---
title: The Mobius meter across digit designs
lead: The Mobius meter across digit designs: exact transfer between scaled columns, the 47-column cancellation census, a power saving on the dense columns under GRH, the pair route with its major-arc lemma and its `l^1` threshold `1/4`, and square-root cancellation as the open exponent.
figure: research-mobius
slug: mobius
---

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](/wiki/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](../lab/rs/mobius-designs/), the divisor section's census numbers by [lab/rs/rho-decoupling](../lab/rs/rho-decoupling/), the GRH section's by [lab/rs/mertens-numerology](../lab/rs/mertens-numerology/) and the pair route's by [lab/py/mobius-region](../lab/py/mobius-region/).

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, **Refuted** means killed here with the witness that kills it.

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](/wiki/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](zeta.md).

## 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](https://doi.org/10.1016/0304-3975(92)00065-Y)) - 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](https://doi.org/10.1215/00127094-2017-0024) (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](https://arxiv.org/abs/1103.4991) 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](https://doi.org/10.1007/s00591-009-0059-5), with position-varying digit rules in [Nathanson 2021](https://arxiv.org/abs/2010.06295)); 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](https://doi.org/10.1006/jnth.1999.2487), carried out for missing digits in [Burnol 2026](https://arxiv.org/abs/2602.19727) and unified in [Allouche, Shallit and Stipulanti 2025](https://arxiv.org/abs/2401.13524)); 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](https://inria.hal.science/inria-00074307), and the oscillation is visible in the series' own numerical moments ([Burnol 2026 oscillations](https://arxiv.org/abs/2604.24754)); the Mobius function itself is automatic in no base, so the Mobius-weighted series inherits none of that continuation ([Coons 2010](https://doi.org/10.5802/jtnb.718)); 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](/wiki/riemann-zeta-function/) 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](https://www.zobodat.at/pdf/SBAWW_106_2a_0761-0830.pdf), and `M(x) = O(x^(1/2 + eps))` for every `eps > 0` is equivalent to the Riemann hypothesis ([Titchmarsh 1986](https://sites.math.rutgers.edu/~zeilberg/EM18/TitchmarshZeta.pdf), Theorem 14.25 (C)), while `limsup |M(x)|/sqrt(x) >= 1.06` unconditionally by [Odlyzko and te Riele 1985](https://doi.org/10.1515/crll.1985.357.138), 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](/wiki/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](/wiki/parity/) is the [Thue-Morse](/wiki/thue-morse-sequence/) 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](https://oeis.org/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:

| `level` | `M_{0,1}` | max | `M_{0,2}` | max | `M_{1,2}` | max |
|---|---|---|---|---|---|---|
| 4 | -2 | 3 | 2 | 3 | -8 | 8 |
| 6 | 2 | 5 | 0 | 3 | -8 | 11 |
| 8 | 2 | 8 | 3 | 7 | -31 | 33 |
| 10 | 5 | 13 | 0 | 11 | -14 | 38 |
| 12 | 56 | 61 | -37 | 40 | -35 | 88 |
| 14 | 11 | 105 | -10 | 67 | -205 | 230 |
| 16 | 149 | 173 | -124 | 152 | 4 | 281 |
| 18 | -30 | 312 | 67 | 249 | -1461 | 1582 |
| 20 | 496 | 539 | -382 | 485 | -3175 | 3255 |
| 21 | 533 | 866 | -194 | 617 | -2005 | 3855 |
| 22 | 1009 | 1089 | -1205 | 1324 | -690 | 3855 |
| 23 | 1824 | 2848 | -2242 | 2942 | -3214 | 3855 |
| 24 | -1886 | 3296 | -133 | 3843 | -3248 | 4113 |

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

| `base` | `F` | `level` | `A_F(base^level)` | `M_F(base^level)` | `Mmax` | `thetamax` | drift |
|---|---|---|---|---|---|---|---|
| 3 | 01 | 24 | 16777216 | -1886 | 3296 | 0.4869 | 0.0452 |
| 3 | 02 | 24 | 16777215 | -133 | 3843 | 0.4962 | 0.0596 |
| 3 | 12 | 24 | 33554430 | -3248 | 4113 | 0.4802 | 0.0754 |
| 4 | 01 | 22 | 4194304 | 34 | 1553 | 0.4819 | 0.0391 |
| 4 | 02 | 22 | 4194303 | -34 | 1553 | 0.4819 | 0.0391 |
| 4 | 03 | 22 | 4194303 | -541 | 1180 | 0.4638 | 0.0488 |
| 4 | 12 | 22 | 8388606 | -855 | 3965 | 0.5197 | 0.1056 |
| 4 | 13 | 22 | 8388606 | -712 | 2631 | 0.4940 | 0.0727 |
| 4 | 23 | 22 | 8388606 | -3255 | 3258 | 0.5074 | 0.0157 |
| 4 | 012 | 14 | 4782969 | -503 | 1057 | 0.4527 | 0.0475 |
| 4 | 013 | 14 | 4782969 | 2313 | 2899 | 0.5183 | 0.0487 |
| 4 | 023 | 14 | 4782968 | -753 | 1166 | 0.4591 | 0.0862 |
| 4 | 123 | 14 | 7174452 | -592 | 1644 | 0.4691 | 0.0729 |
| 5 | 01 | 21 | 2097152 | 153 | 849 | 0.4633 | 0.0485 |
| 5 | 02 | 21 | 2097151 | 250 | 889 | 0.4665 | 0.0268 |
| 5 | 03 | 21 | 2097151 | -116 | 700 | 0.4501 | 0.0311 |
| 5 | 04 | 21 | 2097151 | 0 | 0 | - | - |
| 5 | 12 | 21 | 4194302 | -128 | 1643 | 0.4856 | 0.0732 |
| 5 | 13 | 21 | 4194302 | -2875 | 3533 | 0.5358 | 0.0456 |
| 5 | 14 | 21 | 4194302 | -1511 | 2750 | 0.5193 | 0.0640 |
| 5 | 23 | 21 | 4194302 | 405 | 914 | 0.4471 | 0.0581 |
| 5 | 24 | 21 | 4194302 | 1065 | 2287 | 0.5072 | 0.0540 |
| 5 | 34 | 21 | 4194302 | -2137 | 2538 | 0.5141 | 0.0401 |
| 5 | 012 | 13 | 1594323 | -1016 | 1416 | 0.5080 | 0.0846 |
| 5 | 013 | 13 | 1594323 | -137 | 768 | 0.4652 | 0.0528 |
| 5 | 014 | 13 | 1594323 | 213 | 1005 | 0.4840 | 0.0821 |
| 5 | 023 | 13 | 1594322 | 759 | 858 | 0.4729 | 0.0455 |
| 5 | 024 | 13 | 1594322 | 686 | 959 | 0.4807 | 0.0240 |
| 5 | 034 | 13 | 1594322 | 501 | 1000 | 0.4837 | 0.0847 |
| 5 | 123 | 13 | 2391483 | 88 | 816 | 0.4565 | 0.0489 |
| 5 | 124 | 13 | 2391483 | 1036 | 1613 | 0.5029 | 0.0604 |
| 5 | 134 | 13 | 2391483 | -725 | 981 | 0.4690 | 0.0519 |
| 5 | 234 | 13 | 2391483 | -1926 | 2021 | 0.5182 | 0.1056 |
| 5 | 0123 | 11 | 4194304 | -474 | 1725 | 0.4887 | 0.0732 |
| 5 | 0124 | 11 | 4194304 | 426 | 1494 | 0.4793 | 0.0673 |
| 5 | 0134 | 11 | 4194304 | -644 | 1633 | 0.4852 | 0.0222 |
| 5 | 0234 | 11 | 4194303 | -362 | 2179 | 0.5041 | 0.0794 |
| 5 | 1234 | 11 | 5592404 | 145 | 1101 | 0.4508 | 0.0996 |

Base 10 with one digit excluded, the Kempner designs (`fill = 9`, the sets behind the convergent harmonic series of [Kempner 1914](https://doi.org/10.2307/2972074), revisited at `s = 1` in [Allouche, Hu and Morin 2024](https://arxiv.org/abs/2403.05678)), at `x = 10^8`:

| excluded | `A_F(10^8)` | `M_F(10^8)` | `Mmax` | `thetamax` |
|---|---|---|---|---|
| 0 | 48427560 | 6410 | 8177 | 0.5091 |
| 1 | 43046720 | 4108 | 6069 | 0.4956 |
| 2 | 43046721 | -183 | 3357 | 0.4619 |
| 3 | 43046721 | 455 | 3512 | 0.4644 |
| 4 | 43046721 | 56 | 4957 | 0.4841 |
| 5 | 43046721 | -7614 | 10601 | 0.5273 |
| 6 | 43046721 | -693 | 2564 | 0.4465 |
| 7 | 43046721 | -1411 | 6494 | 0.4994 |
| 8 | 43046721 | 2131 | 4495 | 0.4785 |
| 9 | 43046721 | 2181 | 5234 | 0.4871 |

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)`, `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](https://www.semanticscholar.org/paper/On-Arithmetic-Properties-of-Integers-with-Missing-Erdos-Mauduit/819d346a221f620ec9107933f0acc22cd345928d); 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](../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](/wiki/farey-sequence/) 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](../lab/rs/mertens-numerology/), and the sharpened one-excluded-digit constants of step 3, with the walls they move, by [lab/py/mrly-pairing](../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](../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](../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](https://doi.org/10.1112/jlms/s2-43.2.193), 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](https://doi.org/10.1016/j.ffa.2018.02.005) and in [Zhang 2024](https://arxiv.org/abs/2204.04613). 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](../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](../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](https://link.springer.com/article/10.1007/s00222-019-00865-6) 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.

| `base` | `alpha_base` | `c_base` (proved, up) | `delta_base` (down) | closes, step 3 |
|---|---|---|---|---|
| 1000 | 0.999855 | 0.28087 | -0.03102 | no |
| 2000 | 0.999934 | 0.26335 | -0.01342 | no |
| 3000 | 0.999958 | 0.25430 | -0.00434 | no |
| 3689 | 0.999966 | 0.24997 | -0.00001 | no |
| 3690 | 0.999967 | 0.24997 | 0.00000 | yes |
| 5000 | 0.999976 | 0.24393 | 0.00605 | yes |
| 10^4 | 0.999989 | 0.23141 | 0.01858 | yes |
| 10^5 | 0.999999 | 0.19906 | 0.05094 | yes |
| 10^6 | 0.999999 | 0.17589 | 0.07411 | yes |
| 10^9 | 0.999999 | 0.13305 | 0.11695 | yes |

- **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](https://doi.org/10.1112/jlms/s2-43.2.193) 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](https://arxiv.org/abs/2204.04613), 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](../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.

| `a` | `b(a)` | source | `base_0(a)`, step 3 | chord, any `e_0` | chord, `e_0 in {0, base-1}` | `Q(b)` |
|---|---|---|---|---|---|---|
| 1/2 | 3/4 | both | 3690 | 1499 | 1032 | 723 |
| 13/25 | 1417/1850 | Zhang | 8578 | 3525 | 2459 | 1486 |
| 11/20 | 913/1160 | Zhang | 33547 | 14078 | 10013 | 4754 |
| 4/7 | 4/5 | both | 92317 | - | - | 11221 |
| 3/5 | 4/5 | BH | 92317 | - | - | 11221 |
| 2/3 | 5/6 | BH | 3107080 | - | - | 216023 |
| 3/4 | 7/8 | BH | 6939524168 | - | - | 129458304 |
| 4/5 | 9/10 | BH | <= 3.09358e13 | - | - | 128606353005 |
| 9/10 | 19/20 | BH | <= 3.23663e34 | - | - | <= 1.73431e28 |
| 19/20 | 39/40 | BH | <= 9.24614e83 | - | - | <= 3.30712e68 |

- **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.

| `a` | `b(a)` | source | max `m` |
|---|---|---|---|
| 1/2 | 3/4 | both | 1971 |
| 13/25 | 1417/1850 | Zhang | 1002 |
| 11/20 | 913/1160 | Zhang | 365 |
| 4/7 | 4/5 | both | 176 |
| 3/5 | 4/5 | BH | 176 |
| 2/3 | 5/6 | BH | 8 |

- **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](../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, so `sum_{r mod base} |g_F((t+r)/base)| >= base fill / max_r |g_F| >= base` for every `t`: 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](../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](../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](https://doi.org/10.1016/j.ffa.2018.02.005), 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](https://doi.org/10.1016/j.ffa.2018.02.005) 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](https://arxiv.org/abs/2411.09076)), 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](https://link.springer.com/article/10.1007/s00222-019-00865-6), 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](https://github.com/carlomitchener/carlomitchener/tree/main/research/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.

## 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](https://link.springer.com/article/10.1007/s00222-019-00865-6), 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](coprime.md), and the hypothesis Lemma A' there consumes. Every exponent, region boundary, threshold certificate and census verdict below is printed by [lab/py/mobius-region](../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](coprime.md) 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](https://doi.org/10.1007/978-1-4757-5927-3), 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 `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](coprime.md).
- **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](coprime.md), `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 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 the minor-arc machinery built for the primes on a missing-digit set, and reaches a saving of classical zero-free-region shape with no hypothesis at all. Its whole price is paid in the base.

- **Conjecture.** Let `F` omit exactly one digit, let `base >= 92317`, the ladder rung `b = 4/5` of the table above, let `x = base^level` and let `level` be past a point depending on `base` alone. Then `|M_F(x)| <= C(base) A_F(x) exp(-c(base) sqrt(log x))` with `C(base)` and `c(base) > 0` both effective: an unconditional `o(A_F(x))` on the dense column, the saving a classical `exp(-c sqrt(log x))` rather than a power. The dissection below is written out and every region but one is derived here; the tag is Conjecture and not Proved because the one load-bearing analytic input is carried from memory rather than read at source, and because four bookkeeping items named below are unwritten.
- **The dissection, and which region binds.** Approximate each grid frequency `a/base^level` by a reduced `l/d` with `d <= x^(3/5)`, write `h = |a d - l base^level|` for the height and cut at `Z = exp(C sqrt(log x))`. Region A, `d >= x^(2/5)`, pays the full `l^1` mass of the transform against an unconditional minor-arc bound of strength `x^(4/5)`, so it asks `c_base + 4/5 < alpha_base`, which is exactly the ladder rung `b = 4/5` and its wall `92317`. Region B, `d < x^(2/5)` with `max(d, h) >= Z`, asks the design's `l^1` exponent to sit under `1/4`, the same bar the pair route's headline asks, and closes lower down the base scale than region A does, so region A is the binding one. Region C1, `d < Z` carrying a prime factor outside `base`, runs the perturbed Lemma A' of [coprime](coprime.md) against the trivial `|mu| <= 1` and reads nothing about `mu` at all. Region C2, `d` dividing a power of `base`, is the only place the arithmetic of `mu` enters, and no exceptional zero can live there by the conductor bullet of the pair route above.
- **What it has not got.** The minor-arc bound in `d`-form for `mu` at an arbitrary arc denominator, `|S_mu(theta)| << (x d^(-1/2) + x^(4/5) + (x d)^(1/2)) (log x)^(O(1))` on `|theta - l/d| <= 1/d^2` with `(l, d) = 1`, is quoted from memory and not read at source, and it is the one load-bearing input; the twin of that shape for the von Mangoldt function is read at source in [Maynard 2022](https://doi.org/10.1093/imrn/rnab002), so what is owed is the substitution of `mu` for `Lambda` through Vaughan's identity with `1/zeta` in place of `-zeta'/zeta`, one page and not a new idea. Unwritten besides: the contour bound and the truncated Perron constants, `x` not a power of `base`, the region C1 transfer from an exact fraction `l/d` to the grid point nearest it, and every `m > 1`.
- **The shape is in print for the primes, the object is not.** [Maynard 2022](https://doi.org/10.1093/imrn/rnab002) remarks that Siegel zeros play no role at these highly composite moduli, so the error terms there could be made effective of `exp(-c sqrt(log x))` shape; that puts the shape over a missing-digit set in print for the von Mangoldt function. What is not in print over such a set is the object, a Mobius or Mertens sum, exactly as the GRH section's literature bullet records. 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 against the divisor section.** The imported saving is `exp(-c(base) sqrt(log x))` with `c(base)` falling in the base like a fixed negative power of it, while the level-`x^(alpha_base/2)` bound of the divisor section saves a power of `x` outright, so the imported bound does not overtake the divisor section until `level` is past a threshold that grows like a fixed power of `base`: at every depth any computation reaches, the divisor section is the better number. The import is also silent on every `F` whose `l^1` exponent is at least `1/2`, at every modulus sharing a factor with `base`, and at any fixed `level`.
- **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 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](https://www.semanticscholar.org/paper/On-Arithmetic-Properties-of-Integers-with-Missing-Erdos-Mauduit/819d346a221f620ec9107933f0acc22cd345928d), the ellipsephic almost-primes rest on it ([Dartyge and Mauduit 2000](https://doi.org/10.1006/jnth.1999.2458)), and primes in one-excluded-digit sets took the full circle method at large base in [Maynard 2019](https://link.springer.com/article/10.1007/s00222-019-00865-6), 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](https://doi.org/10.1093/imrn/rnab002) 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](https://arxiv.org/abs/2108.09212) 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](https://arxiv.org/abs/2409.06894) settle [ternary Goldbach](/wiki/goldbach-conjecture/) 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](https://arxiv.org/abs/2411.09076)), 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](../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](../lab/rs/rho-decoupling/) prints every divisor-section number: `CARGO_BUILD_JOBS=4 cargo run --release -p rho-decoupling`.
- [lab/rs/mertens-numerology](../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](../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](../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](coprime.md).
- The Mertens control on the [farey](farey.md) 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](../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.