exceeds the average over all a, which is fill^t + 2w, by fill^(t-2) e^(O(t/base)). Most of that average is its own a = 0 term fill^(2t)/d, 970299/101 of 9803 at base 100 and t = 2, so the average a second moment actually sees, over a nonzero, is (d (fill^t + 2w) - fill^(2t))/(d - 1), smaller again by ~ t/base and 980298/4999 at the same cell: the spread is wider than the exponent states, never narrower. From t = 3 on only a fill^(2-t) fraction of residues can sit near the worst orbit: at a pinned divisor the bad mass is spread, and an average over a is the one handle the supremum gives up.
• The second moment across residues. Proved. sum_{r mod d} (N_F(level; d, r) - fill^level/d)^2 = (1/d) sum_{a not 0 mod d} prod_{j < level} |g_F(a base^j/d)|^2, by Parseval mod d on the orthogonality identity: the mean is the a = 0 term, the variance is the rest, no cross terms survive. It gives up the supremum over r and buys an average over a, which is the one place a saving can survive at a pinned divisor, where every per-factor bound is flat.
• No moment past the second helps at a pinned divisor. Proved. At d = base^t - 1 the 2r-th orbit moment sum_{a mod d} prod_{j < t} |g_F(a base^j/d)|^(2r) is again an additive energy, d times the count of 2r-tuples of length-t strings over F whose two halves have equal value sum mod d, by the same orthogonality as the even moments above; and the pair-count certificate of the bisection bullet below places that energy a factor 4 (base/fill)^(r t) above its own mean fill^(2 r t)/d, a loss growing in r. So the second moment is the only average over a a certificate delivers near that mean, and a chain built on the supremum over a and that second moment is already optimal for the two inputs it has.
• The bisection bound, per divisor. Proved. For every F with fill >= 1, every d >= 2 coprime to base, every level >= 1 and uniformly in r: |N_F(level; d, r) - fill^level/d| <= fill^(level/2) (1 + 2 base^((level+1)/2)/d). Cut the string in the middle and apply the second moment above to each half at depths ceil(level/2) and floor(level/2): at a half's depth b the variance is fill^b (1 + 2 base^b/d), since off the diagonal a congruent pair of length-b strings needs val(f) - val(f') = j d with 0 < |j| <= (base^b - 1)/d and val is injective on strings of one length, so each (f', j) fixes at most one f; then (d + 2 base^(b_1))(d + 2 base^(b_2)) <= (d + 2 sqrt(base) base^(level/2))^2. Two readings follow. At d >= sqrt(base x) it gives max_r |N_F(level; d, r) - fill^level/d| <= 3 fill^(level/2) = 3 fill^level x^(-alpha/2), asking nothing of F and nothing of d past coprimality to base, so the whole top range [sqrt(base x), x], where the orbit machinery says nothing, is covered by one line. And the pair count overshoots its own mean fill^(2b)/d by exactly the factor 2 (base/fill)^b, the wraparound factor every route below inherits.
• Level alpha/2 for the whole block, up to one factor. Proved. Summing that bound over 2 <= d <= D for a divisor cutoff D, against sum_{d <= D} 1/d <= 1 + log D, gives, for every base >= 3, every F with fill >= 1, every level >= 1 and every D >= 2, sum_{2 <= d <= D, (d,base) = 1} max_r |N_F(level; d, r) - fill^level/d| <= fill^level (D fill^(-level/2) + 2 sqrt(base) (1 + log D) (base/fill)^(level/2)), every d and not only the squarefree ones, supremum over the target residue and not only the residue 0. Read at D = x^theta with theta <= alpha/2, the first term is at most 1 and the whole sum is at most 3 sqrt(base) (1 + log x) fill^level x^(m/(2 fill log base)) on a set of m = base - fill excluded digits, since (base/fill)^(level/2) = e^((level/2) log(1 + m/fill)) <= e^(m level/(2 fill)) = x^(m/(2 fill log base)). That is a level of distribution x^(alpha/2 - o(1)) at every theta up to alpha/2 at once, carrying a defect x^(m/(2 fill log base)) that is sub-power in base and a positive power in x: the exponent is 1/(2(base-1) log base) at one excluded digit, under 0.0011 at base 100. The defect does not vanish as level grows at fixed base, so this is a level statement and not an equidistribution statement.
• The assembled theorem across the whole divisor range. Proved. Fix eps in (0,1) and an integer T_0 >= 2, and put base_0(eps, T_0) = max(4^(1/eps), base_1) with base_1 any base satisfying 3 base_1^(-eps)/log base_1 <= eps/(16 T_0). For every base >= base_0, every F = {0..base-1} minus E with 1 <= m <= base^(1-eps)/2, every level >= max(6 T_0, 4/eps) and x = base^level: (i) at every level D <= x, the sum of max_r |N_F(level; d, r) - fill^level/d| over 2 <= d <= D coprime to base with d <= base^(1-eps) or ord_d(base) <= T_0 is at most (T_0 + 2)(1 + log x) fill^level x^(-eps/(8 T_0)); (ii) at level x^(alpha/2) the full sum is at most 3 sqrt(base) (1 + log x) fill^level x^(m/(2 fill log base)); (iii) every d coprime to base with sqrt(base x) <= d <= x has max_r |N_F(level; d, r) - fill^level/d| <= 3 fill^level x^(-alpha/2); and (iv) at one excluded digit the full pinned moduli are damped together, sum_{2 <= t <= level} max_r |N_F(level; base^t - 1, r) - fill^level/(base^t - 1)| <= level fill^(2 - sqrt(2 level)) e^(4 level/base) fill^level. Clause (i) is the level-base^(1-eps) saving and the bounded-order classes summed, proper divisors included; clause (ii) is the block above; clause (iii) is the top range; clause (iv) is the orbit bound of the pinned bullet against the exact a-average there, summed over t. No clause asks d squarefree and every clause is a supremum over the target residue. Clause (iv) is superpolynomial in level and not a fixed power of x: its saving is worst at t ~ sqrt(2 level), where as x^(-c) the exponent c falls to 0 with level.
• No clause in this norm can be a fixed power of x, and clause (iv) is not slack. Proved. At one excluded digit and base >= 10 the two-sided orbit law above supplies the matching lower bound at a single modulus. Take d = base^t - 1 with t = ceil(sqrt(level)), so the orbit closes floor(level/t) times inside level positions and the worst a carries |hat F_level(a/d)| >= fill^level x^(-O(1/sqrt(level))); the second moment across residues then gives max_r |N_F(level; d, r) - fill^level/d| >= |hat F_level(a/d)|/d, and d <= base^(sqrt(level) + 1) = x^(O(1/sqrt(level))) costs only the same shape again. Hence for every fixed theta > 0 and every level >= max(9, 4/theta^2) the sum over 2 <= d <= x^theta coprime to base of max_r |N_F(level; d, r) - fill^level/d| is at least fill^level x^(-O(1/sqrt(level))), carried by that one modulus. Every clause above is in the supremum norm, so no assembly of them reaches a fixed power of x and a level of distribution at a fixed power has to pass through signed sums.
• What those clauses leave, named. Proved. One family survives them at a fixed level: the generic-order middle moduli base^(1-eps) < d <= x^theta coprime to base with ord_d(base) > T_0, where no orbit period closes inside level and the pinned machinery is silent. There only clause (ii) applies, so the block is certified at level alpha/2 up to the single factor x^(m/(2 fill log base)), and that factor is the whole distance between what is proved and a fixed power of x. It is the wraparound overshoot and not slack in a constant: the pair-count certificate, the orbit moment and the additive large sieve over the Farey points share one diagonal, certify pair counts only to one-per-pair precision, and exceed the heuristic fill^(2b)/d by 2 (base/fill)^b, which at the balanced depth b = level/2 the certificate forces is exactly that defect. No rearrangement of cuts, no Cauchy-Schwarz and no divisor bookkeeping tried here removes it, and the reason is circularity rather than looseness: with the pair-count certificate alone both halves of a cut need base^b <= d, so b <= 2 log_base d and the certificate reads 3 d^(1 - alpha) > 1, missing the dip by exactly the sparsity of F, while winning asks the depth b ~ (2/alpha) log_base d at which fill^b ~ d^2 strings meet d classes, and equidistribution there is the statement being proved.

A power saving under GRH at large base

The census above measures cancellation and proves none of it. This section proves some, at the opposite end of the digit scale: not the sparse columns of the census but the dense ones, the base taken large and a single digit removed. There the mass exponent alpha_base = log(fill) / log(base) sits just under 1, the indicator of S_F opens into additive frequencies by the same orthogonality the divisor section uses, and each frequency carries a Mobius exponential sum, which under the generalized Riemann hypothesis is x^(3/4 + eps) uniformly in the frequency. The entire cost of the expansion is one l^1 norm, and past a computable base that cost is smaller than the mass. What comes out is a bound of Mertens shape read against the set's own counting function, conditional and dense-only, with both of those limits proved rather than assumed. Every constant, table row, margin and rung below is printed or test-pinned by lab/rs/mertens-numerology, and the sharpened one-excluded-digit constants of step 3, with the walls they move, by lab/py/mrly-pairing, verb onestep.

Proved under GRH below is the Proved tag with the hypothesis written inside the statement: steps 1, 2, 3 and 5 are derived here from definitions, and step 4 is one published theorem, quoted at its source and used exactly as stated.

• The setting. E is the excluded digit set with m = |E| >= 1, F = {0..base-1} minus E, fill = base - m and alpha_base = log(fill) / log(base), so A_F(x) >>_base x^(alpha_base) at every x by the counting identities of the first section. The digit symbol is g_F(t) = sum_{d in F} e(d t) of the divisor section and D_base(t) = sum_{d = 0}^{base-1} e(d t) is the full Dirichlet kernel, |D_base(t)| = |sin(pi base t)/sin(pi t)|. Write D_level for the digit strings of length level over F and hat F_level(t) = sum_{n in D_level} e(n t) for the transform at that level, which factors as prod_{j < level} g_F(base^j t) because the digits are independent. The one-step constant of the shifted-grid recursion is B_base(F) = sup_t sum_{r mod base} |g_F((t+r)/base)|, bounded above by base PB_base(m) in step 3, with the proved constant PB_base(m) = sqrt(m) + Phi_base/base, where Phi_base = (4/pi) base + (2 base/pi) H(ceil((base-2)/2)) + (1 - 2/pi)(base - 2) + 0.727 and H(n) = ln n + gamma + 1/(2n); the exponent cost is c_base = log PB_base(m)/log base, defined from that proved bound and never from the exact supremum. At m = 1 and base >= 17 step 3 sharpens per excluded digit: with e_0 the single excluded digit and c = e_0 - (base-1)/2, the sharpened bound of the step 3 bullet is base PB_base(1, e_0) with PB_base(1, e_0) = PB_base(1) - 1/2 - (sec(pi c/base)/2 + 0.727 - 2(1 - 2/pi))/base, and the cost it carries is c_base(e_0) = log PB_base(1, e_0)/log base, again read from a proved bound and never from the exact supremum. From base >= 36 the chord bullet of step 3 replaces the kernel constant Phi_base by (4/pi) base + Psi'_base, which is Phi_base - base/2 + 2/pi up to the 0.00023954 of 0.727 at even base, and the second per-digit form is base PB'_base(1, e_0) = (4/pi) base + Psi'_base + base/2 - sec(pi c/base)/2, carrying the cost c'_base(e_0) = log PB'_base(1, e_0)/log base, proved like the first and read like the first. Everything below with m > 1 in it, the corollary and the m-budget table included, runs on PB_base(m) alone.
• Theorem. Proved under GRH. Assume the generalized Riemann hypothesis in its Dirichlet form: L(s, chi) has no zero in the half plane sigma > 1/2, for every Dirichlet character chi of every modulus. Let F omit exactly one digit e_0, let base >= 1499, or base >= 1032 when e_0 is 0 or base - 1, and let eps > 0. Then |M_F(x)| <<_{base,eps} x^(3/4 + c'_base(e_0) + eps) for all x >= 2; and 3/4 + c'_base(e_0) < alpha_base at every such base, so with delta_base(e_0) = (alpha_base - 3/4 - c'_base(e_0))/alpha_base > 0 the same bound reads |M_F(x)| <<_{base,eps} A_F(x)^(1 - delta_base(e_0) + eps), a power saving against the set's own mass. The implied constant depends on base and on eps and on nothing else; no uniformity in base is claimed anywhere. delta_base(e_0) is fixed before eps is chosen, so the statement delivers every fixed delta' < delta_base(e_0) and never the endpoint A_F(x)^(1 - delta_base(e_0)). The two walls are m = 1 only: at m > 1 the corollary below carries the unsharpened PB_base(m) and its own condition.
• Step 1, orthogonality. Proved. For 0 <= n < base^level, 1_{D_level}(n) = base^(-level) sum_{0 <= a < base^level} hat F_level(a/base^level) e(-n a/base^level) by completeness of the additive characters mod base^level, and hat F_level factors over digit positions: the identity of the divisor section with the modulus base^level in place of d, read as an expansion rather than as a count.
• Step 2, the l^1 recursion. Proved. Put c_level = sum_{0 <= a < base^level} |hat F_level(a/base^level)| and split a = a' + s base^(level-1). The transform peels at the position j = 0, hat F_level(t) = g_F(t) hat F_{level-1}(base t), so s moves that factor alone and hat F_{level-1} is 1-periodic; the inner sum over s mod base is a shifted grid of base points, and c_level = sum_{a'} |hat F_{level-1}(a'/base^(level-1))| sum_{s mod base} |g_F((a'/base^(level-1) + s)/base)| <= B_base(F) c_{level-1}. Hence c_level <= B_base(F)^level and the normalized l^1 mass is base^(-level) c_level <= (B_base(F)/base)^level: one constant per digit, no interaction between positions.
• Step 3, the kernel bound. Proved. |g_F| <= |D_base| + |g_E| splits B_base(F) into a kernel part and an excluded part. The base points (t + r)/base are spaced 1/base; writing d_r for the distance of each to Z, |D_base((t+r)/base)| = |sin(pi base d_r)|/sin(pi d_r), the two points nearest the singularity contribute at most (4/pi) base + 0.727 by the two elementary inequalities sin(pi v) <= 4v(1-v) on [0, 1/2] and 1/sin x <= 1/x + 1 - 2/pi on (0, pi/2] (the first because 4x(1-x) - sin(pi x) splits into a concave and a convex piece with the right signs, the second because 1/sin x - 1/x increases), and the remaining base - 2 points pair off at distances >= j/base and contribute at most (2 base/pi) H(ceil((base-2)/2)) + (1 - 2/pi)(base - 2). So sup_t sum_{r mod base} |D_base((t+r)/base)| <= Phi_base. The excluded part is exact rather than estimated: the excluded digits are distinct mod base, so Parseval on the shifted grid gives sum_{r mod base} |g_E((t+r)/base)|^2 = base m for every t, and Cauchy-Schwarz turns that into sum_{r mod base} |g_E((t+r)/base)| <= base sqrt(m). Hence B_base(F) <= base PB_base(m) and the normalized l^1 mass of step 2 is at most base^(level c_base).
• Step 3 sharpened at one excluded digit. Proved. At m = 1, base >= 17 and c = e_0 - (base-1)/2 the triangle split of step 3 is lossy by a fixed share of base, and the loss is taken back with no new input: B_base(F) <= (4/pi) base + Psi_base + base/2 - sec(pi c/base)/2 = base PB_base(1, e_0), where Psi_base = (2 base/pi) H(ceil((base-2)/2)) + (1 - 2/pi) base is the kernel constant less its two-point part and base PB_base(1) = (4/pi) base + Psi_base + base + 0.727 - 2(1 - 2/pi), so the m = 1 mass of step 2 is at most base^(level c_base(e_0)). At one excluded digit the shifted grid is exact, |g_F((t+r)/base)| = |A_r - e(c (t+r)/base)| with A_r = (-1)^r sin(pi t)/sin(pi (t+r)/base), because D_base((t+r)/base) factors as a unimodular phase times (-1)^r sin(pi t)/sin(pi (t+r)/base). Then |a - e(psi)|^2 = (a+1)^2 - 2a(1 + cos psi) with sqrt(1 - X) <= 1 - X/2 produces a correction term, the phase identity sum_{r mod base} (1 + sign(A_r) cos(2 pi c (t+r)/base)) = base + cos(2 pi c (t - 1/2)/base)/cos(pi c/base) sums it and is at least base + 1 since |2 pi c (t - 1/2)/base| <= |pi c/base| < pi/2, |A_r| >= sin(pi t) = s and s/(1 + s) >= s/2 weight it, step 3's own two-point and pairing estimates give the t-dependent kernel bound sum_{r mod base} |D_base((t+r)/base)| <= (4/pi) base + s Psi_base, and h(tau) = cos(pi tau)(Psi_base - base/2) - cos(pi tau) cos(2 pi c tau/base)/(2 cos(pi c/base)) has h' <= 0 on [0, 1/2] once Psi_base >= (1 + pi) base/2, first true at base 17, by sin(pi tau) >= 2 tau, sin y <= y and sec(pi c/base) <= base. The bound falls as |c| rises, so the two extreme digits e_0 in {0, base-1} carry the smallest constant, sec(pi c/base) = 1/sin(pi/(2 base)) there, and the middle digit the largest, sec(pi c/base) = 1 at odd base (lab/py/mrly-pairing, verb onestep).
• Step 3's kernel constant, replaced by its chord. Proved. csc x - 1/x has an all-positive Taylor series on (0, pi/2], so it is convex there and lies under its own chord, 1/sin x <= 1/x + (2/pi)(1 - 2/pi) x, half the flat 1 - 2/pi of step 3 on average and equal only at the endpoint. Pairing r with base - 1 - r sends the shifted grid to the argument pairs pi (t+r)/base and pi (1-t+r)/base at r < floor(base/2), every one inside (0, pi/2] for t in (0, 1/2], and K(t) = sum_{r mod base} |D_base((t+r)/base)| = sin(pi t) sum_{r mod base} 1/sin(pi (t+r)/base) is symmetric about t = 1/2, which carries the rest of the circle. The 1/x half of the r = 0 pair is sin(pi t) base/(pi t (1-t)) <= (4/pi) base and the 1/x half of each r >= 1 pair has t-free maximum 1/r + 1/(r+1), while the chord halves collapse to (1 - 2/pi) base/2 because the paired argument sum is exactly base^2/4 at even base; at odd base the unpaired middle term makes it P(P+1) + t, which is where the odd form's extra 1/(2 base) comes from. So K(t) <= (4/pi) base + sin(pi t) Psi'_base with Psi'_base = (base/pi)(2 H(P-1) - 1 + 1/P) + (1 - 2/pi) base/2 at even base, P = floor(base/2) and H the same harmonic upper bound, and Psi'_base = (base/pi)(2 H(P-1) - 1 + 2/P) + (1 - 2/pi)(base/2 + 1/(2 base)) at odd base; at even base that is exactly Psi_base - base/2 + 2/pi, and at odd base Psi_base - Psi'_base = (2 base/pi)(H(P) - H(P-1)) + (base/pi)(1 - 2/P) + (1 - 2/pi)(base/2 - 1/(2 base)), whose first term is positive at P >= 2 because H(P) - H(P-1) = ln(P/(P-1)) - 1/(2P(P-1)) with ln(P/(P-1)) > 1/(P - 1/2) by the midpoint rule on the convex 1/x, and 1/(P - 1/2) > 1/(2P(P-1)) is 2P^2 - 3P + 1/2 > 0, whose second is nonnegative at P >= 2 and whose third is positive, so Psi'_base < Psi_base at every base >= 5 with no scan. Nothing downstream changes, so the sharpening above runs on it verbatim: B_base(F) <= (4/pi) base + Psi'_base + base/2 - sec(pi c/base)/2 = base PB'_base(1, e_0) wherever the monotone step's hypothesis Psi'_base >= (1 + pi) base/2 holds, first at base 36 on this page's reading H(n) = ln n + gamma + 1/(2n) and at base 37 on the harmonic number itself, an over-estimate that can only keep the kernel lemma true and that moves no printed wall, the lowest being 1032. The hypothesis is sufficient and not necessary: the maximum of h(tau) it exists to place at tau = 0 sits there at every e_0 from base 8 up on the exhaustive scan 4..79. The chord cuts the gap to the exact kernel sup K_base by a factor 5.98, the up-rounded gap columns reading Psi_base - (K_base - (4/pi) base) <= 0.600121 base against Psi'_base - (K_base - (4/pi) base) <= 0.100293 base at base 3690, the lemma's own slack floored to 0.100292 base there, the chord column flat to 1e-5 across base 100, 1000, 2234 and the gap attained at the seat t = 1/2; and nothing measured reaches the new bound, the worst ratio of the exact B_base(F) to it being 0.902124 over every e_0 at base 36..60, 0.941239 at the larger seats and 0.936333 over 4000 seeded draws (Verified for the measured columns, lab/py/mrly-pairing, verb onestep).
• Step 4, the Mobius input. Quoted. Under GRH, max_{theta in [0,1)} |sum_{n <= x} mu(n) e(n theta)| <<_eps x^(3/4 + eps). This is the case a = 1/2 of Baker and Harman 1991, whose hypothesis is exactly that L(s, chi) is zero-free in sigma > a for every Dirichlet character, whose implied constant depends only on eps, and whose maximum is over all real theta; the frequencies this proof uses are the a/base^l, well inside that uniformity. The statement is restated at source in Porritt 2018 and in Zhang 2024. This is the one step not derived here.
• Step 5, assembly and the wall. Proved. If 0 in F, then S_F below x is D_level less {0} intersected with [1, x] at level = ceil(log_base(x+1)), and steps 1 to 4 apply once: |M_F(x)| <= (B_base(F)/base)^level max_theta |sum_{n <= x} mu(n) e(n theta)| <<_{base,eps} x^(3/4 + c_base + eps). If 0 is not in F, then S_F below base^level is the disjoint union of the exact-length blocks l <= level, and summing the per-block bounds is a geometric sum of ratio base^(3/4 + c_base + eps) > 1, so the top block sets the exponent and the answer is the same; the statement holds at all x >= 2 because the implied constant absorbs the bounded range where level is small. Converting to the A_F yardstick needs 3/4 + c_base < alpha_base, equivalently the constant-space certificate gap_base(m) = (base - m) base^(-3/4) - PB_base(m) > 0. At m = 1 that certificate is negative at every 3 <= base < 3690 and positive at base 3690 (Verified, exhaustive in the generator), and steps up at every base >= 723 by the monotone floor below, so base >= 3690 is a half line and not a window. At m = 1 three proved constants read that certificate and each prints its own wall. The step 3 constant PB_base(1) closes at 3690 (lab/rs/mertens-numerology); the phase-sharpened PB_base(1, e_0) closes at 2446 at every excluded digit and at 1812 at e_0 in {0, base-1}, an up-set over the whole scan 17..4 * 10^6 and not a first crossing, the held column printing 3997555 = 4000000 - 2446 + 1 and the five like counts; and the chord form PB'_base(1, e_0) closes at 1499 and at 1032, an up-set over its own scan 36..4 * 10^6, its held column printing 3998502 = 4000000 - 1499 + 1 and five like counts (Verified, lab/py/mrly-pairing, verb onestep). Above those scans the chord bullet's Psi'_base < Psi_base carries them: base PB_base(1) - base PB'_base(1, e_0) = (Psi_base - Psi'_base) + base/2 + sec(pi c/base)/2 + (0.727 - 2(1 - 2/pi)), every term positive, so PB'_base(1, e_0) < PB_base(1) at every base >= 36 and the step 3 certificate itself, positive from 3690 on, makes every sharpened wall a half line too.
• What the proof does not use. Proved. No zero-density input, no restriction of x to a power of base, no multiplicative structure of S_F (there is none: the first section's 4 x 13 witness), and no l^1 bound quoted from the literature. Step 3 is self-contained and explicit at every base, which is what the theorem needs and what the sharper base-10 l^1 bound behind Maynard 2019 does not offer at general base.
• Corollary, m excluded digits. Proved under GRH. With |E| = m, the step 3 constant PB_base(m) and no sharpening, the five steps run unchanged whenever PB_base(m) < (base - m) base^(-3/4), and give |M_F(x)| <<_{base,eps} A_F(x)^(1 - delta_base + eps) with delta_base = (alpha_base - 3/4 - c_base)/alpha_base. Under the proved constants that condition holds for m <= 6 at base 10^4, m <= 78 at base 10^5 and m <= 451 at base 10^6 (Verified, each maximum asserted maximal in the generator), against sqrt(base) = 100, 316, 1000, and it holds asymptotically for m <= base^(1/2)(1 - o(1)) since PB_base(m) is sqrt(m) plus a term of size (2/pi) ln base. The squarefree-digit-gcd hypothesis carried by the exponent conjecture below is automatic in this regime and is not dropped: m < floor(base/2) leaves two consecutive digits in F, so gcd(F) = 1 and the vanishing family of the transfer section cannot occur; at small fill the hypothesis must be stated.
• The shape at large base. Proved under GRH. Phi_base is (2/pi) base ln base up to lower order, so c_base = (ln ln base + ln(2/pi) + o(1))/ln base -> 0 while alpha_base -> 1, hence delta_base -> 1/4 and |M_F(x)| <<_{base,eps} A_F(x)^(3/4 + o(1)): the full-line GRH exponent transplanted verbatim onto the digit-restricted column, measured against that column's own mass. The convergence is logarithmic and nothing better; c_base tracks (ln ln base + ln(2/pi))/ln base to within 0.01 at base 10^12 (Verified, the generator).
• The constants. Verified. The generator prints alpha_base truncated down at six digits, c_base rounded up at five and delta_base rounded down at five, each from the unrounded value with a directional guard of 10^-12, so every printed digit is a true bound in its own direction and alpha_base never prints as 1.000000; the scientific rows carry a relative guard of 10^-10. Both forms of the test, 3/4 + c_base < alpha_base and gap_base(m) > 0, are computed and their agreement asserted at every row and across 3 <= base < 20000. Every column of the table is the step 3 constant PB_base(m): the m = 1 sharpenings move the closing base and not these rows.

• The margin at the step 3 wall. Verified. The saving at base 3690 is far below the fifth printed digit, so the rounded columns cannot display its sign and never certify it. The certificate is the pair of scientific bounds printed from the cancellation-reduced form delta_base = ln(1 + gap_base(m)/PB_base(m))/(alpha_base ln base), which never differences two numbers of size 1 to reach one of size 10^-6: delta_base <= -2.395807653 * 10^-6 and gap_base(1) <= -1.533059397 * 10^-4 at base 3689, against delta_base >= 5.863425182 * 10^-6 and gap_base(1) >= 3.752213034 * 10^-4 at base 3690. Beyond the wall the gap rises at every one of the 96310 steps of 3690..10^5, the smallest step being >= 0.00003172 at the top of that range, where the base^(-3/4) growth of the mass term is nearest the 4/(pi base) jump of the harmonic term.
• The ladder. Proved under Z(a). Write Z(a), for 1/2 <= a < 1, for the hypothesis that L(s, chi) has no zero in sigma > a for every Dirichlet character; Z(1/2) is GRH. Assume Z(a), let F omit exactly one digit and let base >= base_0(a), the least base with PB_base(1) < (base-1) base^(-b(a)); at the three rungs the m = 1 sharpening is scanned at, PB'_base(1, e_0) < (base-1) base^(-b(a)) from the two chord columns of the table on, and the hypothesis reads the smaller wall. Then for every eps > 0 and all x >= 2, |M_F(x)| <<_{base,eps} x^(b(a) + c + eps) with c = c_base under the step 3 wall and c = c'_base(e_0) under the smaller chord wall, and b(a) + c < alpha_base in each case, so |M_F(x)| <<_{base,eps} A_F(x)^(1 - delta + eps) with delta = (alpha_base - b(a) - c)/alpha_base > 0; the cost in the conclusion is the one whose certificate the hypothesis reads, and c_base is not available under a chord wall, b(a) + c_base >= alpha_base at every base < base_0(a) by that wall's own minimality; the corollary runs at m excluded digits whenever PB_base(m) < (base-m) base^(-b(a)). The proof is the one above with a single substitution: step 4 quotes the exponent b(a) that Z(a) buys, and steps 1, 2, 3 and 5 never name an exponent, the geometric sum of step 5 still having ratio above 1. Since alpha_base -> 1 and c_base -> 0 while b(a) < 1 is fixed, every common zero-free half plane for Dirichlet L-functions buys a power saving over the dense column, and GRH is only its first rung: the price of a weaker hypothesis is paid entirely in the base.
• The input b(a), and where it comes from. Proved. b(a) is the smaller of two quoted tables: Baker and Harman 1991 gives a + 1/4 on 1/2 <= a < 11/20, 4/5 on 11/20 <= a < 3/5 and (a+1)/2 on 3/5 <= a < 1, and Zhang 2024, Theorem 1.1, gives (8a - 7a^2)/(4 - 2a) on 1/2 <= a <= 4/7. Where both apply Zhang is smaller and the two meet exactly at the ends of the overlap, by two factorisations: Zhang(a) - (a + 1/4) = -5(a - 1/2)(a - 2/5)/(4 - 2a) is negative on (1/2, 11/20) and Zhang(a) - 4/5 = -7(a - 4/7)(a - 4/5)/(4 - 2a) is negative on [11/20, 4/7), with equality at a = 1/2 (both 3/4) and at a = 4/7 (both 4/5); and b(a) >= 3/4 on the whole range, Baker-Harman by inspection and Zhang by Zhang(a) - 3/4 = -7(a - 1/2)(a - 6/7)/(4 - 2a) > 0 on (1/2, 4/7]. Verified in the generator over every rational of denominator <= 200 inside the overlap, in exact integer arithmetic, b(a) carried as a rational and compared by cross multiplication throughout.
• The rungs. Verified. Each row names its a and the table the exponent comes from; both means the two tables agree there, and a rung is meaningless quoted without them. base_0(a) is the least base >= 3 with gap_base(a, 1) = (base-1) base^(-b(a)) - PB_base(1) > 0 and Q(b) the proved monotone floor below. A wall prints as an exact integer only when it sits below 2^53 and both neighbouring gaps exceed 1024 ulps of the terms differenced; otherwise the row prints <= and a scientific upper bound, which is a bound on the least base and not the least base. Every wall below 4 * 10^6 is reproduced by an exhaustive scan from base 3 against the bisection. The two chord columns are m = 1 only and read PB'_base(1, e_0) in place of PB_base(1); each is the least base of an exhaustive scan from base 36 and an up-set over that whole scan, and a - is a rung the chord has not been scanned at (lab/py/mrly-pairing, verb onestep). The three scans run 36..4 * 10^6, 36..8 * 10^6 and 36..4 * 10^7, each past that rung's own base_0(a), above which the step 3 certificate is positive on its own, so each chord wall is a half line and not a window. Q(b) is the step 3 floor and no chord column touches it.

• The floor is proved, not scanned. Proved. Per step PB_{base+1}(1) - PB_base(1) < 1.291/(base-2) for base >= 40: the harmonic term jumps by at most (4/pi)/(base-2), the (1 - 2/pi)(base-2)/base term adds under 0.017/(base-2) and the 0.727/base term falls, and a step that does not jump the harmonic term is net negative. The mass term (base-1) base^(-b) gains at least (1-b)(base+1)^(-b) per step, so gap_base(a, 1) steps up wherever (1-b)(base-2)(base+1)^(-b) >= 1.291, a quantity strictly increasing in base; Q(b) is the least base >= 40 where it holds, and the gap steps up at every base >= Q(b). Below it nothing closes: gap_base(a, 1) < 0 on 3 <= base < 3690 at every rung (exhaustive), and on [3690, Q(b)] the smooth majorant U(base) = base^(1-b) - PB_base^-(1) dominates the gap and has exactly one interior minimum, since U'(base) = (1-b) base^(-b) - (2/pi)/(base-2) - 2(1 - 2/pi)/base^2 is positive exactly when a quotient falling strictly from +inf to 0 drops below 1, so its maximum on any interval sits at an endpoint and both endpoints are negative. Q(b) < base_0(a) at every rung, so each printed wall is the least base and the gap steps up from it on, with no sweep needed at any rung.
• The floor and the wall hold at every b in [3/4, 1), not only at the printed rungs. Proved. Below 3690 the one exhaustive scan covers every b at once: base^(-b) falls in b, so gap_base(b, 1) <= gap_base(3/4, 1) < 0 on 3 <= base < 3690, that range being cleared exhaustively at b = 3/4. The floor itself rises with b, since (1-b)(base-2)(base+1)^(-b) falls in b at fixed base, so Q(b) >= Q(3/4) = 723. At the floor, minimality of Q = Q(b) bounds (Q-1) Q^(-b) < 1.291/u + 0.015 above, the slack 2 Q^(-b) < 0.015 coming from Q >= 723, and ln(Q+1) > ln(1.291/u)/u below, both in terms of u = 1 - b alone; feeding them into the lower bound PB_Q^-(1) through ln((Q-2)/2) >= ln(Q+1) - ln(1448/721) and (Q-2)/Q >= 721/723 gives gap_Q(b, 1) < [1.291 - (2/pi) ln(1.291/u) - 2.544 u]/u, the coefficient 2.544 assembled from those three ingredients, 0.015, ln(1448/721) and 721/723. Its bracket increases on (0, 1/4] and so is at most its value -0.39014 at u = 1/4, hence gap_Q(b, 1) < -1.56. On [3690, Q] the majorant differs from the gap by under 0.004, so U(Q) < -1.556, while U(3690, b) falls in b with U(3690, 1417/1850) < -0.95, and any b below 1417/1850 has Q(b) <= 1486 < 3690 and an empty range. So gap_base(b, 1) < 0 on [3, Q(b)] and steps up from Q(b) on at every b: base_0(a) exists and exceeds Q(b) at every a, printed rung or not. Constants Verified in the generator on the b-grid 0.75..0.975 in steps of 0.005.
• What a weaker half plane spends first. Verified. The m-budget at base 10^7 is the largest m with PB_base(m) < (base-m) base^(-b(a)), printed by the generator for the rungs whose wall lies below 10^7. Each row carries its a and its source, a rung quoted by b alone being meaningless: two rungs share b = 4/5 from different tables and the budget, not the theorem, is what a wider zero-free half plane costs. The budget is a statement about PB_base(m) at m excluded digits, so the m = 1 sharpening never enters it.

• The cost-out against the Type I defect. Verified. A conditional Type I argument over S_F would run against the level-x^(alpha_base/2) distribution bound of the divisor section above, whose error carries a defect x^(m/(2(base-m) ln base)), and would have to pay that defect out of the saving proved here, so the generator sets the two exponents side by side. They sit on different yardsticks and no derivation joins them: delta_base is normalised to the mass, so as a power of x the saving is x^(alpha_base delta_base) with alpha_base >= 0.99993 on every row compared, while the defect multiplies fill^level. On those rows the saving is below the defect at the wall (5.86342 * 10^-6 against 1.65022 * 10^-5 at base 3690, a factor above 2.8) and above it from base 3692 on, the least such base in a scan of 3690..10^5 where the difference rises at all 96310 steps, monotonicity beyond the scan not being proved; by base 10^9 the saving 1.16951 * 10^-1 clears the defect 2.41275 * 10^-11 by over nine decades, and the tightest corollary row, base 10^6 at m = 451, clears its own defect 1.63296 * 10^-5 at 3.14081 * 10^-5. The whole failure at the wall is the two steps 3690, 3691, so a sharper constant that moves the wall moves the comparison too and is re-costed rather than inherited: the sharpened cost-out block of lab/rs/mertens-numerology prints each proved wall beside its own crossing, the least base at which delta_base exceeds the defect 1/(2(base-1) ln base), reading 3690 and 3692 at the step 3 constant, 2446 and 2450 at the phase sharpening, 1812 and 1815 at its extreme-digit form, 1499 and 1502 at the chord and 1032 and 1036 at the chord's extreme-digit form, each row scanned from its own floor, base >= 3, 17 and 36, and each crossing an up-set to 10^5 within five steps of its own wall, so a lower wall costs out at once and no comparison is inherited. The comparison runs at the GRH rung b = 3/4 and at no rung above it: no rung of the ladder past a = 1/2 is set against the defect anywhere here.
• Conjecture. That such a defect is absorbed at all. The comparison above is two exponents from two unrelated statements on two yardsticks; it is not a necessary condition, no theorem about M_F follows from it, and the string-to-interval bookkeeping and the bilinear half of any such argument are untouched here.
• The l^1 floor, and what it forecloses. Proved. For every digit set, sum_{r mod base} |g_F((t+r)/base)|^2 = base fill exactly, by Parseval on Z/base, so by Cauchy-Schwarz sum_{r mod base} |g_F((t+r)/base)| >= base fill / max_r |g_F| >= base for every t, the floor every later section calls the Parseval floor: the recursion of step 2 never contracts, B_base(F) >= base, and c_base >= 0 at every base and every digit set, so a negative c_base is an arithmetic error and not a discovery. At one excluded digit the floor is higher than base and exact: letting t -> 0 on the shifted grid gives |g_F(0)| = base - 1 and |g_F(r/base)| = 1 at every r != 0, so B_base(F) >= 2(base - 1) and c_base >= log_base(2 - 2/base) > 0 at every base >= 3, and that endpoint is the seat at base 3, by hand and not by a grid: there |g_F((t+r)/3)| = 2 |cos(pi (t+r)/3)|, and with u = pi t/3 in [0, pi/3) the three absolute values collapse to 4 cos u on [0, pi/6) and to 4 cos(u - pi/3) on [pi/6, pi/3), both at most 4 = 2(base-1), attained at u = 0, so B_3(F) = 4 exactly (Proved; the sup read on the t-grid of lab/py/mrly-pairing, verb onestep agrees and is a reading, never the certificate). So at m = 1 no exact constant pushes this decomposition below the base where (base - 1) base^(-3/4) > 2 - 2/base, which is base^(1/4) > 2 and so base >= 17: at m = 1 the method needs fill > (2 - 2/base) base^(3/4) and not fill > base^(3/4). The other end of the same lever is measured and not proved: the exact B_base(F) read on the grid would close the GRH certificate at 927 at every excluded digit, last failure base 926 at e_0 = 462, and at 304 at e_0 in {0, base-1}, last failure 303, against the proved 1499 and 1032, but both are readings of Sigma(1/2) with no upper certificate on the supremum and no monotonicity in base, so they bound nothing and enter no statement (Verified, lab/py/mrly-pairing, verb onestep). That higher floor is still too weak to ask fill > 2 base^(3/4), since at base 17 the base fill = 16 lies between (2 - 2/base) base^(3/4) = 15.759 and 2 base^(3/4) = 16.744. The consequence is a hard limit on this decomposition, not on the problem: at every digit set, m = 1 included, it needs alpha_base > 3/4, that is fill > base^(3/4), so every column at fixed digit count is out of its reach, F = {0,1} at base 3 included, under GRH or without it. The dense columns this section proves something about and the sparse columns the census measures do not overlap.
• Nothing unconditional follows in this decomposition. Proved. Put Davenport's unconditional max_theta |sum_{n <= x} mu(n) e(n theta)| <<_A x (log x)^(-A), carried at source in Porritt 2018, into step 4: by the l^1 floor the result is at best of size x (log x)^(-A), which exceeds A_F(x) by the power x^(1 - alpha_base). An unconditional power saving here would need an unconditional uniform power-saving input, which is itself of zero-free-strip strength; an unconditional route has to split arcs and use the structure of mu in progressions mod base^j, which this decomposition never touches.
• The l^2 route is worse than trivial. Proved. Cauchy-Schwarz with Parseval on both factors, sum_{a mod base^level} |hat F_level(a/base^level)|^2 = base^level fill^level and sum_{a mod base^level} |sum_{n <= base^level} mu(n) e(n a/base^level)|^2 of size (6/pi^2) base^(2 level), gives exponent (1 + alpha_base)/2 > alpha_base. The supremum over frequencies paid against the l^1 mass is the only arrangement of this decomposition that saves anything.
• The ceiling, and the endpoint. Proved. Even with the conjectured x^(1/2 + eps) in step 4, the exponent Porritt 2018 records as the expected one, the floor c_base >= 0 still forces alpha_base > 1/2, that is fill > base^(1/2): the exponent conjecture below, which is about fixed fill, is beyond every version of this method and not merely beyond its conditional form. And within the dense regime the endpoint stays out: delta_base is fixed before eps, so what is proved is A_F(x)^(1 - delta') for every fixed delta' < delta_base and never A_F(x)^(1 - delta_base), a distinction that is the whole claim at base 3690, where delta_base >= 5.863425182 * 10^-6.
• What this is, against the literature. Verified. As far as the sources in REFS.md are read, none of them carries a Mobius or Mertens sum over a digit-restricted set: the nearest multiplicative function computed over a missing-digit set is the divisor function (Kim 2024), whose own framing is that the set's lack of multiplicative structure blocks the standard approaches, and the nearest arithmetic-function theorem over such a set is the prime count of Maynard 2019, which enters through the set's level of distribution and not through a Mobius bound. The theorem above is of Mertens shape: a power of the set's own counting function, A_F(x)^(1 - delta') for every fixed delta' < delta_base, at every x >= 2, for one excluded digit at every base >= 1499, and at every base >= 1032 when the excluded digit is 0 or base - 1, and for m excluded digits under the stated condition, the bound it beats being the trivial |M_F(x)| <= A_F(x) on those columns. The card carries both halves: it is conditional on GRH, it yields nothing unconditional inside this decomposition, and it says nothing whatever in the sparse regime fill <= base^(3/4) where the census and the exponent conjecture live.
• The lane. This section is written up on the shelf as sparse-mertens-under-grh: the theorem, the corollary, the ladder and the l^1 floor with full proofs, and its scripts/verify.py recomputes every step 3 constant, table row, margin and rung above from the formulas alone, independently of lab/rs/mertens-numerology, in under three seconds. The m = 1 sharpenings of step 3, their walls and the higher l^1 floor at one excluded digit are not carried there, so the shelf's wall 3690 is the step 3 wall and the theorem's 1499 and 1032 on this page supersede it.

The pair route

The section above buys a power saving on the dense columns under GRH and states plainly that nothing unconditional follows from that decomposition: an unconditional route has to split arcs and use the structure of mu in progressions mod base^j. This section is that route, laid out as far as it goes. It follows the only existing proof that counts a thin arithmetic sequence on a missing-digit set, Maynard 2019, and asks what changes when the sequence counted is mu rather than the primes. Most of that chain never looks at the sequence at all; the two steps that do are proved here; what is left is a region in the three exponents the digit set owns. No theorem about M_F comes out of it. A criterion does: its region is exactly two inequalities, its gate is fill >= base^(3/4), it is refuted at base 10 at every excluded digit, and the one design of the census that clears it is base 21 missing the digit 0.

Notation as in the divisor section, with x = base^level, D_level the digit strings of length level over F, hat F_level(t) = sum_{n in D_level} e(n t) = prod_{j < level} g_F(base^j t), alpha = log(fill) / log(base) the mass exponent, S_mu(t) = sum_{n <= x} mu(n) e(n t), rad(base) the product of the primes dividing base and omega(n) the number of them. Write alpha_1 for the l^1 exponent of the transform in its sup-over-shift form, sup_s sum_{a < Y} |hat F_l(s + a/Y)| << fill^l Y^(alpha_1), the supremum over real shifts s, which is 27/77 at base 10 with one digit excluded. That is the strength the source's own l^1 lemma carries and the strength Farey spacing consumes; it dominates the bare grid exponent, so every lower bound on the grid exponent below transfers up to it, and the threshold it is asked to clear is correspondingly the stronger ask. Write m_t for the l^t exponent of the normalised transform F_x(t) = fill^(-level) |hat F_level(t)| on the grid and beta = inf_{1 <= t < 2} m_t/(2 - t) for the exceptional-set threshold, the third exponent the design owns. Call e base-smooth when rad(e) divides rad(base), and write (E1) for the hypothesis that every prime dividing the gcd of the digit differences of F divides base: the one-dimensional form of condition (E) of coprime, and the hypothesis Lemma A' there consumes. Every exponent, region boundary, threshold certificate and census verdict below is printed by lab/py/mobius-region.

• The bilinear half of the chain never sees the coefficients. Verified. The Type II estimate of that proof is stated for arbitrary 1-bounded sequences with one support constraint, that every counted integer carries a divisor in a prescribed dyadic range; its proof applies Cauchy-Schwarz in the long variable first and then drops all four coefficient factors by the triangle inequality, leaving a sum over pairs of frequencies with no coefficient in it at all, which a geometry-of-numbers argument places near a rank-2 lattice or on a line. Residue sums of the coefficient side occur exactly once in that proof, on the major arcs at moduli below a fixed power of log x. So a Type II estimate on a digit set is not a hypothesis about cancellation of the coefficients in progressions, and the whole range above that cut transfers from primes to mu unread, every sentence of this bullet read at its source.
• The l^1 floor is the shifted-grid floor iterated. Proved. The l^1 floor of the GRH section, sum_{r mod base} |g_F((t+r)/base)| >= base fill / max_r |g_F| >= base for every t, is one digit position of the same statement; iterating it over level positions through the peeling recursion of that section's step 2, or reading it off the grid directly by sum_a |z_a| >= (sum_a |z_a|^2)/max_a |z_a| with Parseval sum_{a mod base^l} |hat F_l(a/base^l)|^2 = base^l fill^l and the maximum fill^l at a = 0, gives sum_{a mod base^l} |hat F_l(a/base^l)| >= base^l and hence alpha_1 >= 1 - alpha at every base and every digit set. One mechanism, stated once there per position and once here per exponent. Two consequences: an l^1 exponent below 1/2 forces fill > sqrt(base), so the sparse columns of the census are outside this route exactly as they are outside the route of the GRH section; and alpha + alpha_1 >= 1 always, which is what makes the scale sum in the level-of-distribution statement below geometric with ratio at least 1. That reach fill > sqrt(base) is true and unsharp: the same Parseval identity puts the same floor on the exceptional-set threshold, and the route's own window condition lifts the gate to fill >= base^(3/4) below.
• No exceptional character sits at a base-smooth modulus. Proved. Every real primitive Dirichlet character of base-smooth modulus has conductor dividing 8 rad(base), and the conductors in play number exactly 2^omega(base_1) at odd base and 3 * 2^omega(base_1) at even base, base_1 the odd part of base, while the characters number 2^omega(base_1) at odd base and 4 * 2^omega(base_1) at even base. A real primitive character of conductor f > 1 is the Kronecker symbol of a fundamental discriminant of absolute value f, so writing f = 2^u f_1 with f_1 odd, f_1 is squarefree and u is 0, 2 or 3; base-smoothness forces f_1 | rad(base), hence f | 8 rad(base). Conversely every 2^u f_1 of that shape occurs, and the two counts differ: exactly one of +-f_1 is 1 mod 4, giving one character at u = 0; exactly one of +-f_1 is 3 mod 4, giving one at u = 2; and both of +-2 f_1 are 2 mod 4 and squarefree, giving two at u = 3, so four characters sit over three conductors for each odd squarefree f_1 dividing rad(base), while at odd base only u = 0 is available and the counts coincide. That is the whole content of the remark in the source that its major-arc moduli are too composite for Siegel zeros to matter: an exceptional zero belongs to a real primitive character, a real primitive character has a fundamental discriminant for a conductor, a fundamental discriminant is squarefree away from a factor 4 or 8, and a power of the base is as far from squarefree as an integer gets. The conductors in play run over a set of size bounded in terms of base alone rather than to infinity, so Siegel's theorem is never invoked and the constants below are effective.
• The major arcs for mu, with the exponent they deliver. Proved. Let base >= 3, let F satisfy (E1), let C > 0, and put T = (log x)^C and M(C) = {a mod x : |a/x - b/d| <= T/x for some d <= T and some b coprime to d}. Then there are c > 0 and x_0, both depending only on base, fill and C and both effectively computable, with |x^(-1) sum_{a in M(C)} hat F_level(a/x) S_mu(-a/x)| <= fill^level exp(-c sqrt(log x)) for x >= x_0. The proof splits M(C) at the base-smooth denominators. Off them the modulus carries a factor d_2 > 1 coprime to base, and the perturbed Lemma A' of coprime gives |hat F_level(a/x)| <= fill^level exp(-c' log x / log log x) against the trivial |S_mu| <= x. On them x = base^level makes every such b/d an exact grid point, so the arcs are intervals of consecutive integers and no Dirichlet approximation enters; there |hat F_level| <= fill^level is trivial, partial summation strips the shift, and what is left is sum_{n <= u, n = r mod e} mu(n) at a base-smooth e <= T, which the classical zero-free region for L(s, chi) bounds by u exp(-c'' sqrt(log u)) (Davenport, chapters 14 and 20; read at source as Montgomery and Vaughan 2007, Theorem 11.3 and Exercise 8 of Section 11.3, M(x, chi) <<_A x exp(-c_1 sqrt(log x)) at q <= (log x)^A) with the only ineffective ingredient, the exceptional real zero, removed by the conductor bound above and the effective Landau-Page bound of the same chapter 14. There is no main term at any arc, the frequency a = 0 included, where the contribution is fill^level M(x)/x. The saving is exp(-c sqrt(log x)). It is not compared with the (log x)^(-C) the source states for the prime analogue, which is an asymptotic with a main term where this is a bound with none; what is worth stating is that the main term is absent at every arc and that the prime number theorem is what puts the a = 0 term inside the error.
• The level of distribution on an initial segment. Proved. Assume (E1) and the large sieve the design supplies, sum_{d <= Q} sum_{(b,d) = 1} |hat F_m(b/d)| << fill^m (Q^(2 alpha_1) + Q^2 base^(-m(1 - alpha_1))) at every m <= level, which follows from the l^1 exponent by Farey spacing alone and reads Q^(54/77) + Q^2 Y^(-50/77) at base 10. Then for every B > 0 there is C with sum_{d <= Q, (d,base) = 1} max_{y <= x} |#{n in D_level : n <= y, d | n, (n,base) = 1} - (1/d) #{n in D_level : n <= y, (n,base) = 1}| <= fill^level (log x)^(-B) at every Q <= x^(1 - alpha_1) (log x)^(-C). The initial segment costs nothing in the level and one power of log x in the saving, for two reasons. D_level below y is a disjoint union of blocks {P base^m + t : t in D_m}, at most fill of them per scale whatever y is; and the error the transform gives for #{t in D_m : t = r mod d} is uniform in the target residue r, so a shifted target is exactly as cheap as the residue 0 the source asks for. Above the cut the large sieve pays, below it Lemma A' pays, and the scale sum is dominated by its top scale because alpha + alpha_1 >= 1.
• What the base's own divisors cost. Proved. For d = d_1 d_2 with d_1 base-smooth and (d_2, base) = 1, the split of the divisor section carries the level to d: the low digits fix n mod d_1 and reach the rest only through an invertible multiplier, so the count reduces to the same transform estimate in d_2. What does not carry is the main term. It is a digit-string count times 1/d_2 and not 1/d, reading fill^(-v) against a naive base^(-v) at d_1 = base^v, so a Type I sum with coefficients c_d produces sum_d c_d rho_F(d) where the coprime case produces sum_d c_d / d, and nothing here shows the first small. Nor is the coprimality peeled off in general: sum_{n in S_F, n <= x} mu(n) = sum_{w | rad(base)} mu(w) sum_{n' : w n' in S_F, (n', base) = 1} mu(n') is an identity, and whether it reduces the problem depends on the inner sets. Sometimes it does - at base 10 and F = {0,1} the carry-free scaling of the transfer section gives {n : 2 n in S_F} = S_{0,5} and {n : 5 n in S_F} = S_{0,2}, both designs, and that column sits below this route's own l^1 floor in any case, fill = 2 < sqrt(10). Sometimes it does not: at base 10 and F = {0,1,2} the set {n : 2 n in S_F} begins 1, 5, 6, 10, 11, 50, 51, 55, 56, 60, 61, 100, 101, 105, and it is a digit design at no base tested, the base-10 digit set it forces being {1,5,6}, which misses 10, or {0,1,5,6}, which wrongly admits 15 because 30 leaves S_F. The hypothesis (n, base) = 1 therefore stays inside the criterion below.
• From strings to the design. Proved. The two statements above count D_level, the padded strings, while the criterion counts S_F. With 0 in F the two agree below base^level but for the element 0, which carries mu(0) = 0. With 0 outside F, S_F below base^level is the disjoint union of the exact-length blocks, each of them a D_l, so both statements sum over l with the top block setting the exponent, the geometric sum having ratio fill > 1; that the level-of-distribution statement holds on an initial segment is what makes the sum legitimate at every l.
• The window and the criterion, as arithmetic. Proved. The exceptional set is E = {a : |hat F_level(a/x)| >= fill^level x^(-beta)}. Both places the source spends it reduce to m_t < (2 - t) beta for some t in [1, 2), so the least admissible threshold is inf_{1 <= t < 2} m_t/(2 - t), which at the source's own t = 235/154 and m_t = 59/433 is 9086/31609 = 0.287449, rounding up to its 23/80. The Type II window is then [(5/4) beta, 1 - 2 beta], and by the symmetry of the phase in its two variables also [2 beta, 1 - (5/4) beta]; at beta = 23/80 that is [9/25, 17/40]. Decomposing mu by a Heath-Brown identity of order above 1/alpha_1, a piece with a free variable above x^(alpha_1) is Type I at the level above, and otherwise greedy accumulation lands in the window under two conditions, beta <= 1/4, which merges the two windows into one interval, and alpha_1 + (5/2) beta <= 1, the greedy overshoot. The bilinear estimate itself asks five more, listed in the lattice bullet below; every one of them is free under alpha_1 < 1/3, beta <= 1/4 and the l^1 floor. Taking t = 1, so beta = alpha_1 + eps, the binding condition is alpha_1 < 1/4.
• The hybrid bound the lattice branch needs holds at every base, with the exponent the digit set owns. Proved. The published proof reaches its bilinear estimate through one bound whose whole purpose is to beat the plain l^1 exponent in the modulus aspect: for D, E, Y, Q_1 powers of base with D E << Y, e_1 ~ Q_1 coprime to base and d ~ D base-smooth, sum_{e_2 ~ Q_2, (e_2,base) = 1} sum_{a < d e_1 e_2, (a, d e_1 e_2) = 1} sum_{|eta| <= E/Y, (eta + a/(d e_1 e_2)) Y in Z} F_Y(a/(d e_1 e_2) + eta) << (D E)^(alpha_1) (Q_1 Q_2^2)^(1 - alpha) + E^(alpha_1 + alpha/2) D^(1 + alpha/2) Q_1 Q_2^2 Y^(-alpha/2), where F_Y is the transform normalised by its own mass. The proof is the source's, carried in general parameters: the product identity F_(Y_1 Y_2)(t) = F_(Y_1)(t) F_(Y_2)(Y_1 t) and the monotonicity F_Y <= F_U for U <= Y split the sum, the Chinese remainder theorem sends the residues through complete reduced systems exactly once, the l^1 exponent and the shifted large sieve it supplies by Farey spacing pay three of the four factors, and the fourth is pure Parseval on a window R = base^r, int_0^1 F_R^2 = R^(-alpha) and int_0^1 (F'_R)^2 << R^2 R^(-alpha), the first exact when 0 is in F and two-sided up to constants otherwise, in the direction used either way. So the two exponents are the digit set's own dimension and nothing else: the modulus exponent is 1 - alpha and the saving exponent is alpha/2. At base 10 with one digit excluded the source prints them as 1/21 and 10/21 on the single check 20/21 < log 9 / log 10, and 1 - alpha = 0.045757 sits under 1/21 = 0.047619 while alpha/2 = 0.477121 sits over 10/21 = 0.476190: both roundings are safe and both are lossy. By the l^1 floor above, 1 - alpha <= alpha_1 at every base and every digit set, so this bound never loses to the plain one in the modulus aspect, and the reading that a general base must fall back on alpha_1 there mistakes the floor's equality case for the general value.
• The lattice branch transfers to every base, and the conditions it asks are free below 1/3. Proved. With that bound the whole lattice half of the bilinear estimate runs in general parameters: for x = base^level and the window N K >= x^(1 - 2 beta), delta >= N/x, Q <= x^(1/2), the sum of F_x(a_1/x) F_x(a_2/x) over pairs whose large contribution comes from a rank-2 lattice is << (log x)^5 (Q + E)^(-eps/4) x/(N K), the source's own log power, the count of pairs of base-smooth moduli being O(Q_0^(eps/2)) at every fixed base. Five inequalities close it: 2 alpha_1 < alpha; (2 - alpha) 2 beta < 1 - alpha_1; some u in (0, min(1, 2 alpha_1/alpha)] has 2 beta (alpha_1 (3 - u) + u - 1) < u alpha/2; 5 beta < 1 + alpha/2; and 2 beta < (1 - alpha_1)(1 - alpha_1 + alpha/2). Their provenance is not uniform: the source writes a numerical check for the second and the fourth, while the first, third and fifth are read off steps it performs silently under the phrase that the exponents have been simplified for an upper bound. Its third written check renames an exponent and imposes nothing past the fourth condition, so beta < alpha/2 is not a hypothesis of the branch. All five are monotone, worse as beta or alpha_1 grows and better as alpha grows, so the corner alpha = 1 - alpha_1, beta = 1/4 decides them all, and there they read alpha_1 < 1/3, 1/3, 1/3, 1/2 and 1 - 1/sqrt(3) = 0.422649. Under alpha_1 < 1/3, beta <= 1/4 and the l^1 floor alpha + alpha_1 >= 1 every one of the five holds, and 1/3 is sharp: three of them are equalities there. The floor and the threshold on beta alone are not enough, as alpha_1 = 0.40, alpha = 0.60, beta = 1/4 shows, where the first three read 0.8 < 0.6, 0.7 < 0.6 and 0.4 < 0.3. So the criterion's own alpha_1 < 1/4 clears the branch with room, and the pair alpha_1 < 1/3 and (1 + alpha_1) 2 beta < 1 - alpha_1, which a reading of the first two through the plain l^1 exponent produces, is not a pair of separate demands.
• The threshold obeys the same Parseval floor, and the gate is fill >= base^(3/4). Proved. The normalised transform is at most 1 pointwise, so m_t is non-increasing in t; and m_2 = 1 - alpha exactly, since two digit strings of length level congruent modulo base^level are equal, which is Parseval on the grid again. Hence m_t >= 1 - alpha for every t <= 2, and 2 - t <= 1 for t >= 1 gives beta >= 1 - alpha at every base and every digit set, the floor alpha + alpha_1 >= 1 read on the third exponent. The route's window condition beta <= 1/4 then forces alpha >= 3/4, that is fill >= base^(3/4), with equality pinning the grid l^1 exponent to 1 - alpha as well, the floor's own equality case. Dropping that window condition does not widen the route, it narrows it, and the step that shows it is beta <= alpha_1: taking t = 1 in the infimum gives beta <= m_1, and the grid sum is one shift of the supremum, so m_1 <= alpha_1. The single-window branch carries the greedy step on its own whenever alpha_1 <= 1 - (13/4) beta, which against beta <= alpha_1 asks beta <= 4/17 = 0.235294..., strictly under the 1/4 it replaces, and with the two floors asks alpha >= 13/17 = 0.764705..706, so the gate rises to fill >= base^(13/17). That step is load-bearing and the alpha coordinate alone does not replace it: alpha = 0.9, alpha_1 = 0.154, beta = 0.26 meets both floors, all five lattice conditions, 2 alpha_1 < alpha among them, the greedy cap beta <= (2/5)(1 - alpha_1) and the branch alpha_1 <= 1 - (13/4) beta with slack on each, at beta > 1/4, and only beta <= alpha_1 kills it. The floor beta >= 1 - alpha is an equality of Parseval and improves at no design, so the only movable numbers in the gate are the window floor max((5/4) beta, (5 beta - 1/2)/3), which the line branch imposes, and the window ceiling 1 - 2 beta; sharpening the first, or finding a greedy step that crosses the gap between the two windows at beta > 1/4 without merging them, is what would break 3/4, and no other parameter of the region can. The gate arithmetic is exact from the two floors; verb check of lab/py/mobius-region prints those floors at four designs and beta <= alpha_1 at t = 1 at the base-21 recompute, and verb boundary prints the branch cap beta <= 4/17 and the gate alpha >= 13/17 themselves, with that witness beside them. This 3/4 is also not the fill > base^(3/4) of the GRH section, which comes from that section's l^1 floor B_base(F) >= base: the two numbers meet for different reasons and share no proof.
• The region the route asks for is exactly two inequalities. Proved. Of the eight inequalities the whole chain asks, one is a ceiling on alpha_1 alone, 2 alpha_1 < alpha, and seven are caps on beta at fixed (alpha, alpha_1); four of the seven fall in alpha_1 and three are constant in it, so each takes its minimum over the region at the wall alpha_1 = alpha/2. At that wall the cap (2 - alpha) 2 beta < 1 - alpha_1 reads exactly 1/4 at every alpha, the cap 2 beta < (1 - alpha_1)(1 - alpha_1 + alpha/2) reads (2 - alpha)/4 and the cap 5 beta < 1 + alpha/2 reads (1 + alpha/2)/5, three identities in alpha and not roundings, the last two strictly above 1/4 for alpha in (1/2, 1). The u-condition 2 beta (alpha_1 (3 - u) + u - 1) < u alpha/2 splits at alpha = 2/3 and is 1/4 only above it: at the wall its coefficient is c(u) = (3 alpha/2 - 1) + u (1 - alpha/2), and u/c(u) rises in u exactly when 3 alpha/2 - 1 > 0, so for alpha >= 2/3 the best u is u = 1 with c(1) = alpha and the cap is exactly 1/4, while for alpha in (1/2, 2/3) the coefficient is negative at small admissible u, every beta passes and the condition is vacuous rather than 1/4. Vacuous or 1/4, it never cuts, and none of the four ever cuts. For alpha in (1/2, 1) the region is therefore alpha_1 < alpha/2 together with beta <= min(1/4, (2/5)(1 - alpha_1)), the greedy cap cutting below 1/4 exactly from alpha_1 = 3/8 and from nowhere else, a threshold free of alpha; the admissibility cap beta <= 2/5 never binds beside the window cap. The three identities and the u-condition's split are proved and need no sweep; a sweep of alpha in [67/100, 999/1000] by 1/1000 and alpha_1 in (0, alpha/2] by alpha/400, which lies entirely above 2/3, corroborates them at 0 of 66000 cells and 0 of 264000 cap tests, with both wall equalities holding at each of 330 rational alpha (Verified, lab/py/mobius-region, verbs region and boundary). The headline alpha_1 < 1/4 is the t = 1 proxy of the window cap and clears the whole region with room.
• The line branch and the two bookkeeping steps, at every base. Proved. With x = base^level, a threshold beta admissible and at most 2/5, which with the Parseval floor beta >= 1 - alpha already asks alpha >= 3/5 of the design, delta >= N/x, N K >= x^(1 - 2 beta), K above the absolute constant of the pair dichotomy, and N >= x^(eps + max((5/4) beta, (5 beta - 1/2)/3)), the pair sum of F_x(a_1/x) F_x(a_2/x) over the pairs whose large contribution lies on a line is << (log x)^(O(1)) x^(-eps') x/(N K) for x past a point depending on base, fill and eps alone, with eps' a function of eps and the implied constant depending on those three alone. The statement asks nothing of the l^1 exponent and asks of the dimension only what the admissibility of beta already encodes, so the whole l^1 content of the route sits in the lattice half and the greedy step. Two bookkeeping steps complete it, and neither is a new idea. For coefficients bounded by the j-fold divisor function, orthogonality on the grid with tau_j^2 <= tau_(j^2) gives #{a mod x : |sum_n c_n e(n a/x)| >= x/C} <<_j C^2 (log x)^(j^2 - 1), so a Heath-Brown piece costs a power of log x where a 1-bounded sequence costs none. And Cauchy-Schwarz in the long variable gives |Sigma|^2 <= (x/N) sum_{(a_1, a_2) in E^2} F_x(a_1/x) F_x(a_2/x) T(a_1, a_2) with T(a_1, a_2) = sum_{l_1, l_2 <= N} min(x/N, ||(a_1 l_1 - a_2 l_2)/x||^(-1)), which is the exact step at which all four coefficient factors leave by the triangle inequality; the dyadic split into level sets and pair-mass classes costs two more log powers. Those two steps are derived here; the line estimate itself is the source's, carried in general parameters, and its close is read once here and owes a second reading.
• Nothing in the bilinear half is base-10 mathematics. Refuted. No step of it needs base 10 numerically, and no step of it needs the coefficient side past 1-boundedness. The lattice and line sections never see the polytope, the coefficients or the sequence weighting the frequencies; those are spent one section earlier and survive only as the two window numbers. Base 10 enters in exactly three places and all three are names rather than arithmetic: the set of integers all of whose primes divide the base, the coprimality to the base, and dyadic parameters that are powers of the base. Every printed exponent in those sections is a rounding of alpha, alpha_1 or beta: 1/21 rounds 1 - alpha up, 10/21 rounds alpha/2 down, 27/77 is alpha_1, 50/77 is 1 - alpha_1, 9/8 rounds 1 - alpha_1 + alpha/2 down, 3/16 rounds alpha/2 - beta down, 17/40 is 1 - 2 beta, 9/25 rounds (5/4) beta up, and 23/80 is beta.
• The criterion. Conjecture. A digit set satisfying (E1) whose l^1 exponent obeys alpha_1 < 1/4 has sum_{n in S_F, n <= x, (n,base) = 1} mu(n) = O_B(A_F(x) (log x)^(-B)) for every B. This is a program, not a theorem, and every part of it is named. The two steps that are about mu rather than about primes are the two Proved statements above. The lattice and line estimates, the geometry of numbers and the exceptional-set bookkeeping are set-only or coefficient-free and transfer as read, and the lattice half, the line half and both bookkeeping steps are written out at general base in the bullets above, so the three write-outs the program lists as owed are written. What stands in their place is arithmetic rather than machinery, and it is one item: the level of distribution at base-divisible moduli, needed only to drop (n, base) = 1. The level itself carries, by the split of the divisor section, but the main term does not, being a digit-string count times 1/d_2 and not 1/d, so a Type I sum with coefficients c_d produces sum_d c_d rho_F(d) where the coprime case produces sum_d c_d/d, and nothing here shows the first small. The hypothesis (n, base) = 1 is therefore exactly what keeps every Type I modulus coprime to the base, since a Heath-Brown factorisation of an integer coprime to base has every factor coprime to base. Of the source reading itself, the lattice branch is re-derived line by line and the line branch's close is read once; a second reading of that close is the only source work left.
• Base 10 is refuted at every excluded digit. Refuted. A published or certified moment exponent is an upper bound, so it bounds beta from above and can never show the criterion fails: the published beta <= 23/80 = 0.2875 against 1/4 prices a gap of 3/80 and refutes nothing. The threshold is bounded from below by two monotonicities. F_x <= 1 pointwise makes m_t non-increasing in t, so on a cell [t_0, t_1] every t in it has m_t/(2 - t) >= m_(t_1)/(2 - t_0); and above a cut the Parseval value m_2 = 1 - alpha alone forces the ratio past 1/4. With the moment bounded below by the infimum window matrix, adaptive chains of 25 to 53 cells certify beta > 1/4 at all ten one-missing-digit sets of base 10, the certified lower bounds running 0.2502716 to 0.2541480, so no admissible threshold clears the window condition there. The single-window branch is no escape and needs no second certificate: it asks alpha_1 <= 1 - (13/4) beta, which with beta <= alpha_1 reads beta <= 4/17 < 1/4, the harder of the two asks, so one certificate of beta > 1/4 kills both branches at once and the route is dead at base 10 at every digit rather than merely unreached. The refuting certificates do not order the columns: the brackets [0.2510933, 0.2625620] at the digit 9 and [0.2515026, 0.2875159] at the digit 4 overlap. Run at the target 0.2626 the same chain certifies beta >= 0.2632014 at each of the eight non-extreme digits, up to 0.2645208 at the digit 7, above both extreme upper bounds, while the digits 0 and 9 come back undecided as they must, and that settles the two extreme digits as strictly the cheapest columns of the base. The miss is at most 0.0125620 at the cheapest column and at least 0.0139557 at the digit 4, and the factor 2.99 between the two printed excesses over 1/4, 0.0375159 against 0.0125620, is a ratio of upper bounds and not of misses (lab/py/mobius-region, verbs threshold and threshold 0.2626). A second miss is Refuted. Reading the l^1 exponent against 1/3 substitutes the plain l^1 exponent for the modulus exponent of the hybrid bound; with the true modulus exponent 1 - alpha = 0.045757 base 10 clears every one of the five lattice conditions as published, at 0.701299 < 0.954242, 0.601311 < 0.649350, 0.304275 < 0.350649 at u = 0.734926, 1.4375 < 1.477121 and 0.575 < 0.731475, all five arithmetic from the three brackets that lab/py/mobius-region, verb params prints, with the u admissible against alpha rounded up. The greedy cap asks only beta <= 20/77 = 0.259740 there, so the window cap 1/4 is what binds, and it is the cap base 10 provably fails: the obstruction there is the exceptional-set threshold alone.
• The census of the criterion, and the least base that clears. Verified. Over the 38 proper digit sets of base 3, 4, 5, the ten base-10 one-missing-digit columns and base 21 missing 0, one design clears the criterion, 47 are refuted and one is open, the open cell being base 5 with F = {0,1,3,4}, where the transform vanishes inside a window cell, the infimum matrix loses a row and the machine returns no positive lower bound rather than a false one. A pass is decided at the pessimistic corner, alpha low and alpha_1, beta high, and a failure at the optimistic one, every cap being monotone in each parameter (Proved). The one design that clears is base 21 missing the digit 0, at alpha_1 in [0.2499715, 0.2499822] from a second implementation of the window method against the five-digit [0.2499715, 0.2499821] already certified: the two agree on the lower bound to all seven printed digits and differ by one unit in the last on the upper, so the agreement witnesses transcription and the upper-bound gap is the only independent information (lab/py/mobius-region, verbs criterion and params). The least base whose whole one-missing-digit family clears is base 34, certified with its brackets and the bases it beats in coprime.
• The shape the route can deliver, at best. Proved. The conclusion is a log saving and not a power, and the binding step is the level of distribution rather than the arcs: the major-arc lemma gives exp(-c sqrt(log x)) and the minor arcs give a power, while the Type I input gives (log x)^(-B). So this route decides whether M_F(x) = o(A_F(x)) on the columns it reaches and says nothing whatever about the exponent theta(F) below. It reaches only the columns with fill >= base^(3/4), by the Parseval floor on the threshold above, and only under (n, base) = 1, by the split above: three limits stated rather than assumed. The weaker reading fill > sqrt(base), which follows from alpha_1 < 1/2 alone, stays true and is simply not sharp.
• What would break this route against the census, and that it does not. Verified. The criterion's conclusion is a log saving, so it caps theta(F) at 1 in A_F units, far above every measured running-maximum exponent of the census below, 0.4465 to 0.5358: neither statement can break the other. The chain that does print an exponent is the GRH one of coprime, theta(F) <= 1 - (1/4 - alpha_1)/alpha, and it reads above 1 at every base-10 column, 1.1054746 at the digit 4, and 0.9999819 at base 21 missing 0, a saving under 2 * 10^(-5) in the exponent against the trivial bound. So no proved conditional bound sits below a measured exponent at any design of the census, in either chain, and a design whose measured exponent rose above its own proved ceiling would refute one of them (lab/py/mobius-region, verb criterion).
• The paper. The l^1 census of this route, the least base 21 missing 0, the family floor 34, the digit-uniform bound from 125 and the Mertens bar under GRH are written up as The First Base Below a Quarter.

The unconditional dissection

The GRH section buys a power saving on the dense columns and the pair route buys a log saving through the Type I level of distribution. A third route splits the frequency grid by Dirichlet approximation instead of by arithmetic, imports an unconditional minor-arc bound for mu, and reaches a saving of classical zero-free-region shape with no hypothesis at all. Its whole price is paid in the base. The wall, the block and region checks and the falsification are printed by lab/py/mobius-dissection.

• The theorem. Proved. Let E be the excluded digits, m = |E| >= 1, F the rest and fill = base - m, let F keep two consecutive digits, and let it carry a shifted-grid l^1 certificate below 1/5: constants C_F >= 1 and alpha_1 < 1/5 with sum_(a < base^i) |hat F_i(s + a/base^i)| <= C_F fill^i base^(i alpha_1) at every i >= 0 and every real s. Then there are C > 0 and c > 0, depending on base and F alone and both effectively computable, with |M_F(x)| <= C A_F(x) exp(-c sqrt(log x)) at every x >= 2: an unconditional o(A_F(x)) on the dense column, of classical zero-free-region shape and not a power. Every one-missing-digit set from base 584 carries such a certificate, by the digit-uniform chain of coprime The dissection for the von Mangoldt function, every one of base 301..583 by the window certificates Verified there, and every one of base 115..300 by the per-digit window certificates of the bullet The least base below a fifth is 115 on the same page, also Verified; so the bound is Proved at every one-missing-digit set from base 584 and holds at every one from base 115 on those certificates. 115 is the floor of this route at one missing digit: every base 3..114 carries a missing digit certified alpha_1 > 1/5 there, base 114 missing 56 among them, so no certificate below 1/5 exists for that set, while single sets clear lower, the first base 65 missing 0. The wall condition (W), PB_base(m) < fill base^(-4/5), or at m = 1 and base >= 36 its chord form PB'_base(1, e_0) < (base - 1) base^(-4/5), is a certificate with C_F = 1 that forces the consecutive pair, and at m > 1 it is the one this page carries, the m-budget of the GRH section read at b = 4/5, which admits m <= 176 at base 10^7. The bullets below are the proof: the blocks, the dissection, the minor-arc input, the four regions, the arithmetic input and the assembly.
• What the per-digit constant alone gives. Verified. At one excluded digit (W) is the ladder rung b = 4/5 read with the chord constant, and it closes far above the chain's 584; its walls are the one place on this section where a float scan, and not a proof, fixes a number, so they are Verified and never Proved. An exhaustive float scan of 36..2 * 10^5 puts the least base at 39363 over every e_0 and at 28352 over e_0 in {0, base-1}, each an up-set of the scan (held 160638 = 200000 - 39363 + 1 and 171649), and the least float gap above each wall sits at the wall itself; re-read at 40 digits the gap (base - 1) base^(-4/5) - PB'_base(1, e_0) is <= -2.3195 * 10^-6 at 39362 against >= 2.3677 * 10^-5 at 39363, and <= -1.3539 * 10^-5 at 28351 against >= 1.8837 * 10^-5 at 28352. Above the scan the chord bullet's PB'_base(1, e_0) < PB_base(1) and the step 3 wall 92317 of the ladder, a half line by the proved floor, carry it, so both walls are half lines. The step 3 constant alone closes at 92317 at every digit, and 1/5 - alpha_1 is at least 2.6965 * 10^-7 at 39363, 2.3643 * 10^-7 at 28352 and 1.8712 * 10^-8 at 92317 (lab/py/mobius-dissection, verb wall).
• The blocks, and x off the powers of base. Proved. For k >= 0, y = base^k and P >= 0 the block P y + D_k has sum Sigma(P, k) = sum_(t in D_k) mu(P y + t) = y^(-1) sum_(a mod y) hat F_k(a/y) S_P(-a/y) with S_P(theta) = sum_(P y <= n < (P+1) y) mu(n) e(n theta), by step 1 of the GRH section shifted by P y, whose phase e(P a) is 1. Let x have L digits x_j. Reading n <= x against the digits of x from the top splits S_F below x into the blocks P base^k + D_k whose prefix P copies the digits of x above position k, all of them in F, and then takes a digit f < x_k of F, with f >= 1 at the top position; the shorter elements, which are D_(L-1) less {0} when 0 is in F and the blocks D_l, 1 <= l < L, when it is not; and the point x itself, with mu(0) = 0 read wherever a block at P = 0 holds 0. So there are at most fill + 1 blocks at each scale k < L, and A_F(x) >= fill^(L-1) - 1. With K = ceil(kappa sqrt(log x)) the blocks at scales k < L - K weigh at most ((fill + 1)/(fill - 1)) fill^(L-K) <= 6 fill^(1-K) A_F(x) together, and every other block has x/y < base^K, so the theorem follows from the block bound |Sigma(P, k)| <= C' fill^k exp(-c' sqrt(log x)) at x/y < base^K, which is what the bullets below prove; the same split removes the case 0 outside F, which no block sees. The split, the count per scale and the mass floor are checked exactly at 400 random x below 2 * 10^6 and at every base^e - 1 in five families (verb blocks).
• The dissection. Fix a block with x/y < base^K, put Q = y^(3/5) and Z = exp(C_0 sqrt(log x)) with C_0 <= 1/2. Every a mod y has a reduced l/d with d <= Q and |a/y - l/d| <= 1/(d Q), by Dirichlet; fix one and write h = |a d - l y| <= y^(2/5) for its height. Region A is d >= y^(2/5); region B is d < y^(2/5) with max(d, h) >= Z; region C1 is d < Z and h < Z with d carrying a prime outside base; region C2 is d < Z and h < Z with d dividing a power of base. The four partition the residues, and mu is real, so |S_P(-a/y)| = |S_P(a/y)| and each region is bounded by y^(-1) sum_a |hat F_k(a/y)| |S_P(a/y)| over its own residues. Throughout, C_F and alpha_1 are the certificate of the theorem. Under (W), B_base(F) <= base PB with PB its constant, and step 2 of the GRH section, run on a shifted grid since B_base(F) is a supremum over shifts, gives sum_(a < V) |hat F_i(s + a/V)| <= (base PB)^i = fill^i V^(alpha_1) at every V = base^i and every real s, the certificate with C_F = 1 and alpha_1 = 1 + c_W - alpha_base, c_W = log_base PB, so that (W) reads alpha_1 < 1/5.
• The minor-arc input, at both approximations. Proved. Basak, Robles and Zaharescu 2023, Theorem 1.4, read at source, gives |S_mu(X, theta)| <<_eps X^(4/5 + eps) + X d^(-1/2) (log X)^3 + (X d)^(1/2) (log X)^3 for S_mu(X, theta) = sum_(n <= X) mu(n) e(n theta), |theta - l/d| <= 1/d^2, (l, d) = 1, X >= 2 and fixed eps > 0, and nothing else; its proof is Vaughan's identity at U = V = min(X^(2/5), d, X/d) against the Type I and Type II estimates of Koukoulopoulos 2019, Theorems 23.5 and 23.6, read at source in the author's preliminary text, sum_(n <= x) (f * log^v)(n) e(n alpha) << (y + x/q + q)(log x)^(v+1) ||f||_inf at every v >= 0 and sum_(n <= x) (f * g)(n) e(alpha n) << (q + y + z + yz/q)^(1/2) sqrt(log 2q) ||f||_2 ||g||_2, both with absolute constants, together with the divisor bound, so the constant depends on eps alone and is effective; the one Siegel-Walfisz input of that paper sits in its major-arc section and is not used, and its restatement of the Type I estimate at v >= 2, applied at v = 0, is covered by the book's v >= 0. Since S_P is the difference of two such sums of length at most x and x/y < base^K = x^(o(1)), region A's data y^(2/5) <= d <= y^(3/5) and |a/y - l/d| <= 1/(d Q) <= 1/d^2 give |S_P(a/y)| <<_eps x^(4/5 + 2 eps). Below y^(2/5) there is a second approximation: when h >= 1, Dirichlet at level 2y/h gives l'/d' with |a/y - l'/d'| <= h/(2 d' y), which is not l/d since l/d sits at distance h/(d y), so 1/(d d') <= h/(d y) + h/(2 d' y), that is y <= h d' + h d/2, and y/(2h) <= d' <= 2y/h because d h <= y; at l'/d' the theorem's last two terms are x (2h/y)^(1/2) <= x^(7/10 + o(1)) and (2 x y/h)^(1/2) <= 2^(1/2) x h^(-1/2). So on regions B, C1 and C2, |S_P(a/y)| <<_eps x^(4/5 + 2 eps) + x (log x)^3 max(d, h)^(-1/2).
• Region A. Proved. Its residues carry at most the whole unshifted mass, y^(-1) sum_(a mod y) |hat F_k(a/y)| <= C_F fill^k y^(alpha_1 - 1), the certificate at s = 0, against |S_P| <<_eps x^(4/5 + 2 eps), so region A is <<_eps C_F fill^k y^(alpha_1 - 1/5 + 3 eps), a power saving at eps = (1/5 - alpha_1)/4. It is the only region that pays the whole l^1 mass against x^(4/5), and so the one that sets the wall.
• The hybrid l^1 bound. Proved. Let D >= 1 and H >= 0 with 16 base^2 D (D + H) <= y, and let R(D, H) be the residues whose data has D <= d < 2D and h < 2H, or h = 0 when H = 0. Let V_1 = base^(i_1) be the least power of base at least 4 D^2 and V_2 = base^(i_2) the least at least 4H/D + 1, so V_1 V_2 < 16 base^2 D (D + H) <= y. Then sum_(a in R(D, H)) |hat F_k(a/y)| <= C_F^2 (1 + pi base^2 C_F/fill) fill^k (V_1 V_2)^(alpha_1), and (1 + pi base) fill^k (V_1 V_2)^(alpha_1) under (W). Split hat F_k(t) = hat F_(i_1)(t) hat F_(k - i_1 - i_2)(V_1 t) hat F_(i_2)(y t/V_2) and bound the middle factor by fill^(k - i_1 - i_2). At t = a/y the top factor is hat F_(i_2)(a/V_2), and the residues attached to one fraction l/d are at most 4H/D + 1 <= V_2 consecutive integers, distinct mod V_2, so they pay at most the grid sum C_F fill^(i_2) V_2^(alpha_1), the certificate at s = 0. The bottom factor is at most its supremum over J_(l,d) = [l/d - 1/(8 D^2), l/d + 1/(8 D^2)], which holds every residue of l/d because 16 D H <= y, and the J_(l,d) are disjoint because distinct fractions of denominator below 2D differ by more than 1/(4 D^2), fractions read mod 1 so that 0/1 and 1/1 are one, whose residues are then consecutive mod y and, since V_2 divides y, still distinct mod V_2; so by |f(t)| <= |J|^(-1) int_J |f| + int_J |f'| on each J the suprema sum to at most 4 D^2 ||f||_1 + ||f'||_1 with f = hat F_(i_1). The certificate at the shift u/V_1, integrated over u, gives ||f||_1 <= V_1^(-1) C_F fill^(i_1) V_1^(alpha_1). Differentiating the product one factor at a time costs |g_F'| <= pi base (base - 1) at position j, and the product with that factor removed is |hat F_j(t)| |hat F_(i_1 - j - 1)(base^(j+1) t)|, whose integral splits at the period of the second factor into base shifted grids of the first and is at most C_F^2 (fill base^(alpha_1 - 1))^(i_1 - 1); with sum_(j < i_1) base^j < V_1/(base - 1) that gives ||f'||_1 <= (pi base^2/fill) C_F^2 fill^(i_1) V_1^(alpha_1), so the bottom pays C_F (1 + pi base^2 C_F/fill) fill^(i_1) V_1^(alpha_1). Under (W) the peeling recursion of step 2 run inside the integral does better, ||f||_1 <= V_1^(-1) B_base(F)^(i_1) and ||f'||_1 <= pi base^2 B_base(F)^(i_1 - 1) <= pi base B_base(F)^(i_1) by the l^1 floor B_base(F) >= base, whence 1 + pi base. The certificate is the whole input, at every shift for the bottom factor and at s = 0 for the top, and no large sieve is quoted. At three small designs the C_F = 1 form of the bound, with alpha_1 from the step 3 constant, holds at every class meeting the two conditions its proof uses, V_1 V_2 <= y and 16 D H <= y, a wider set than its hypothesis, 73, 73 and 86 classes; past the class of a = 0, where |hat F_k(0)| = fill^k makes the ratio a closed form, the largest ratio is at most 0.001356, and the Gallagher step itself, at the 71, 71 and 75 of those classes other than that of a = 0 with V_1 <= 10^5, the fractions' largest |hat F_(i_1)| summed against 4 D^2 ||f||_1 + ||f'||_1 with both norms read on a grid, reads at most 0.110814, a check that can fail (verb regions).
• Region B. Proved. Group its residues into classes D <= d < 2D, and H <= h < 2H or h = 0, over powers of two D < y^(2/5) and H <= y^(2/5), so 16 base^2 D (D + H) <= 32 base^2 y^(4/5) <= y once y is large, every class has max(D, H) > Z/2, and there are at most (log_2 y + 2)^2 classes. By the minor-arc bullet and the hybrid bound a class weighs <<_eps y^(-1) (x^(4/5 + 2 eps) + x (log x)^3 max(D, H)^(-1/2)) fill^k (32 base^2 max(D, H)^2)^(alpha_1). With max(D, H) <= y^(2/5) the first term sums to << fill^k y^((4/5)(alpha_1 - 1/4) + 3 eps) (log y)^2, a power saving, and the second, since 2 alpha_1 - 1/2 < -1/10 and max(D, H) > Z/2, to << fill^k base^K (log x)^5 Z^(-1/10). So region B asks only alpha_1 < 1/4, with kappa log base < C_0 (1/2 - 2 alpha_1) for the block depth, and with the per-digit constant alpha_1 < 1/4 is PB < fill base^(-3/4), the GRH section's own certificate at b = 3/4, closing from 1499 at one excluded digit; region A, which pays the whole l^1 mass against x^(4/5), is the one that binds.
• Region C1 at the grid point. Proved. A residue of C1 sits at a/y = l/d + eta with |eta| = h/(d y) < Z/y, where d = e m' with e dividing a power of base, m' > 1 coprime to base, and l is nonzero mod m' because (l, d) = 1. That is the perturbed Lemma A' of coprime at level k, stated at t/d + eta and so needing no passage from the exact fraction, read at dim = 1, where its proof runs verbatim: for d = e m' as above, l nonzero mod m', a digit set with two consecutive digits and |eta| < base^(-2k/3)/(4 base (base - 1)), |hat F_k(l/d + eta)| <= fill^k c'^(floor(2k/(3 m_d))), the pair being the two consecutive digits; its hypothesis |eta| < y^(-2/3)/(4 base (base - 1)) holds once 4 base^2 Z <= y^(1/3), and its condition (E) holds because F carries two consecutive digits, which the theorem assumes and (W) forces (the assembly bullet). It gives |hat F_k(a/y)| <= fill^k c'^(floor(2k/(3 m_d))) with c' = 1 - (2/fill)(1 - cos(pi/(4 base))) and m_d <= log_base Z + 1, and with log y >= (log x)/2 that exponent is at least sqrt(log x)/(6 C_0) - 1 for x large. C1 holds at most sum_(d < Z) phi(d)(2Z/d + 1) <= 3 Z^2 residues and |S_P| <= y, so region C1 is at most 3 c'^(-1) fill^k exp((2 C_0 - |log c'|/(6 C_0)) sqrt(log x)), which saves once C_0^2 < |log c'|/12; |log c'| is about pi^2/(16 fill base^2), tiny and explicit. The bound is met at 3000 seeded grid points at level 30 in base 10 and 3000 at level 45 in base 5, the logarithm of the bound exceeding that of |hat F_k(a/y)| by at least 27.9 at every one (verb regions).
• Region C2 and the Perron constants. Proved. Each prime power in a C2 denominator d < Z has exponent below log_2 Z < k, so d divides y, a = l y/d + j with |j| = h/d < Z/d, and partial summation of e(n j/y) along the block gives |S_P(a/y)| <= (1 + 2 pi |j|) max_u |sum_(P y <= n <= u) mu(n) e(n l/d)| <= 2 (d + 2 pi Z) max_(u <= x, r) |M(u; d, r)| <= 16 Z max_(u <= x, r) |M(u; d, r)|, the partial sum being sum_(r mod d) e(r l/d) times a difference of two M, with M(u; d, r) = sum_(n <= u, n = r mod d) mu(n). C2 holds at most 3 Z N_Z residues, N_Z <= (1 + log_2 Z)^omega(base) the number of base-smooth d < Z, so region C2 is at most 48 fill^k base^K Z^2 N_Z x^(-1) max |M(u; d, r)|, and the one arithmetic input is (M): |M(u; d, r)| <= C_1 x exp(-c_1 sqrt(log x)) at every u <= x, every r and every d dividing a power of base with d <= exp(sqrt(log x)/2), C_1 and c_1 effective. Proof. At u <= x exp(-sqrt(log x)) it is trivial. Otherwise put g = (r, d): every n = r mod d has (n, d) = g, so M = 0 unless g is squarefree, and then (r/g, d/g) = 1, mu(g n') = mu(g) mu(n') at (n', g) = 1, and M(u; d, r) = mu(g) sum_b M(u/g; q, b) over at most rad(base) classes b coprime to q = (d/g) s, s the primes of g not dividing d/g, a base-smooth modulus below d rad(base). Orthogonality gives |M(v; q, b)| <= max_(chi mod q) |M(v, chi)|, and mu chi is mu chi* convolved with chi* on the integers built from the primes of q, chi* the primitive character behind chi, so M(v, chi) = sum_w chi*(w) M(v/w, chi*) over at most (1 + log_2 v)^omega(base) such w, the lengths v/w below sqrt v bounded trivially and the rest at least x^(1/4), with conductor q* <= rad(base) exp(sqrt(log x)/2), below rad(base) exp(sqrt(log v/w)). At q* = 1 the sum is the Mertens function and << v exp(-c_2 sqrt(log v)) is Koukoulopoulos 2019, Exercise 8.4, read at source with its hint, which is the argument that follows with zeta in place of L. At q* >= 3 Perron's formula, Theorem 7.2 there, at alpha = 1 + 1/log v and T = exp(sqrt(log v)) gives M(v, chi*) = (2 pi i)^(-1) int_(alpha - iT)^(alpha + iT) v^s ds/(s L(s, chi*)) + O(v (log v)/T + 1). If chi* has no exceptional zero, Chang and Martin 2019, Lemma 2.1, read at source, makes L(s, chi*) zero-free in sigma >= 1 - c_3/log(q* (|tau| + 4)) and |log L(s, chi*)| <= log log(q* (|tau| + 4)) + O(1) in half that width, c_3 effective and absolute, so |1/L| <= exp |log L| << log(q* (|tau| + 4)), which is their Lemma 2.4, |w^z| <= e^(L |z|) when |log w| <= L, at z = -1; if it has one, chi* is real, since a zero beta of a complex chi* would be a zero of its conjugate too and the lemma allows one character, and so its conductor divides 8 rad(base) by the conductor bullet of the pair route, and Proposition 2.3 of the same paper makes L(s, chi*) zero-free in sigma >= 1 - c_0/(1 + log^+ |tau|) with c_0 = eta_0/(q*^(1/2) log^2 q*) and |log L| <= (1/2) log q* + 3 log log(q* (|tau| + 4)) + O(1) there, the paper's log log^+(q tau) read with q* (|tau| + 4), a larger argument, eta_0 effective, a finite family whose c_0 is bounded below by base alone. Moving the segment to sigma_1 = 1 - (c_3/2)/log(q* (T + 4)), or to 1 - c_0/(1 + log T), inside the region costs v^(sigma_1) (log v)^4 + v (log v)^4/T, and log(q* (T + 4)) <= 3 sqrt(log v) makes it << v exp(-(c_3/7) sqrt(log v)), or <<_base v exp(-(c_0/3) sqrt(log v)). Summing back over w, b and g gives (M). So region C2 saves exp(-(c_1 - 2 C_0 - kappa log base) sqrt(log x)) up to powers of log x. Siegel's theorem is never invoked: the only exceptional zeros in play belong to a finite family fixed by base.
• Assembly, and every m. Proved. Take C_0 = min(1/2, sqrt(|log c'|/24), c_1/4) and kappa = C_0/(20 log base), so base^K <= base exp((C_0/20) sqrt(log x)). Then region A saves a power of x, region B saves exp(-(C_0/20) sqrt(log x)), region C1 saves exp(-(|log c'|/(12 C_0)) sqrt(log x)) and region C2 saves exp(-(c_1/2 - C_0/20) sqrt(log x)), each up to powers of log x and constants depending on base and F, and the discarded scales of the blocks bullet save fill^(1-K) <= fill exp(-kappa log(fill) sqrt(log x)); the theorem follows with c = kappa log(fill)/2, and every x below the effective point where the steps marked for x large take hold is absorbed into C. No step above used m = 1 except through the certificate: region A asks alpha_1 < 1/5, region B asks alpha_1 < 1/4, the block split counts at most fill + 1 blocks per scale at every m, C2 never sees F, and C1 asks (E), which holds as soon as F has two consecutive digits. That is automatic at m = 1, and at m > 1 (W) forces it: PB_base(m) >= sqrt(m) and PB_base(m) > 2, the latter from Phi_base >= (4/pi) base, so (W) asks base^(1/5) > 2 and sqrt(m) < base^(1/5), whence m < base^(2/5) < (base - 1)/2, and fewer than (base - 1)/2 excluded digits leave two consecutive ones. That is the fourth bookkeeping item; the other three are the C2 bullet's Perron constants, the blocks bullet's x off the powers of base and the C1 bullet's passage to the grid point.
• The falsification. Verified. At every grid point of three small designs, base 10 missing 5 and missing 0 at level 6 and base 5 missing 2 at level 9, with Z = 16 and 12: the four regions sum to the exact sum_(n in D_k) mu(n) to float precision; C1 and C2 stay under 3 Z^2 and 3 Z N_Z; every C2 denominator divides y with |j| d = h < Z; the second approximation lands in [y/(2h), 2y/h] at all 74778 and 128942 points of region B with h >= 1; and the hybrid bound holds at every class meeting the two conditions its proof uses, 73 classes at each base-10 design and 86 at base 5, its Gallagher step at the 71, 71 and 75 of them other than the class of a = 0 with V_1 <= 10^5, reading at most 0.110814 of its bound. The minor-arc bound carries an unstated constant, so its readings on the grid are printed and prove nothing (verb regions, five seconds).
• What lowers the wall. Proved. Only region A sets it, through two numbers: the minor-arc exponent 4/5 on y^(2/5) <= d <= y^(3/5) and the l^1 certificate, which the digit-uniform chain of coprime carries down to 584, its window certificates to 301 and the per-digit window certificates to 115, the floor at one missing digit, from the per-digit constant's 39363, a float-scanned wall. A per-digit constant bounds the shifted-grid supremum B_base(F), which has a floor at one excluded digit, B_base(F) >= 2(base - 1) from the l^1 floor bullet of the GRH section, so region A run on any such constant needs (base - 1) base^(-4/5) > 2 - 2/base, that is base^(1/5) > 2, and no constant bounding B_base(F) brings this dissection below base 33. The chain bounds the sums at every depth at once and meets no such floor, and region A pays only the unshifted mass c_k = sum_(a mod base^k) |hat F_k(a/base^k)|, whose only proved floor is Parseval's base^k; the readings c_k/c_(k-1) = 74.1654 at base 33 missing 16 and 70.5661 missing 0, at k = 4, against 2(base - 1) = 64, and 35.5234 at base 17 against 32, put it above that floor and prove nothing (verb wall). The exponent 4/5 is the one Maynard 2022 calls a limit of its own method, and that paper carries no Type I or Type II estimate at all, as REFS.md records at source, so nothing imported from it moves the wall; the family floor 34 of the pair route is reached through a level of distribution and a bilinear estimate, not through a minor-arc exponent.
• The shape is in print for the primes, the object is not. Maynard 2022, Theorem 1.1, proves the von Mangoldt asymptotic at one missing digit for base > 2 * 10^6, remarks that a more involved calculation reaches base > 2500 by the same method, and remarks that Siegel zeros play no role at these highly composite moduli, so its error terms could be made effective of exp(-c sqrt(log x)) shape. Its two inputs, a minor-arc bound for Lambda and primes in progressions to moduli dividing a power of the base, have standard mu analogues, so a Mobius version at base > 2 * 10^6 is a routine adaptation, and it is not written there or in any source REFS.md lists. What this section adds is that object written out, mu over the design with an effective constant, from base 584, and from 115 on the window certificates, below both the source's written 2 * 10^6 and its remarked 2500, the von Mangoldt count being carried from the same bases on coprime; the shape is in print for Lambda, and the card would be first in its object and never in its shape.
• Dead at fixed digit count. Proved. At F = {0,1} in base 3 the design's l^1 exponent is log 2 / log 3 = 0.630929, above every bar the dissection sets, region B's 1/4 and region A's own ask included: the route is dead there at every strength of every input. This is the l^1 floor of the GRH section again, read arc by arc instead of position by position, and it is why the wall sits in the base and not in the depth.
• Its reach. The saving is exp(-c sqrt(log x)) with c effective but tiny, at most sqrt(|log c'|/24)/40 by the assembly, |log c'| being about pi^2/(16 fill base^2); the theorem says nothing at any F without a certificate below 1/5, so nothing at any column of the census, and nothing about the exponent theta(F). The divisor section's power saving is a level-of-distribution statement about a different sum and is not a rival bound on M_F.
• Vaughan's identity is circular here at power strength. Proved. In the mu_{<=U} * mu_{<=U} * 1 piece of Vaughan's identity the main term over S_F is fill^level M_1(U)^2 with M_1(y) = sum_{f <= y} mu(f)/f, so bounding the pieces one by one at power strength already forces M_1(U) << U^(-delta), which continues 1/zeta into sigma > 1 - delta. The method is blocked at power strength by the same circularity that leaves Vaughan's identity for the classical Mertens function at log strength. Nothing about M_F itself is decided by this: what it asks for is an identity whose pieces do not isolate M_1(U), and this page carries none.
• The paper. The theorem, its walls, the hybrid l^1 bound, the characters and the floor 33 are written up for an outside reader as An Unconditional Mertens Bound at Large Base.
• The demo. The dissection demo runs the objects of this section on a small base and one missing digit, base 10 missing 7 and its friends: M_F(x) against A_F(x) and A_F(x)^(1/2), psi_F(x) against kappa_F A_F(x), the grid a/base^k cut into A, B, C1 and C2 with each region's share of the l^1 mass, and the chain's alpha_1 by base against 1/5 and 1/4 with the walls drawn.

The half interval

The dissection above spends its base on one number, the l^1 cost of the digit transform, and asks nothing else of the digit set but a consecutive pair. Half the digits kept as an interval is the set where that cost is a Dirichlet kernel's: logarithmic in the base, where the proved one-step bound of the GRH section, sqrt(m) + Phi_base/base per digit, pays sqrt(m), a power of the base at m = fill - 1; the exact one-step constant is at most (3/2) lambda by the certificate below, logarithmic as well, and only the bound is lost. Every number below is printed by lab/py/interval-digits.

• The object. Proved. At an odd base base = 2 fill - 1 the half interval is F = {0..fill-1}: m = fill - 1 excluded digits, all consecutive, and S_F holds the positive integers whose digits are all at most (base-1)/2. Doubling carries nowhere, so 2 S_F is the design on the even digits {0, 2, .., base-1}, which keeps no consecutive pair and has only even members. At a prime base = p, S_F is the set of k >= 1 with p not dividing C(2k,k): Legendre's formula gives v_p(C(2k,k)) = (2 s_p(k) - s_p(2k))/(p-1), s_p the digit sum, and s_p(2k) = 2 s_p(k) - (p-1) c with c the number of carries in k + k, so v_p(C(2k,k)) = c, zero exactly when every digit of k is at most (p-1)/2.
• Where the digit set enters. Proved. The theorem of the section above holds at any number of excluded digits and reads the digit set at six points: the shifted-grid certificate, at s = 0 in region A and at every shift in the hybrid bound of region B; two consecutive digits in region C1, through Lemma A' of coprime; fill in the constant c' = 1 - (2/fill)(1 - cos(pi/(4 base))) of C1, which the assembly carries into C_0 and c; at most fill + 1 blocks per scale; 0 in F, which gives the mass floor A_F(x) >= fill^(L-1) - 1 of the block split directly, the case 0 outside F taking the shorter blocks; and, for the prime count only, kappa_F, read on the last digit. Every region bound is fill^k times a saving, so the mass exponent alpha_base = log(fill)/log(base) enters only through the Parseval floor alpha_1 >= 1 - alpha_base; that floor makes alpha_base > 4/5 necessary for any certificate below 1/5, and the half interval, where 1 - alpha_base = log(2 base/(base + 1))/log base < log 2/log base, meets it at every odd base >= 27 and fails it at 25 (verb wall). The obstruction a half-density set might meet is not met: the Type II information sits inside the minor-arc bound for mu itself, the l^infinity input is Lemma A' under the pair alone, and no large sieve is used, the hybrid l^1 bound reading the certificate at every shift. So the theorem, and the prime count of coprime, hold on the half interval, which keeps the pair {0, 1} and the digit 0, at every odd base where it carries a certificate below 1/5, and the rest of this section supplies one.
• The transfer operator. Proved. For phi of period 1 put (T phi)(t) = sum_(r < base) |hat F((t+r)/base)| phi((t+r)/base), and write Sigma_i(s) = sum_(a < base^i) |hat F_i(s + a/base^i)|. Peeling the lowest digit, a = a' + base^(i-1) r puts s + a/base^i = (t + r)/base with t = base s + a'/base^(i-1), and the higher factors form hat F_(i-1)(t), free of r; so Sigma_i(s) is the level-(i-1) sum at shift base s weighted by T 1, and by induction Sigma_i(s) = (T^i 1)(base^i s) exactly. Hence if 1 <= phi <= Phi and T phi <= lambda phi everywhere, Sigma_i(s) <= Phi lambda^i at every i and every real s: a certificate with C_F = Phi and base^(alpha_1) = lambda/fill, and phi = 1 is the peel of the GRH section, lambda = B_base(F) = max T 1. Write G = T 1. Since |hat F(u)| sin(pi u) = |sin(pi fill u)|, the weight |sin(pi t)| goes to S(t) = sum_(r < base) |sin(pi fill (t+r)/base)|, and fill is the inverse of 2 mod base, so fill r runs over every residue and the geometric sum gives S(t) = cos(pi (sigma - 1/2)/base)/sin(pi/(2 base)) <= csc(pi/(2 base)), sigma the fractional part of fill t.
• The certificate. Proved. At every odd base >= 3, with c_P = sqrt 2 - 4/pi and X_0 = (base/pi)(log(base + 3) + gamma + log tan(3 pi/8 + pi/(4 base))) + c_P (base + 1)^2/(8 base), the half interval has T phi <= lambda phi for phi(t) = 1 + |sin(pi t)|/2 and lambda = fill + X_0 + csc(pi/(2 base))/2, so Sigma_i(s) <= (3/2) lambda^i at every i and every real s. Proof. Both sides have period 1 and are even about t = 1/2, so take t in [0, 1/2], a = sin(pi t/2) and b = cos(pi t/2). The points (t + r)/base lie at distance delta from Z, with delta = (t + r)/base for r < fill, the low points, and delta = (1 - t + r)/base for r <= fill - 2, the high points, every delta <= 1/2; there |hat F| = |sin(pi fill delta)|/sin(pi delta) with fill delta = (t + r + delta)/2 or (1 - t + r + delta)/2. So with P(delta) = 1/(2 sin(pi delta/2)) and Q(delta) = 1/(2 cos(pi delta/2)) a point weighs aP + bQ at a low even r >= 2, at most fill at r = 0, bP - aQ at a low odd r, bP + aQ at a high even r, and |bQ - aP| at a high odd r, which is bQ - aP where delta > t and at most aP where delta <= t. The aQ terms sum to at most 0, the high even point (2j + 1 - t)/base sitting below the low odd point (2j + 1 + t)/base against the increasing Q. On (0, 1/2], 1/(pi delta) <= P(delta) <= 1/(pi delta) + c_P delta, the chord of the convex csc z - 1/z on (0, pi/4]. The bP terms carry at most (base/pi) sum_(j <= J) (1/(2j+1+t) + 1/(2j+1-t)) + c_P (base+1)^2/(8 base), J = floor((fill-2)/2), and 1/(m+t) + 1/(m-t) = 2/m + 2t^2/(m(m^2 - t^2)) with sum_(j <= J) 2/(2j+1) = psi(J + 3/2) + gamma + 2 log 2 <= log(base + 3) + gamma by psi(x) <= log x - 1/(2x), the rest at most K_2 t^2 with K_2 = 2(4/3 + (36/35)(7 zeta(3)/8 - 1)) < 2.7733. The bQ terms of the low even and high odd points, by Hermite-Hadamard on the convex Q at spacing 2/base inside [0, 1/2 + 1/base], carry at most (base/pi) log tan(3 pi/8 + pi/(4 base)). The aP terms carry a U with U = sum_LE P + sum_(HO, delta <= t) P - sum_(HO, delta > t) P, and U < X_0. The low even points are delta = (t + 2k)/base, 1 <= k <= K = floor((fill-1)/2) <= (base-1)/4, each delta <= 1/2, so they carry at most (base/pi) sum_(k <= K) 1/(2k) + c_P K/2 <= (base/(2 pi))(log((base - 1)/4) + 1) + c_P (base - 1)/8 by H_K <= log K + 1. The high odd points are delta = (2k - t)/base, 1 <= k <= K, with delta <= t exactly at k <= X = t(base + 1)/2; put k* = min(floor(X), K). The points up to k* carry at most (base/pi)(2/3 + (1/2) log((4k* - 1)/3)) + c_P X t when k* >= 1, from 1/(2k - t) <= 1/(2k - 1/2) <= (1/2) int_(k-1)^k dx/(x - 1/4) at k >= 2, and the points above k* at least (base/(2 pi)) log((K + 1)/(k* + 1)), from P >= 1/(pi delta) and 1/(2k - t) >= 1/(2k) >= (1/2) int_k^(k+1) dx/x. With (4k* - 1)(k* + 1) <= 4X^2 + 3X, X <= (base + 1)/4, K + 1 >= fill/2 = (base + 1)/4 and X t <= (base + 1)/8, the signed high odd sum is at most (base/pi)(2/3 + (1/2) log((base + 4)/3)) + c_P (base + 1)/8, and at k* = 0 it is at most 0. So U <= (base/pi)((1/2) log((base - 1)(base + 4)/12) + 7/6) + c_P base/4 <= (base/pi)(log(base + 4) - 0.0757) + c_P base/4, since 7/6 - (1/2) log 12 < -0.0757; and X_0 - U >= (base/pi)(1.4585 + 0.0757 - 1/(base + 3)) - c_P base/4 > 0, by log tan(3 pi/8) = log(1 + sqrt 2), gamma + log(1 + sqrt 2) > 1.4585 and log((base + 3)/(base + 4)) >= -1/(base + 3), so a U <= a X_0, also when U < 0. So G(t) <= fill + (a + b) X_0 + (base/pi) K_2 t^2 <= fill + X_0 + sin(pi t)(X_0/2 + K_2 base/(4 pi)), by a + b <= 1 + ab = 1 + sin(pi t)/2, which is (1 - a)(1 - b) >= 0, and t^2 <= sin(pi t)/4. Then T phi = G + S/2 <= lambda + (lambda/2) sin(pi t), because K_2 base/(2 pi) < fill. □
• The constant is 2/pi. Proved. At every odd base >= 101, lambda/fill <= (2/pi) log base + c_inf + 3.4/base with c_inf = 1 + 2/pi + (2/pi)(gamma + log(1 + sqrt 2)) + c_P/4 <= 2.60043004, by log(base + 3) <= log base + 3/base, the slope 2/sin(2z) < 2.9 of log tan z on [3 pi/8, 3 pi/8 + pi/(4 base)] and csc z <= 1/z + (2/pi)(1 - 2/pi) z, and checked directly at 120 bits at every odd base 101..3001 and at 94939, 200001, 10^6 + 1 and 10^8 + 1, smallest gap >= 1.3332 * 10^-7 (verb wall). In the other direction G(t) >= (base/pi) log((base + 2)/5) - fill/4 at every t: the low odd and high even points carry b (base/pi)(log(base + 1) + gamma - 2/fill) - fill/4 by psi(x) >= log x - 1/x, the low even points a (base/(2 pi)) log((base + 2)/5), and b + a/2 >= 1 on [0, 1/2]. That gives min G/fill >= (2 base/(pi (base + 1))) log((base + 2)/5) - 1/4 >= (2/pi) log base - 1.31 at every odd base >= 9, the last step certified at 120 bits at every odd base 9..9999 and, from 101, the difference being at least 1.06 - (2/pi) log 5 - (2/pi) log((base + 2)/5)/(base + 1) >= 0.0165; the factor (2/pi) log base - 1.31 is positive from 9 and negative at 3, 5 and 7. Since T^i 1 >= (min G)^i pointwise and min G >= 0, every grid sum Sigma_i(s), shifted or not, is at least (((2/pi) log base - 1.31) fill)^i at every odd base >= 9, and at least base^i at every base, because Parseval on Z/base gives sum_r |hat F((t+r)/base)|^2 = base fill and |hat F| <= fill, so G >= base fill/fill = base. So the l^1 cost of the half interval per digit, in the normalisation of the certificate, is (2/pi) log base + O(1) from both sides, (1/pi) log base measured against base itself. The one-step constant B_base(F) = max G reads (2 sqrt2/pi) log base + 1.19 at t = 1/2, so a one-step certificate pays a factor sqrt 2 in the leading term, and G(0)/fill - (2/pi) log base reads 1.962430 at base 100001 against 1 + gamma' = 1.962523, gamma' the Lebesgue constant of coprime, readings of verb check.
• The wall. Proved. alpha_1 = log_base(lambda/fill) < 1/5 at every odd base >= 94939. The tail bound clears 1/5 from base 94946 and stays clear, since base^(1/5) - (2/pi) log base - c_inf - 3.4/base increases wherever base^(1/5) > 10/pi, from 327; the closed form lambda/fill is certified at 120 bits at every odd base 94939..94947 and does not clear at 94937, gap <= -2.6733 * 10^-5. At the wall 1/5 - alpha_1 >= 1.3678 * 10^-8, and alpha_1 <= 0.1993872 at 100003, 0.1761232 at 1000003 and 0.1331636 at 10^9 + 7. Against 1/4 the same certificate clears at every odd base >= 3789, certified at 3789..3793 with the tail bound from 3793, increasing from 43, not clearing at 3787, and 1/4 - alpha_1 >= 7.9625 * 10^-6 at 3789 (verb wall, under a second). The wall is the certificate's and not the route's: the true rate of T reads (2/pi) log base + 2.26 near 7 * 10^4 and meets base^(1/5) between 70001 and 80001, and the one-step constant would clear 1/5 only from 317063 and 1/4 from 7075, readings of verb rate that bound nothing.
• The theorem on the half interval. Proved. At every odd base >= 94939, with F = {0..fill-1} and kappa_F = (base/phi(base)) #{1 <= f < fill : gcd(f, base) = 1}/fill, there are C, c > 0 computable from base alone with |M_F(x)| <= C A_F(x) exp(-c sqrt(log x)) and |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: the theorem of the section above and the prime count of coprime, its subsection The dissection for the von Mangoldt function, each run on the certificate above and the consecutive pair {0, 1}. At a prime p >= 94939, kappa_F = p/(p+1), so the Mobius function cancels on {k : p does not divide C(2k,k)}, and the sum of Lambda(k) over it up to x is p A(x)/(p+1) within C A(x) exp(-c sqrt(log x)). Proved under GRH, in the form of the GRH section: at every odd base >= 3789 the bullet What the bar 1/4 buys on coprime gives |M_F(x)| <<_(base, eps) A_F(x)^(1 - delta + eps) with delta = (1/4 - alpha_1)/alpha_base > 0, and delta -> 1/4 as alpha_1 -> 0 and alpha_base -> 1, so the exponent tends to 3/4.
• The constant of the prime count is base/(base + 1). Proved. At every odd base >= 3, kappa_F = (base/phi(base)) #{f in F : gcd(f, base) = 1}/fill = base/(base + 1). The residues 1 <= f <= base - 1 coprime to base pair as f and base - f, which is coprime to base too and never equal to f, since f = base/2 is not an integer at odd base; the two sum to base, so exactly one of them is at most (base - 1)/2 = fill - 1, and f = 0 is not coprime to base >= 3. So exactly phi(base)/2 coprime residues lie in F, and kappa_F = (base/phi(base))(phi(base)/2)/fill = base/(2 fill) = base/(base + 1), which is p/(p + 1) at a prime.
• What is in print. Verified at source. Maynard 2022, Theorem 1.3, read in arXiv v1, proves sum_(n < q^k) Lambda(n) 1_B(n) asymptotic with error O_A((q - s)^k (log q^k)^(-A)) when the s excluded digits are consecutive and q - s >= q^(4/5 + eps), q sufficiently large in terms of eps, by a sketch in its Section 9 whose constant alpha_(q,s) = log((2 + 2/log q)(q/(q - s)) log q)/log q clears 1/5 on the half interval only from q = 7777884825, the same reading giving its one-missing-digit crossing 1520573 (verb wall). The half interval has q - s = (q + 1)/2, inside that range, so the prime asymptotic on it at sufficiently large base, with an unquantified base and a log-power error, is the content of that theorem, and at a prime base so is the prime count on {k : q does not divide C(2k,k)}. The theorem prints its main term as q(phi(q) - s')/((q - 1) phi(q)) (q - s)^k, s' the excluded digits coprime to q, which at the half interval of a prime q = p is p/(2(p - 1)) times the count, against the p/(p + 1) of the theorem here; with q - s in place of q - 1, the form of the 2019 constant kappa_B = q(phi(q) - t)/(phi(q)(q - s)) and of the Section 9 instruction to replace q - 1 by q - s, it is p/(p + 1), so the printed form reads as a misprint against the 2019 constant, and the proof here gives the q - s form. Maynard 2019, Theorem 1.2, read in arXiv:1604.01041v2, gives the order of magnitude X^(log(q-s)/log q)/log X of the primes avoiding the top block B = {q - s..q-1} at large q and s <= q - q^(57/80), which holds the half interval, and remarks the asymptotic for the primes avoiding the bottom block B = {0..s-1} at s <= q - q^(3/4 + delta) with an o(1). Neither carries a Mobius sum. After its Theorem 1.1, Maynard 2022 remarks that Siegel zeros play no role at its highly composite moduli, so its error terms could be replaced by effective ones of size O((q-1)^k exp(-c k^(1/2))), and its Section 9 carries the argument to Theorem 1.3; so for Lambda the effective shape is remarked at source. What this section adds is the written proof of that shape at the half interval, the Mobius bound, and the explicit base 94939 for both sums; the gain over the Section 9 constant lies wholly in the certificate, (2/pi) log base per digit against its 4 log base.
• Uniform square-root cancellation over the bases implies RH. Proved. The implication is trivial, the digits playing no part in it. If for every eps > 0 some C_eps gives |M_F(x)| <= C_eps A_F(x)^(1/2 + eps) on the half interval at every odd base and every x >= 1, the Riemann hypothesis holds: S_F contains every integer below fill, so at base = 2 ceil(x) + 1 the sum up to x is the Mertens function M(x) and A_F(x) = floor(x), and M(x) = O(x^(1/2 + eps)) for every eps is RH by the Titchmarsh bullet of the first section. The range x < fill carries the whole implication, where the statement is RH itself; RH says nothing about x >= fill, and no converse is claimed.
• The falsification. Verified. At every odd base 3..401 and at 1001, 10001 and 100001 on 201 shifts, the three sums of the certificate's proof stay under their bounds, the bP sum by at least 3.99 * 10^-5; at every odd base 3..401 on 801 shifts of [0, 1/2] and at 1001, 4001, 10001, 30001 and 100001 on 401, the closed form of S meets the direct sum, G stays under its bound, at most 0.999857 of it and tight at t = 0 by design, T phi under lambda phi, at most 0.999866, and G over its lower bound, at least 1.534638 of it; the grid sums Sigma_i(s) at 40 digits at nine bases 3..21, four shifts and every level with base^i <= 9261 stay under (3/2) lambda^i, at most 0.557408 of it, and the transfer identity holds at level 2 to 10^-15 (verb check, 13 seconds). Far below the wall the meter reads max |M_F|/A_F^(1/2) at 1.267, 1.423 and 1.152 at the prime bases 101, 1009 and 10007 below 10^7, and sum log p/(kappa_F A_F) at 0.9975, 0.9979 and 0.9990, sanity prints that are evidence for nothing (verb meter).
• What stays open. The even-digit design 2 S_F is outside the theorem: its meter is -sum mu(k) over the odd k of S_F, and at odd base the parity of k is that of its digit sum, so that sum is the twist of M_F by e(k/2); the twist reads the certificate at the shift 1/2, which it covers, but near a fraction l/d with d dividing a power of base it sees the transform at l/d + 1/2, so regions C1 and C2 must be rerun with 2 base in place of base, and that is not written. Below 94939 a sharper certificate moves the wall toward the true rate's crossing between 70001 and 80001, and if the unshifted mass that region A pays grows at that rate, as the readings show, no certificate passes the crossing; the crossing and the rate are readings, and a proof below 94939 is not written.
• The paper. The theorem, the certificate with every step, the constant 2/pi from both sides, the walls and the reduction to the dissection are written up for an outside reader as An Unconditional Mertens Bound on the Half Interval.

What the proof imports

The dissection's bullets above name each input at its source; none of them reads the digit set, so nothing differs from one missing digit, and the half interval meets each one by meeting the base. Effective here means computable from base and F, not explicit: the constant of the minor-arc bound at the eps used, the zero-free constants of the character bounds and the constants C_1, c_1 of (M) are unstated at source, so C and c of the theorem are computable and not printed.

• The minor arcs for mu: Basak, Robles and Zaharescu 2023, Theorem 1.4, with Theorems 23.5 and 23.6 of Koukoulopoulos 2019 inside its proof. It asks |theta - l/d| <= 1/d^2 with (l, d) = 1 and fixed eps, a condition on the frequency alone, met at both Dirichlet approximations of the minor-arc bullet. Digit set: nothing. Met.
• The Mertens function and the characters of region C2: Exercise 8.4 and Theorem 7.2 of Koukoulopoulos 2019, and Lemma 2.1, Lemma 2.4 and Proposition 2.3 of Chang and Martin 2019. They hold at every modulus q >= 3; the restriction to moduli dividing a power of base below exp(sqrt(log x)/2) is the dissection's own, and it is what confines any exceptional zero to a real primitive character of conductor dividing 8 rad(base), a finite family fixed by base, so Siegel's theorem is never used. Digit set: nothing. Met at every odd base.
• The prime count: Lemma 4.2 of Maynard 2022 on the minor arcs for Lambda, and Theorems 12.3, 12.4 and 12.8 of Koukoulopoulos 2019 on progressions to the same moduli. Digit set: nothing. Met.
• Under GRH: Baker and Harman 1991 at a = 1/2, step 4 of the GRH section, uniform in the frequency. Digit set: nothing. Met.
• No large sieve and no hybrid estimate is imported: the hybrid l^1 bound is proved in the dissection from the certificate at every shift and |g_F'| <= pi base (base - 1), which holds at every digit set. What the digit set must supply is in-house: the certificate of this section, and the consecutive pair {0, 1} that Lemma A' asks. Met.

The exponent, tagged honestly

• Conjecture. For every digit set F with 2 <= fill <= base - 1 whose digit gcd is squarefree, theta(F) = 1/2: square-root cancellation against the set's own mass, the RH shape transplanted to the sparse column. The census is consistent with this and proves none of it: the 47 running-maximum exponents sit in [0.4465, 0.5358] with per-family drifts of 0.0157..0.1056 over the last five levels, and the full-set controls - whose limiting exponent is 1/2 under RH - read 0.4413..0.4517 at the same depths. A slope is a fit; the exact integers above are the claim, the exponent is not.
• The believable refutation targets are one family with a proved exponent below 1/2 (excess cancellation) or a proved omega-result (a family whose meter provably tracks its mass). The scaling mechanism produces neither: the vanishing family {0,4} at base 5 is total cancellation for the trivial reason 4 | n, and its reduced column {0,1} carries the open question unchanged.
• The base-free carrier of the two-digit column is exact. Proved. For F = {0,1} the meter is base-free: it is one function nu* on the nonzero polynomials of Z[x] up to units, supported on the monoid M* of products of 0/1 polynomials and summed in the order the base fixes, and the whole ladder rests on the classes of nonzero constant term, since nu* vanishes at every polynomial divisible by x^2 and nu*(x b) = -nu*(b). Its design Mertens at the level boundary, M(base^level) = sum_(deg P < level) nu*(P) + nu*(x^level), is pinned to -1 at every level >= 2 from the graded mass 1 - 2t, and reads -1 at every level 2 to 23 where it is computed. The running maximum reads 1, 1, 2, 3, 4, 7, 15, 23, 45, 86, 162, 331, 741, 1665, 3173, 7508, 17753, 36147, 79645, 182432, 427806, 858703, 2026147 at level 1..23, always far under the design mass 2^level: the ratio max/2^level bottoms at 0.079102 at level 11, falls for the last time at level 15, and rises at every step from there to 0.241536 at level 23. What runs ahead of the mass is the rate, and its estimate is window-unstable: at depth 23 the geometric mean step reads 2.245836, 2.202419, 2.242075 over the last 4, 6, 8 levels but 1.996559 over the last 20, a hull straddling the mass rate 2 because the long windows open inside the levels where the ratio was still falling, while at the short windows every reading from depth 20 to 23 sits above 2.18 and above its own depth-18 value. That the rate exceeds the mass rate 2, on a census of twenty-three levels with log_2(max)/level climbing to 0.910883 without settling, stays Conjecture.
• The technology gap is real: distribution of digit-restricted sets in residue classes is Erdos, Mauduit and Sarkozy 1998, the ellipsephic almost-primes rest on it (Dartyge and Mauduit 2000), and primes in one-excluded-digit sets took the full circle method at large base in Maynard 2019, whose Type I input (Proposition 7.1) is the set's own level of distribution in base 10: moduli coprime to 10 up to X^(50/77), the residue 0, a saving of any power of log X, with 50/77 = 1 - 27/77 for the l^1 exponent 27/77 of the digit transform (Lemma 10.3), itself the Markov eigenvalue bound lambda_(1,4) < 2.24190 < 10^(27/77) of that paper's (10.5). Among Maynard 2019, Maynard 2022, Nath 2024 and the sources REFS.md lists, that proposition is the one Type I statement for such a set, and it is stated for base 10 with one excluded digit: the general-base multi-digit version is a substitution sketched in its Section 16, reaching s <= base^(23/80) excluded digits and s <= base - base^(57/80) when they are consecutive, and Maynard 2022 reaches s < base^(1/5 - eps) and base - s >= base^(4/5 + eps) through the four Fourier norms of its Section 5 and the sketch of its Section 9. On the prime side Nath 2024 proves Bombieri-Vinogradov theorems for Lambda(n) 1_A(n) at large base: unweighted with a maximum over residues only to level X^(1/3 - delta), and near X^(1/2) only against well-factorable weights, never unweighted; the set enters that proof through four norms of its transform (l^1, large sieve, hybrid, l^infinity) and never through a progression count, its only set-level fact being the count of the set in one last-digit class. Leng and Sawhney 2025 settle ternary Goldbach on the one-missing-digit set with the l^1 bound g^(eps fill) of the digit transform. The nearest multiplicative function computed over a missing-digit set is the divisor function (Kim 2024), and a proved theta for any restricted column sits at or beyond that frontier; a whole-text search of the three circle-method sources finds the word Mobius once, as an inversion step inside the proof of Proposition 7.1, Liouville nowhere, and Mertens only as Mertens' theorem on a product over primes, so none of them carries a Mobius or Mertens sum over the set. What the divisor section above adds against that is a power saving in x where Proposition 7.1 saves a power of log X, uniform over up to base^(1 - eps)/2 excluded digits, on the modulus range d <= base^(1 - eps): sharper in saving type and in digit count, far shorter in range, and not a first level-of-distribution statement for such sets. Verified at source for Maynard 2019, Maynard 2022, Nath 2024 and Leng and Sawhney 2025. The census stands as the falsifiable record the eventual theorem must match.

Generators

• lab/rs/mobius-designs prints every row, identity check, slope, distribution and band above: CARGO_BUILD_JOBS=4 cargo run --release -p mobius-designs.
• lab/rs/rho-decoupling prints every divisor-section number: CARGO_BUILD_JOBS=4 cargo run --release -p rho-decoupling.
• lab/rs/mertens-numerology prints every constant, table row, margin, rung and cost-out number of the GRH section: CARGO_BUILD_JOBS=4 cargo run --release -p mertens-numerology; its 34 tests pin every rendered row as a string, the sign change of the certificate at 3689 -> 3690, the exhaustive sweep of 3690..10^5 with its smallest step, the kernel bound against the exact shifted-grid sum on a 4001-point grid, each ladder wall below 4 * 10^6 against a scan from base 3, and the constants of the general-b floor bound.
• lab/py/mrly-pairing prints the exact one-step constant at one excluded digit, the phase identity behind the sharpening, the two sharpened step 3 bounds and the walls they move: uv run python research/lab/py/mrly-pairing/pairing.py onestep, its runtime printed in that study's README. Its sup is read on a t-grid of cut 1/4000 over [0, 1/2], where the seat sits at t = 1/2 at every family printed from base 100 up and interior at base 11 and base 13 with e_0 = 0, so every measured B_base(F) there is a reading and never a certificate; the statements above use the sharpened bound and not the reading.
• lab/py/mobius-region prints the pair route's three exponents, its region and corner identities, its threshold certificates and its census: uv run python research/lab/py/mobius-region/mobius_region.py check in 27 seconds, region and boundary in under a second each, threshold in 41 seconds and threshold 0.2626 in 80, criterion in 155 seconds over the 49 designs, and params 21 123456789abcdefghijk in 12. The caps are exact rational functions evaluated in Fraction, alpha is bracketed by integer comparison of fill^b against base^a at whole a and b and never by a float logarithm, and a window cell whose infimum falls to zero prints no lower bound rather than a false one. The rest of the section's numbers are exact rational arithmetic carried out in the sentence that prints them, exponents quoted from the source named there, or certified base thresholds whose generator is named on coprime.
• lab/py/mobius-dissection prints the unconditional dissection's wall at the rung b = 4/5, its block split and its region falsification: uv run python research/lab/py/mobius-dissection/dissection.py wall in under a second, blocks in 18 seconds and regions in five.
• lab/py/interval-digits prints the half interval's certificate walls, the density floor and the lower-bound check in 120-bit interval arithmetic, the falsification of the certificate's proof, the rate readings and the sanity meter: uv run python research/lab/py/interval-digits/interval.py wall in under a second, check in 13 seconds, rate in about 20 and meter in 2.
• The Mertens control on the farey page is rendered by lab/py/mertens-meter; the checkpoint controls here are the same function read at powers of the base.
• lab/rs/carry-free-mobius prints the base-free ladder, its M(base^level) by level, its running maxima, the census of M* by degree and the rate above: uv run python research/lab/rs/carry-free-mobius/carry_free.py exponent is the pinned reference to level 18 and CARGO_BUILD_JOBS=4 cargo run --release -p carry-free-mobius -- ladder 22 2.5 carries it to level 23, asserting the Python numbers at every level to 18 before it prints a deeper one.