The Mobius meter across digit designs

Fix a base q >= 3 and a digit set F inside {0..q-1} with k = |F| >= 2. The digit-restricted set S_F holds the positive integers whose base-q digits 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/mobius-designs, the divisor section's by lab/rho-decoupling, and the GRH section's by lab/mertens-numerology.

Tags as everywhere in this tree: Proved means derived here from definitions, Verified means recomputed exactly and checked against an independent path, Conjecture is labelled belief.

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(q^L) = k^L - 1 when 0 in F (plus 1 when 1 in F too, for the boundary element q^L itself), and A_F(q^L) = (k^(L+1) - k)/(k - 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_q k) 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(q^L) itself, and the running maximum max of |M_F(x)| over x <= q^L, 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_q k), 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 a q-automatic sequence, 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 <= k <= q - 1.
• The Dirichlet series over S_F is built territory, and this page claims nothing about it: the abscissa is log_q k (Kohler and Spilker 2009, with position-varying digit rules in Nathanson 2021); the series continues meromorphically to C with simple poles among s = log_q k - m + 2 pi i j / log q (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 q 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 not k-automatic for any k, 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 q = 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 q^L} |hat F_L(a/q^L)|^(2r) = q^L E_r(L) with E_r(L) the number of 2r-tuples of length-L digit strings with n_1 + ... + n_r = n_(r+1) + ... + n_(2r) mod q^L, by orthogonality, and E_r(L) is counted by a carry DP on the carry pairs of the two sides, so each moment is C-finite in L of order at most r(r+1)/2 and its growth constant Lambda(2r) = q rho is an algebraic number, rho the Perron root of the transfer matrix, certified in exact rationals (lab/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(k^4, q k^2), q k^3] and a hair above k^4 at the dense families (log_q(Lambda(4)/k^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/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 q, 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/rho-decoupling the arcs lines).
• The multiplicative energy of a digit column has no exponent of its own. Proved. With E_x(L) = #{(n_1, n_2, n_3, n_4) in D_L^4 : n_1 n_2 = n_3 n_4} and K = k^L, the two diagonals give 2K^2 - K <= E_x(L), and E_x(L) = sum_m r(m)^2 <= K^2 max_m r(m) with r(m) <= d(m) gives E_x(L) = k^(2L) x^(o(1)) for every base and digit set (the census reads 58760487 at {0,1} base 3, L = 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 (q^i u, q^j v, q^(i') u, q^(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, L = 12 (lab/rho-decoupling, the menergy module).
• 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/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/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/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, L = 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 k^L 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..q-1}, then m -> am maps S_F' bijectively onto S_F preserving digit length: am = sum (a d_j) q^j and each a d_j <= q - 1, so no carry occurs and the digit string scales digitwise. Hence A_F(q^L) equals the string count of F' at the same depth, and M_F(q^L) = sum of mu(am) over m in S_F' with at most L digits.
• Vanishing. If a has a square factor then mu(am) = 0 for every m, so M_F is identically zero: at q = 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')(q^L) = -sum of mu(m) over the m in S_F' not divisible by p. At q = 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 q = 4, M_{0,2}(4^L) = -M_{0,1}(4^L) exactly: since 4 | q, 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^L - 1)/3 and 4^L the running maxima agree level by level as well. The census confirms both at all 22 levels, e.g. meters -110/110 at L = 15, -342/342 at L = 17, 34/-34 at L = 22, and Mmax = 1553 for both at L = 22.

Verified: the generator recomputes all eight scaled census families ({0,2} at q = 3; {0,2}, {0,3} at q = 4; {0,2}, {0,3}, {0,4}, {2,4}, {0,2,4} at q = 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(q^L) 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 (q = 3, F = {1,2}, L = 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^L 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 (k = 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 q = 3 the three columns read 0.4802, 0.4869, 0.4962; at q = 4 the ten run 0.4527 to 0.5197; at q = 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(L; d, r) for the number of length-L digit strings over F whose value sum_j f_j q^j is r mod d, and N_F(L; d) = N_F(L; d, 0). The value map is injective on strings of one length, so with 0 in F this counts the multiples of d below q^L whose padded digits lie in F, and with 0 outside F it is the length-L block of S_F, the blocks l <= L partitioning S_F below q^L. Write k = |F|, e(x) = exp(2 pi i x), g_F(t) = sum_{f in F} e(f t), Delta_F for the gcd of the digit differences, gamma_F(d) = max over a not 0 mod d of |g_F(a/d)|/k, and normalized error for d |N_F(L; d) - k^L/d| / k^L. 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. Every number is printed by lab/rho-decoupling.

• Orthogonality. Proved. N_F(L; d, r) = (1/d) sum_{a mod d} e(-a r/d) prod_{j < L} g_F(a q^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 k^L/d and every bound below is a bound on the rest.
• The uniform geometric bound. Proved. For k >= 2, d >= 2, (d, q) = 1 and gcd(d, Delta_F) = 1, every r and every L >= 1: |N_F(L; d, r) - k^L/d| <= ((d-1)/d) k^L (1 - 8/(k^2 d^2))^L <= k^L exp(-8 L/(k^2 d^2)). Coprimality to q keeps a q^j nonzero mod d at every position, |g_F(a/d)|^2 = k^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/(k^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 q = 100, F = {0,1}, L = 16. The exponent d^(-2) is not slack: at d | q - 1 with F an arithmetic progression of common difference m', taking a m' = 1 mod d gives |g_F(a/d)|/k = sin(pi k/d)/(k sin(pi/d)) = 1 - Theta(k^2/d^2).
• The dense-digit bound. Proved. For F = {0..q-1} minus E with m = |E|, k = q - m and (d, q) = 1: gamma_F(d) <= (d/2 + m)/k, since g_F is the full Dirichlet kernel less g_E, |D_q(a/d)| <= 1/(2||a/d||) <= d/2 and |g_E| <= m; hence for d/2 + m < k the error is at most k^L ((d/2 + m)/k)^L, uniform in r.
• A power saving at level q^(1-eps). Proved. Fix eps in (0,1) and take q >= 4^(1/eps), m <= q^(1-eps)/2, L >= 4/eps. Every 2 <= d <= q^(1-eps) coprime to q then has per-digit factor (d/2 + m)/k <= q^(-eps/2), so sum over those d of |N_F(L; d) - k^L/d| <= k^L q^(1 - eps L/2) <= k^L x^(-eps/4) at x = q^L: 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 q = 3, 4, 5, 10, 100 as k runs 2, 3, 4, 9, 99, against per-factor ceilings 0.9010, 0.7490, 0.5617, 0.2002, 0.0221; for F = {0,1} at q = 100 it stays 0.4992 with ceiling 0.9010, the same ceiling F = {0,1} has at q = 3.
• The split across the base's own divisors. Proved. For d = d1 d2 with d1 | q^m for some m <= L and (d2, q) = 1, the low m digits fix the value mod d1 and reach the rest only through the invertible multiplier q^m mod d2, so N_F(L; d) = sum over w in F^m with d1 | val(w) of N_F(L - m; d2, r_w), r_w = -val(w) (q^m)^(-1) mod d2. The density splits exactly, rho_F(d1 d2) = (N_F(m; d1)/k^m) (1/d2), and the base part is a digit-string count rather than 1/d1: a divisor sharing a factor with q is read off the digits, never off a density. Checked against direct enumeration at q = 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 q = 3, F = {0,2}, d = 2 every value is even, N_F(L; 2) = k^L, and the normalized error is exactly 1 at every L. Over d <= 200 the unrestricted worst error for that family reads 1.0483 at L = 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 L only, at a rate the digit count controls, and the census sees nothing better: F = {0,1} at q = 100 has worst normalized error 28.593, 14.590, 9.0340, 7.2034 at L = 16, 32, 64, 96 over d <= 500, per-digit factor 0.9929, with argmax d = 481 | q^3 - 1 at the first two depths and d = 303 | q^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 q^j mod d visits only ord_d(q) points, so no averaging happens across positions, and at every family's deepest level the sweep argmax has ord_d(q) <= 8, hence divides q^t - 1 with t <= 8 (d = 164 at q = 3, d = 143 at q = 10, d = 101, 303, 481 at q = 100); shallow depths can stray, d = 199 with ord = 99 at q = 10, L = 6.
• The signed pinned sum does not cancel. Verified. Weight each squarefree pinned modulus e = (q^t - 1)/g, g | q - 1, e >= 2, t <= L, by mu(e), with T_L(e) = N_F(L; e) - k^L/e, and set the signed sum Sigma_L = sum mu(e) T_L(e) against the absolute sum Abs_L = sum |T_L(e)| over the same moduli. Printed at every L = 3..40 by the generator, the ratio Sigma_L/Abs_L swings across [-1, 1] (-1.00 at L = 5, 6 in the first family) with no decay: -0.211, -0.123, +0.069, -0.498 at L = 10, 20, 30, 40 for F = {0,1} at q = 3 and +0.812, -0.495, -0.127, -0.192 for one excluded digit at q = 10, while Abs_L/k^L reads 2.1 * 10^-4 and 3.9 * 10^-12 at L = 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. Every N_F(L; 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 L at every t; mu(e) is read off a complete factorisation of every q^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_L|/Abs_L reading 0.498 and 0.192 at L = 40, and that factor does not grow with depth at any depth computed.
• Conjecture. For F = {0..q-1} minus one digit and d = q^t - 1, the worst orbit-mean damping is k^(-1/t) (1 + o(1)): the orbit carries t - 1 undamped points and one damped by ~ 1/k. At q = 100 and L = 12 the single-divisor probes read orbit mean 0.1059 at d = q^2 - 1 against k^(-1/2) and 0.2369 at d = q^3 - 1 against k^(-1/3), with the proper divisor d = 3367 | q^3 - 1 better at 0.0549 and d = 101 | q + 1 pinned but harmless at 0.0261. Two values of t on one base are a check, not a law.
• The second moment across residues. Proved. sum_{r mod d} (N_F(L; d, r) - k^L/d)^2 = (1/d) sum_{a not 0 mod d} prod_{j < L} |g_F(a q^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.

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_q = log_q k 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/mertens-numerology.

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..q-1} minus E, k = q - m and alpha_q = log k / log q, so A_F(x) >>_q x^(alpha_q) 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_q(t) = sum_{d = 0}^{q-1} e(d t) is the full Dirichlet kernel, |D_q(t)| = |sin(pi q t)/sin(pi t)|. Write D_L for the length-L digit strings over F and hat F_L(t) = sum_{n in D_L} e(n t) for the level-L transform, which factors as prod_{j < L} g_F(q^j t) because the digits are independent. The one-step constant of the shifted-grid recursion is B_q(F) = sup_t sum_{r mod q} |g_F((t+r)/q)|, bounded above by q PB_q(m) in step 3, with the proved constant PB_q(m) = sqrt(m) + Phi_q/q, where Phi_q = (4/pi) q + (2q/pi) H(ceil((q-2)/2)) + (1 - 2/pi)(q - 2) + 0.727 and H(n) = ln n + gamma + 1/(2n); the exponent cost is c_q = log PB_q(m)/log q, defined from that proved bound and never from the exact supremum.
• 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 q >= 3690, let F omit exactly one digit, and let eps > 0. Then |M_F(x)| <<_{q,eps} x^(3/4 + c_q + eps) for all x >= 2; and 3/4 + c_q < alpha_q at every such q, so with delta_q = (alpha_q - 3/4 - c_q)/alpha_q > 0 the same bound reads |M_F(x)| <<_{q,eps} A_F(x)^(1 - delta_q + eps), a power saving against the set's own mass. The implied constant depends on q and on eps and on nothing else; no uniformity in q is claimed anywhere. delta_q is fixed before eps is chosen, so the statement delivers every fixed delta' < delta_q and never the endpoint A_F(x)^(1 - delta_q).
• Step 1, orthogonality. Proved. For 0 <= n < q^L, 1_{D_L}(n) = q^(-L) sum_{0 <= a < q^L} hat F_L(a/q^L) e(-n a/q^L) by completeness of the additive characters mod q^L, and hat F_L factors over digit positions: the identity of the divisor section with the modulus q^L in place of d, read as an expansion rather than as a count.
• Step 2, the l^1 recursion. Proved. Put c_L = sum_{0 <= a < q^L} |hat F_L(a/q^L)| and split a = a' + s q^(L-1). The transform peels at the position j = 0, hat F_L(t) = g_F(t) hat F_{L-1}(q t), so s moves that factor alone and hat F_{L-1} is 1-periodic; the inner sum over s mod q is a shifted grid of q points, and c_L = sum_{a'} |hat F_{L-1}(a'/q^(L-1))| sum_{s mod q} |g_F((a'/q^(L-1) + s)/q)| <= B_q(F) c_{L-1}. Hence c_L <= B_q(F)^L and the normalized l^1 mass is q^(-L) c_L <= (B_q(F)/q)^L: one constant per digit, no interaction between positions.
• Step 3, the kernel bound. Proved. |g_F| <= |D_q| + |g_E| splits B_q(F) into a kernel part and an excluded part. The q points (t + r)/q are spaced 1/q; writing d_r for the distance of each to Z, |D_q((t+r)/q)| = |sin(pi q d_r)|/sin(pi d_r), the two points nearest the singularity contribute at most (4/pi) q + 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 q - 2 points pair off at distances >= j/q and contribute at most (2q/pi) H(ceil((q-2)/2)) + (1 - 2/pi)(q - 2). So sup_t sum_{r mod q} |D_q((t+r)/q)| <= Phi_q. The excluded part is exact rather than estimated: the excluded digits are distinct mod q, so Parseval on the shifted grid gives sum_{r mod q} |g_E((t+r)/q)|^2 = q m for every t, and Cauchy-Schwarz turns that into sum_{r mod q} |g_E((t+r)/q)| <= q sqrt(m). Hence B_q(F) <= q PB_q(m) and the normalized l^1 mass of step 2 is at most q^(L c_q).
• 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/q^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_L less {0} intersected with [1, x] at L = ceil(log_q(x+1)), and steps 1 to 4 apply once: |M_F(x)| <= (B_q(F)/q)^L max_theta |sum_{n <= x} mu(n) e(n theta)| <<_{q,eps} x^(3/4 + c_q + eps). If 0 is not in F, then S_F below q^L is the disjoint union of the exact-length blocks l <= L, and summing the per-block bounds is a geometric sum of ratio q^(3/4 + c_q + 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 L is small. Converting to the A_F yardstick needs 3/4 + c_q < alpha_q, equivalently the constant-space certificate gap_q(m) = (q - m) q^(-3/4) - PB_q(m) > 0. At m = 1 that certificate is negative at every 3 <= q < 3690 and positive at q = 3690 (Verified, exhaustive in the generator), and steps up at every q >= 723 by the monotone floor below, so q >= 3690 is a half line and not a window.
• What the proof does not use. Proved. No zero-density input, no restriction of x to a power of q, 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 q.
• Corollary, m excluded digits. Proved under GRH. With |E| = m and the same constants, the five steps run unchanged whenever PB_q(m) < (q - m) q^(-3/4), and give |M_F(x)| <<_{q,eps} A_F(x)^(1 - delta_q + eps) with delta_q = (alpha_q - 3/4 - c_q)/alpha_q. Under the proved constants that condition holds for m <= 6 at q = 10^4, m <= 78 at q = 10^5 and m <= 451 at q = 10^6 (Verified, each maximum asserted maximal in the generator), against sqrt(q) = 100, 316, 1000, and it holds asymptotically for m <= q^(1/2)(1 - o(1)) since PB_q(m) is sqrt(m) plus a term of size (2/pi) ln q. The squarefree-digit-gcd hypothesis carried by the exponent conjecture below is automatic in this regime and is not dropped: m < floor(q/2) leaves two consecutive digits in F, so gcd(F) = 1 and the vanishing family of the transfer section cannot occur; at small k the hypothesis must be stated.
• The shape at large base. Proved under GRH. Phi_q is (2/pi) q ln q up to lower order, so c_q = (ln ln q + ln(2/pi) + o(1))/ln q -> 0 while alpha_q -> 1, hence delta_q -> 1/4 and |M_F(x)| <<_{q,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_q tracks (ln ln q + ln(2/pi))/ln q to within 0.01 at q = 10^12 (Verified, the generator).
• The constants. Verified. The generator prints alpha_q truncated down at six digits, c_q rounded up at five and delta_q 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_q never prints as 1.000000; the scientific rows carry a relative guard of 10^-10. Both forms of the test, 3/4 + c_q < alpha_q and gap_q(m) > 0, are computed and their agreement asserted at every row and across 3 <= q < 20000.

• The margin at the wall. Verified. The saving at q = 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_q = ln(1 + gap_q(m)/PB_q(m))/(alpha_q ln q), which never differences two numbers of size 1 to reach one of size 10^-6: delta_q <= -2.395807653 * 10^-6 and gap_q(1) <= -1.533059397 * 10^-4 at q = 3689, against delta_q >= 5.863425182 * 10^-6 and gap_q(1) >= 3.752213034 * 10^-4 at q = 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 q^(-3/4) growth of the mass term is nearest the 4/(pi q) 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 q >= q_0(a), the least base with PB_q(1) < (q-1) q^(-b(a)). Then for every eps > 0 and all x >= 2, |M_F(x)| <<_{q,eps} x^(b(a) + c_q + eps) and b(a) + c_q < alpha_q, so |M_F(x)| <<_{q,eps} A_F(x)^(1 - delta_q(a) + eps) with delta_q(a) = (alpha_q - b(a) - c_q)/alpha_q > 0; the corollary runs at m excluded digits whenever PB_q(m) < (q-m) q^(-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_q -> 1 and c_q -> 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. q_0(a) is the least q >= 3 with gap_q(a, 1) = (q-1) q^(-b(a)) - PB_q(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 q and not the least q. Every wall below 4 * 10^6 is reproduced by an exhaustive scan from q = 3 against the bisection.

• The floor is proved, not scanned. Proved. Per step PB_{q+1}(1) - PB_q(1) < 1.291/(q-2) for q >= 40: the harmonic term jumps by at most (4/pi)/(q-2), the (1 - 2/pi)(q-2)/q term adds under 0.017/(q-2) and the 0.727/q term falls, and a step that does not jump the harmonic term is net negative. The mass term (q-1) q^(-b) gains at least (1-b)(q+1)^(-b) per step, so gap_q(a, 1) steps up wherever (1-b)(q-2)(q+1)^(-b) >= 1.291, a quantity strictly increasing in q; Q(b) is the least q >= 40 where it holds, and the gap steps up at every q >= Q(b). Below it nothing closes: gap_q(a, 1) < 0 on 3 <= q < 3690 at every rung (exhaustive), and on [3690, Q(b)] the smooth majorant U(q) = q^(1-b) - PB_q^-(1) dominates the gap and has exactly one interior minimum, since U'(q) = (1-b) q^(-b) - (2/pi)/(q-2) - 2(1 - 2/pi)/q^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) < q_0(a) at every rung, so each printed wall is the least q 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: q^(-b) falls in b, so gap_q(b, 1) <= gap_q(3/4, 1) < 0 on 3 <= q < 3690, that range being cleared exhaustively at b = 3/4. The floor itself rises with b, since (1-b)(q-2)(q+1)^(-b) falls in b at fixed q, 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_q(b, 1) < 0 on [3, Q(b)] and steps up from Q(b) on at every b: q_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 q = 10^7 is the largest m with PB_q(m) < (q-m) q^(-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 cost-out against a hypothetical Type I defect. Verified. A conditional Type I argument over S_F would run against a level-x^(alpha_q/2) distribution bound for the digit strings whose error carries a defect x^(m/(2(q-m) ln q)), a bound this page does not state, 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_q is normalised to the mass, so as a power of x the saving is x^(alpha_q delta_q) with alpha_q >= 0.99993 on every row compared, while the defect multiplies k^L. On those rows the saving is below the defect at the wall (5.86342 * 10^-6 against 1.65022 * 10^-5 at q = 3690, a factor above 2.8) and above it from q = 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 q = 10^9 the saving 1.16951 * 10^-1 clears the defect 2.41275 * 10^-11 by over nine decades, and the tightest corollary row, q = 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 must be re-costed rather than inherited.
• 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 q} |g_F((t+r)/q)|^2 = q k exactly, so sum_{r mod q} |g_F((t+r)/q)| >= q k / max_r |g_F| >= q for every t: the recursion of step 2 never contracts, B_q(F) >= q, and c_q >= 0 at every base and every digit set, so a negative c_q is an arithmetic error and not a discovery. The consequence is a hard limit on this decomposition, not on the problem: it needs alpha_q > 3/4, that is k > q^(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_q). 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 q^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 q^L} |hat F_L(a/q^L)|^2 = q^L k^L and sum_{a mod q^L} |sum_{n <= q^L} mu(n) e(n a/q^L)|^2 of size (6/pi^2) q^(2L), gives exponent (1 + alpha_q)/2 > alpha_q. 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_q >= 0 still forces alpha_q > 1/2, that is k > q^(1/2): the exponent conjecture below, which is about fixed k, is beyond every version of this method and not merely beyond its conditional form. And within the dense regime the endpoint stays out: delta_q is fixed before eps, so what is proved is A_F(x)^(1 - delta') for every fixed delta' < delta_q and never A_F(x)^(1 - delta_q), a distinction that is the whole claim at q = 3690, where delta_q >= 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_q, at every x >= 2, for one excluded digit at every q >= 3690 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 k <= q^(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 constant, table row, margin and rung above from the formulas alone, independently of lab/mertens-numerology, in under three seconds.

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 q^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 threshold on one number the digit set owns. No theorem about M_F comes out of it. A criterion does, and the criterion fails at base 10 on one number, the exceptional-set threshold, and holds from base 21.

Notation as in the divisor section, with x = q^L, D_L the length-L digit strings over F, hat F_L(t) = sum_{n in D_L} e(n t) = prod_{j < L} g_F(q^j t), alpha = log_q k the mass exponent, S_mu(t) = sum_{n <= x} mu(n) e(n t), rad(q) the product of the primes dividing q 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_beta sum_{a < Y} |hat F_l(beta + a/Y)| << k^l Y^(alpha_1), 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. Call e base-smooth when rad(e) divides rad(q), and write (E1) for the hypothesis that every prime dividing the gcd of the digit differences of F divides q: the one-dimensional form of condition (E) of coprime, and the hypothesis Lemma A' there consumes.

• 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 q} |g_F((t+r)/q)| >= q k / max_r |g_F| >= q for every t, is one digit position of the same statement; iterating it over L 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 q^l} |hat F_l(a/q^l)|^2 = q^l k^l and the maximum k^l at a = 0, gives sum_{a mod q^l} |hat F_l(a/q^l)| >= q^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 k > sqrt(q), 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.
• No exceptional character sits at a base-smooth modulus. Proved. Every real primitive Dirichlet character of base-smooth modulus has conductor dividing 8 rad(q), and the conductors in play number exactly 2^omega(q_1) at odd q and 3 * 2^omega(q_1) at even q, q_1 the odd part of q, while the characters number 2^omega(q_1) at odd q and 4 * 2^omega(q_1) at even q. 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(q), hence f | 8 rad(q). 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(q), while at odd q 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 q 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 q >= 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 q, k and C and both effectively computable, with |x^(-1) sum_{a in M(C)} hat F_L(a/x) S_mu(-a/x)| <= k^L 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 q, and the perturbed Lemma A' of coprime gives |hat F_L(a/x)| <= k^L exp(-c' log x / log log x) against the trivial |S_mu| <= x. On them x = q^L 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_L| <= k^L 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) 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 k^L 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)| << k^m (Q^(2 alpha_1) + Q^2 q^(-m(1 - alpha_1))) at every m <= L, 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,q) = 1} max_{y <= x} |#{n in D_L : n <= y, d | n, (n,q) = 1} - (1/d) #{n in D_L : n <= y, (n,q) = 1}| <= k^L (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_L below y is a disjoint union of blocks {P q^m + t : t in D_m}, at most k 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, q) = 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 k^(-v) against a naive q^(-v) at d_1 = q^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(q)} mu(w) sum_{n' : w n' in S_F, (n', q) = 1} mu(n') is an identity, and whether it reduces the problem depends on the inner sets. Sometimes it does - at q = 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, k = 2 < sqrt(10). Sometimes it does not: at q = 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, q) = 1 therefore stays inside the criterion below.
• From strings to the design. Proved. The two statements above count D_L, the padded strings, while the criterion counts S_F. With 0 in F the two agree below q^L but for the element 0, which carries mu(0) = 0. With 0 outside F, S_F below q^L 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 k > 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. Write beta for the exceptional-set threshold, E = {a : |hat F_L(a/x)| >= k^L x^(-beta)}, and m_t for the l^t exponent of the transform. Both places the source spends the exceptional set 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 q with D E << Y, q_1 ~ Q_1 coprime to q and d ~ D base-smooth, sum_{q_2 ~ Q_2, (q_2,q) = 1} sum_{a < d q_1 q_2, (a, d q_1 q_2) = 1} sum_{|eta| <= E/Y, (eta + a/(d q_1 q_2)) Y in Z} F_Y(a/(d q_1 q_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 = q^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 ≍ 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 = q^L 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.
• Nothing in the bilinear half is base-10 mathematics. Refuted. No step of it needs q = 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,q) = 1} mu(n) = O_B(A_F(x) (log x)^(-B)) for every B. This is a program, not a theorem, and its parts are 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. The lattice branch, which was the one named gap, is closed by the two Proved bullets above. Three things are owed in its place and none of them is a new idea: the line branch at general base, whose two lemmas are set-free and coefficient-free but whose own conditions m_t < (2 - t) beta and N >= x^max((5/4) beta, (5 beta - 1/2)/3) are not gathered into a statement anywhere here; and the write-out at general base of the two bookkeeping steps between the window and the pair geometry, the Parseval count of large frequencies for the Heath-Brown pieces and the dyadic reduction of the bilinear sum to the pair sum, both stated at source for arbitrary 1-bounded sequences. The hypothesis (n, q) = 1 is exactly what keeps every Type I modulus coprime to the base, since a Heath-Brown factorisation of an integer coprime to q has every factor coprime to q; dropping it needs the level of distribution at base-divisible moduli, whose main term the bullet above prices and does not cancel.
• Base 10 fails on one number, the moment. Proved. The threshold is beta = 23/80 = 0.2875 against 1/4, a miss of 3/80, which asks m_t < (2 - t)/4 at some t in [1, 2) and at the source's own t = 235/154 asks m_t < 73/616 = 0.118506 against the published 59/433 = 0.136258, a required drop of 4735/36344 = 0.130282 of the value, more than the numerical slack in the eigenvalue bound that produces it. The greedy condition alpha_1 + (5/2) beta <= 1 asks only beta <= 20/77 = 0.259740 there, so the threshold 1/4 is what binds and the miss is 3/80 exactly. 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 the condition is 2 alpha_1 < alpha, which base 10 clears at 0.701299 against 0.954242, and the companion condition (2 - alpha) 2 beta < 1 - alpha_1 clears at 0.601311 against 0.649350. Every one of the five lattice conditions holds at base 10 as published, as it must. The threshold read here is the sharp one, beta = inf_t m_t/(2 - t) <= 1/4; the headline alpha_1 < 1/4 is its t = 1 proxy and base 10 misses that proxy too. So the obstruction there is the exceptional-set threshold alone, and closing it would close the criterion at base 10.
• The least base that clears, and the family floor. Verified. The least base carrying a one-missing-digit set with alpha_1 < 1/4 is q = 21 missing the digit 0, and the least base whose whole one-missing-digit family clears is q = 34; both are certified, with their brackets and the bases they beat, 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 k > sqrt(q), by the l^1 floor above, and only under (n, q) = 1, by the split above: three limits stated rather than assumed.

The exponent, tagged honestly

• Conjecture. For every digit set F with 2 <= k <= q - 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 q = 5 is total cancellation for the trivial reason 4 | n, and its reduced column {0,1} carries the open question unchanged.
• 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 <= q^(23/80) excluded digits and s <= q - q^(57/80) when they are consecutive, and Maynard 2022 reaches s < q^(1/5 - eps) and q - s >= q^(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 k) 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 q^(1 - eps)/2 excluded digits, on the modulus range d <= q^(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/mobius-designs prints every row, identity check, slope, distribution and band above: CARGO_BUILD_JOBS=4 cargo run --release --manifest-path research/lab/Cargo.toml -p mobius-designs.
• lab/rho-decoupling prints every divisor-section number: CARGO_BUILD_JOBS=4 cargo run --release --manifest-path research/lab/Cargo.toml -p rho-decoupling.
• lab/mertens-numerology prints every constant, table row, margin, rung and cost-out number of the GRH section: CARGO_BUILD_JOBS=4 cargo run --release --manifest-path research/lab/Cargo.toml -p mertens-numerology; its 29 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 q = 3, and the constants of the general-b floor bound.
• The pair route has no generator of its own and needs none: every number in it is either exact rational arithmetic carried out in the sentence that prints it, an exponent quoted from the source named there, a certified base threshold whose generator is named on coprime, or a finite list that the sentence's own definition unrolls and a reader checks by hand.
• The Mertens control on the farey page is rendered by lab/mertens-meter; the checkpoint controls here are the same function read at powers of the base.