# The digit-restricted Mobius meter

- 2026-09-01 [Proved] Carry-free scaling ties the digit designs' Mobius meters together: for digit sets `F = a F'` inside `{0..base-1}`, `m -> a m` is a digit-length-preserving bijection `S_F' -> S_F` (each scaled digit stays below `base`, so no carry occurs), giving `M_F(base^level) = sum mu(a m)`; a square factor in `a` kills the meter identically (`F = {0,4}` at `base = 5`: zero at all 21 levels), and prime `a = p` gives `M_(pF')(base^level) = -sum_(p not | m) mu(m)`, the `{0,2}` column at `base = 3` reading as the `{0,1}` column twisted by the Thue-Morse sign of the binary index; asserted at every level on all eight scaled census families. Witness: mobius.md, lab/rs/mobius-designs.
- 2026-09-01 [Proved] The base-4 anti-symmetry `M_{0,2}(4^level) = -M_{0,1}(4^level)`: `4 | base` forces every element of `S_{0,1}` to `0` or `1 mod 4`, so even elements carry `mu = 0` and `M_{0,2}(x) = -M_{0,1}(x/2)` at every real `x`, running maxima included since `S_{0,1}` is empty strictly between `(4^level - 1)/3` and `4^level`; exact at all 22 levels, `-110/110` at `level = 15`, `34/-34` and shared `Mmax = 1553` at `level = 22`. Witness: mobius.md, lab/rs/mobius-designs.
- 2026-09-01 [Proved] No Euler product for a digit design: `S_F` is not multiplicatively closed, witness `4 = 11_3` and `13 = 111_3` in `S_{0,1}` at base 3 with `4 x 13 = 52 = 1221_3` outside, so `M_F` is not the coefficient sum of an inverse Dirichlet series; the series itself is built literature (abscissa Kohler and Spilker 2009, continuation and poles Burnol 2026) and carries no Mobius sum anywhere. Witness: mobius.md, REFS.md.
- 2026-09-01 [Verified] The digit-restricted Mobius census: exact `M_F(base^level)` and running maxima `max |M_F(x)|` for all 38 digit sets with `2 <= fill <= base - 1` at `base = 3, 4, 5` (depths 24, 22, 14, 21, 13, 11 by class), the ten base-10 one-digit-excluded columns to `10^8`, and full-set controls to `3^17`, `4^13`, `5^11`, `10^8`; factorization and sieve agree on the `base = 3` `{1,2}` family at every level to `level = 16`, the base-10 control reproduces A084237, and an independent second-language recompute matched 99 sampled rows exactly. Witness: lab/rs/mobius-designs, mobius.md, A084237.
- 2026-09-03 [Proved] A power saving for the Mobius meter on the dense digit columns, under GRH: assume `L(s, chi)` has no zero in `sigma > 1/2` for every Dirichlet character `chi`, let `F` omit exactly one digit `e_0`, and let `base >= 1499`, or `base >= 1032` when `e_0` is `0` or `base - 1`; then for every `eps > 0` and all `x >= 2`, `|M_F(x)| <<_{base,eps} x^(3/4 + c'_base(e_0) + eps)` with `c'_base(e_0) = log PB'_base(1, e_0)/log base`, `PB_base(1) = 1 + Phi_base/base` and `Phi_base = (4/pi) base + (2q/pi) H(ceil((base-2)/2)) + (1 - 2/pi)(base-2) + 0.727`, and `3/4 + c_base < alpha_base = log(base-1)/log base`, so `|M_F(x)| <<_{base,eps} A_F(x)^(1 - delta_base + eps)` with `delta_base = (alpha_base - 3/4 - c_base)/alpha_base > 0`, every fixed `delta' < delta_base` delivered and the endpoint never; orthogonality mod `base^level`, the shifted-grid `l^1` recursion `c_level <= B_base(F) c_{level-1}`, the kernel bound `B_base(F) <= base PB_base(1)` from `sin(pi v) <= 4v(1-v)` and `1/sin x <= 1/x + 1 - 2/pi` with Parseval exact on the excluded digit, and the assembly with its geometric sum are derived, and the uniform `max_theta |sum_{n <= x} mu(n) e(n theta)| <<_eps x^(3/4 + eps)` of Baker and Harman 1991 is quoted at source; the corollary at `m` excluded digits runs whenever `PB_base(m) < (base-m) base^(-3/4)`, which holds at `m <= 6, 78, 451` at `base = 10^4, 10^5, 10^6` and asymptotically for `m <= base^(1/2)(1-o(1))`, and `c_base -> 0` gives `delta_base -> 1/4`. The attempt to break it drives the chain below the wall, where the failure is quantified rather than hidden (`c_base = 0.28087` against `alpha_base = 0.999855` at `base = 1000`), checks the exponent test against the constant-space certificate `gap_base(m) = (base-m) base^(-3/4) - PB_base(m) > 0` at every `3 <= base < 20000`, the cancellation-reduced and direct forms of `delta_base` against each other to `10^-9` relative at every printed base, and `Phi_base` against the exact shifted-grid kernel sum on a `4001`-point grid at `base = 50, 101, 200`, where it is loose by under `20%`. Witness: mobius.md a power saving under GRH at large base, lab/rs/mertens-numerology.
- 2026-09-03 [Proved] The ladder above that theorem, and its floor: for `1/2 <= a < 1`, if `L(s, chi)` has no zero in `sigma > a` for every Dirichlet character then the same five steps give `|M_F(x)| <<_{base,eps} A_F(x)^(1 - delta_base(a) + eps)` with `delta_base(a) = (alpha_base - b(a) - c_base)/alpha_base > 0` at every `base >= base_0(a)`, `b(a)` the smaller of the Baker and Harman 1991 table and Zhang 2024 Theorem 1.1 (Zhang strictly smaller inside `(1/2, 4/7)` and equal at both ends, by the factorisations `-5(a - 1/2)(a - 2/5)/(4 - 2a)` and `-7(a - 4/7)(a - 4/5)/(4 - 2a)`, with `b(a) >= 3/4` throughout), so every common zero-free half plane buys the saving and GRH is only its first rung, the price of a weaker hypothesis being paid entirely in the base; the wall `base_0(a)` exists and is a true least base at every `a`, since `PB_{base+1}(1) - PB_base(1) < 1.291/(base-2)` for `base >= 40` while the mass term gains `(1-b)(base+1)^(-b)` per step, so the gap steps up at every `base >= Q(b)`, the least `base` with `(1-b)(base-2)(base+1)^(-b) >= 1.291`, and below that it is negative: exhaustively on `3 <= base < 3690`, and on `[3690, Q(b)]` by a majorant with one interior minimum whose endpoint values are both negative. The attempt to break it looks for a rung the floor misses and finds none: at every `b` in `[3/4, 1)`, printed rung or not, minimality of `Q(b)` gives `gap_{Q(b)}(b, 1) < -1.56` and a majorant below `-0.95` at both ends, with any `b < 1417/1850` forcing `Q(b) <= 1486` and an empty range, the constants reproduced on a `b`-grid across the whole interval. Witness: mobius.md a power saving under GRH at large base, lab/rs/mertens-numerology.
- 2026-09-03 [Verified] The rungs of that ladder: `(a, b(a), source, base_0(a), Q(b))` reads `(1/2, 3/4, both, 3690, 723)`, `(13/25, 1417/1850, Zhang, 8578, 1486)`, `(11/20, 913/1160, Zhang, 33547, 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)`, then `(4/5, 9/10, BH, <= 3.09358e13, 128606353005)`, `(9/10, 19/20, BH, <= 3.23663e34, <= 1.73431e28)` and `(19/20, 39/40, BH, <= 9.24614e83, <= 3.30712e68)`, a wall printing as an exact integer only below `2^53` with both neighbouring gaps above `1024` ulps and otherwise as an upper bound on the least `base`; the GRH rung reproduces the wall `3690` and the margin there is `delta_base <= -2.395807653 * 10^-6` at `base = 3689` against `delta_base >= 5.863425182 * 10^-6` at `base = 3690`, with `gap_base(1) <= -1.533059397 * 10^-4` and `>= 3.752213034 * 10^-4`; the `m`-budget at `base = 10^7` falls `1971, 1002, 365, 176, 176, 8` along the rungs below that base. The attempt to break them reproduces every wall under `4 * 10^6` by an exhaustive scan from `base = 3` against the bisection, requires `Q(b) < q_0(a)` at every rung, sweeps `3 <= base < 3690` for an early close at every rung and finds none, and pins each rendered row as a string. Witness: lab/rs/mertens-numerology, mobius.md a power saving under GRH at large base.
- 2026-09-03 [Proved] The `l^1` floor is a wall on the method, not on the problem: `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 shifted-grid recursion never contracts, `B_base(F) >= base` and `c_base >= 0` at every base and every digit set; hence the decomposition needs `alpha_base > 3/4`, that is `fill > base^(3/4)`, and every fixed-`fill` column, `F = {0,1}` at `base = 3` included, is beyond it with or without GRH, so it never meets the census or the exponent conjecture. The same floor kills the two neighbouring routes: Davenport's unconditional `x (log x)^(-A)` in the quoted step exceeds `A_F(x)` by the power `x^(1 - alpha_base)`, so no unconditional saving follows inside this decomposition without an input of zero-free-strip strength, and Cauchy-Schwarz with Parseval on both factors gives exponent `(1 + alpha_base)/2 > alpha_base`, worse than trivial. The attempt to break it hunts a negative `c_base` over `3 <= base < 5000` and a `PB_base(1)` below `1` and finds neither, Parseval forbidding both. Witness: mobius.md a power saving under GRH at large base, lab/rs/mertens-numerology.
- 2026-09-03 [Verified] The cost-out of that saving against a hypothetical Type I defect: with the saving `delta_base` set beside the defect exponent `m/(2(base-m) ln base)` carried by a level-`x^(alpha_base/2)` distribution bound for the digit strings, a bound no page here states, the saving is below the defect at the wall (`5.86342e-6` against `1.65022e-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` in which the difference rises at all `96310` steps, monotonicity beyond the scan unproved; at `base = 10^9` it is `1.16951e-1` against `2.41275e-11`, and the tightest corollary row `base = 10^6`, `m = 451` reads `3.14081e-5` against `1.63296e-5`. The attempt to break it checks the crossover for a premature crossing at `base = 3690, 3691` and for a single down-step in the scan and finds none, and holds the yardsticks apart: `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`. Witness: lab/rs/mertens-numerology, mobius.md a power saving under GRH at large base.
- 2026-09-03 [Conjecture] That the defect `x^(m/(2(base-m) ln base))` of a level-`x^(alpha_base/2)` distribution bound for the digit strings, a bound no page here states, is absorbed by the GRH saving at all: the cost-out sets two exponents from two unrelated statements on two yardsticks side by side and no derivation joins them, so it is neither a necessary condition nor a proof that a Type I estimate for `M_F` follows, the string-to-interval bookkeeping and the bilinear half of any such argument being untouched; the comparison is decided at the wall and nowhere else, lost there by a factor under `3` and won two steps later, so any sharper constant that moves `q_0` must be re-costed rather than inherited. Witness: lab/rs/mertens-numerology, mobius.md a power saving under GRH at large base.
- 2026-09-06 [Verified] The coefficient sequence a real Vaughan decomposition hands the bilinear sum is not the sequence that beats the method's diagonal floor: at the eight swept boxes with both sides above `x^(2/5)`, the boxes the identity produces, the Type II coefficient `sum_{d | l, d <= x^(2/5)} mu(d)` takes values in `{-1, 0, 1}` at seven of the eight and its full quadratic form sits in `[0.6929, 1.2045]` of its own diagonal, where a sign vector engineered against the column reads `0.2043` on such a box; over all eighty coefficient cells of the census, sixteen boxes, four families, two depths, two cuts and five real sequences, the form over the diagonal stays in `[0.3138, 52.6676]` with none below `0.1` and every departure from the swept band upward. Witness: lab/rs/rho-decoupling section menergy signed vaughan.
- 2026-09-06 [Proved] The large-values refinement of the moment route is the `l^2` route itself: splitting the grid at `|hat F_level(a/base^level)| >= fill^level x^(-eta)`, bounding the large set by its `l^2` mass under the fourth moment and Parseval and the rest by the threshold, all against Parseval on the bilinear side, gives the exponent `max(min(alpha + 1/2 - eta, (1 + alpha)/2), min((1 + alpha)/2, alpha + (nu_4 + 2 eta)/2)) = (1 + alpha)/2` identically at every `eta >= 0` and every digit set with `alpha < 1`, and the large-sieve constant of any grid subset for `base^level` consecutive frequencies is `base^level` exactly, so no spacing enters. Witness: lab/rs/rho-decoupling section riesz large values chain, 66 rows with `c = -(1 - alpha)/2`.
- 2026-09-06 [Verified] The large frequencies are adjacent or isolated grid points, `407` in `331` runs at `{0,1}` base 3 `level = 12` `eta = eta_4` against the fourth-moment count `4096`, least gap `1/base^level`, large-sieve constant on `D_level` in `1.06009e5..2.13280e5` nearly at its `l^2` floor `1.05611e5`; at the eight dense census cells the Type II sum at `a_m = b_l = 1` on the box `M = N = floor(x^(1/2)/2)` is the representation count, `0.26` to `0.41` of `fill^level`, and the balanced sum at `a_m = 1_(base | m)`, `b_l = 1` a fixed share of `fill^level`, so no bound uniform over bounded coefficients holds there; the sparse cell `{0,1}` base 100 `level = 3` is void at the box. Witness: lab/rs/rho-decoupling sections riesz large values and riesz large values witness.
- 2026-09-06 [Proved] The second-largest grid value of the digit transform is `max_(a != 0) |hat F_level(a/base^level)| = fill^(level-1) max_(b != 0 mod base) |g_F(b/base)|`, equal to `fill^(level-1)` at one excluded digit and at `{0,1}` base 3, so the large set is the zero frequency alone exactly below `eta_1(level) = log(1/gamma_1)/(level log base)`, a threshold that vanishes with depth. Witness: lab/rs/rho-decoupling section riesz large values cells, eight cells at `1/fill`.
- 2026-09-06 [Conjecture] Whether a Vaughan decomposition's coefficient sequence, a convolution and not a free sign vector, can be steered near the engineered sign vector that beats the Cauchy-Schwarz diagonal floor by a factor thirty-eight at a top box; and whether the arc regime `M, N >= x^(2/5)` carries a dyadic box with `R = x^(alpha - o(1))`, the middle-divisor question on which the balanced route's refutation for `alpha < 2/5` is conditional. Witness: lab/rs/rho-decoupling, sections menergy signed engineered and menergy type II.
- 2026-09-06 [Refuted] The adversarial pass on the census: an independent linear-sieve recompute in a second language rebuilt 99 rows - nine families, four controls, one excluded-digit column, meters, counts and running maxima - and first DISAGREED on eleven `{0,1}`-family rows, traced to the recompute itself double-counting the boundary `base^l` its length filter had already caught; fixed, it agrees on all 99. Two generator runs differ in zero of 784 shared rows, and a first-draft page table assembled by hand was wrong in multiple cells before every page table was switched to script extraction from the generator's printed rows. Witness: lab/rs/mobius-designs.
- 2026-09-06 [Refuted] That the Mobius signs cancel the digit column's off-diagonal multiplicative correlation better than an unstructured sign vector on the same support: over sixteen boxes `|Sigma_mu|` is `0.0913` to `0.7178` of the random-sign root mean square against `0.0359` to `1.5048` for the support-matched controls, the split against those controls is `3, 9, 4` at chi-square `0.375` against the uniform-rank null, and the fifteen-of-sixteen advantage over Liouville is the support of the Mobius function; a sign vector engineered against a known column drives the same Cauchy-Schwarz bound to `0.0265` of its diagonal floor, so the census refutes the arithmetic of the coefficients and not the method. Witness: lab/rs/rho-decoupling, sections menergy signed, menergy signed summary and menergy signed engineered.
- 2026-09-07 [Proved] The major-arc input for `mu` on a digit design is effective. Let `F` be a digit set with `fill >= 2` every prime of whose digit-difference gcd divides `base`, condition (E) in one dimension, and let `x = base^level`. Every real primitive Dirichlet character whose modulus has all its primes dividing `base` has conductor dividing `8 rad(base)`, so the possible exceptional zeros run over a set of size bounded in `base` and Siegel's theorem is never invoked; with that, `x^(-1) Sum_{a in M} hat F_level(a/x) S_mu(-a/x)` is at most `fill^level exp(-c sqrt(log x))` with `c` effectively computable, over the arcs `|a/x - b/d| <= (log x)^C/x` with `d <= (log x)^C`, and there is no main term at any arc. Witness: mobius.md The pair route.
- 2026-09-07 [Proved] Under condition (E) in one dimension and the large sieve `Sum_{d <= Q} Sum_{gcd(b,d)=1} |hat F_m(b/d)| << fill^m (Q^(2 alpha_1) + Q^2 base^(-m(1 - alpha_1)))` at every scale `m <= level`, with `alpha_1 < 1/2` the sup-over-shift `l^1` exponent, a digit design's level of distribution survives restriction to an initial segment: `Sum_{d <= Q, gcd(d,base)=1} max_{y <= x} |#{n in D_level : n <= y, d | n, gcd(n,base)=1} - (1/d) #{n in D_level : n <= y, gcd(n,base)=1}| << fill^level (log x)^(-B)` at `Q <= x^(1 - alpha_1)(log x)^(-C)`, the same level as the full-range statement and one power of `log x` less saving, because the transform's error is uniform in the target residue and the segment splits into at most `fill` blocks per scale. Witness: mobius.md The pair route.
- 2026-09-07 [Proved] The hybrid bound that carries the digit-restricted bilinear estimate holds at every base with the digit set's own dimension as its exponent. Let `F` be a digit set with `fill = abs(F) >= 2`, `alpha = log(fill)/log(base)`, sup-over-shift `l^1` exponent `alpha_1`, and assume the shifted and perturbed large sieve it supplies by Farey spacing, `sup over shifts of Sum_{a <= d} sup_{abs(eta) <= delta} F_Y(a/d + shift + eta) << (1 + delta d)(d^(alpha_1) + d Y^(-(1 - alpha_1)))` at every scale. For `D, E, Y, Q_1` powers of `base` with `D E << Y`, `Q_2 >= 1`, `q_1 ~ Q_1` coprime to `base` and `d ~ D` all of whose primes divide `base`, the sum of `F_Y(a/(d q_1 q_2) + eta)` over `q_2 ~ Q_2` coprime to `base`, over `a < d q_1 q_2` coprime to `d q_1 q_2`, and over `abs(eta) <= E/Y` with `(eta + a/(d q_1 q_2)) Y` an integer is `<< (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)`. Both exponents come from Parseval on a window `base^r`, where `int F^2 = base^(-r alpha)` exactly when `0` is in `F` and `≍` otherwise, so the base-10 values `1/21` and `10/21` are `1 - alpha` rounded up and `alpha/2` rounded down. Since `alpha + alpha_1 >= 1` at every design, this never loses to the plain `l^1` bound in the modulus aspect. Witness: mobius.md The pair route.
- 2026-09-07 [Proved] The lattice half of the digit-restricted bilinear estimate transfers to every base, and the five inequalities it asks are free below `1/3`. With `x = base^level`, the window `N K >= x^(1 - 2 beta)`, `delta >= N/x` and `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, whenever `2 alpha_1 < alpha`, `(2 - alpha) 2 beta < 1 - alpha_1`, `2 beta (alpha_1 (3 - u) + u - 1) < u alpha/2` for some `u` in `(0, min(1, 2 alpha_1/alpha)]`, `5 beta < 1 + alpha/2` and `2 beta < (1 - alpha_1)(1 - alpha_1 + alpha/2)`. The source writes a numerical check for the second and the fourth only; the first, third and fifth are read off steps it performs silently. All five are monotone in the three exponents, so the corner `alpha = 1 - alpha_1`, `beta = 1/4` decides them, and every one holds under `alpha_1 < 1/3`, `beta <= 1/4` and the `l^1` floor `alpha + alpha_1 >= 1`, with `1/3` sharp since three become equalities there. The floor and the threshold on `beta` alone do not suffice, as `alpha_1 = 0.40`, `alpha = 0.60`, `beta = 1/4` shows. Base 10 clears all five as published. Witness: mobius.md The pair route.
- 2026-09-07 [Proved] The pair route's eight inequalities are two. Write `alpha = log(fill)/log(base)` for a digit set's dimension, `alpha_1` for the sup-over-shift `l^1` exponent of its transform and `beta` for the exceptional-set threshold. Of the eight inequalities the route 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 those fall in `alpha_1` and two are constant in it, so each takes its minimum over the region at the wall `alpha_1 = alpha/2`. At that wall the lattice cap `(2 - alpha) 2 beta < 1 - alpha_1` and the geometric-mean condition read exactly `1/4`, the last lattice cap reads `(2 - alpha)/4` and the fourth `(1 + alpha/2)/5`, all identities in `alpha`, so none of them ever cuts below the window threshold `1/4` inside the wall. For `alpha` in `(1/2, 1)` the region is therefore exactly `alpha_1 < alpha/2` and `beta <= min(1/4, (2/5)(1 - alpha_1))`, the greedy cap cutting from `alpha_1 = 3/8` and from nowhere else. A sweep of `66000` cells, `264000` cap tests, finds no exception, and the two wall equalities hold at each of `330` rational `alpha`. Witness: lab/py/mobius-region verb boundary.
- 2026-09-07 [Proved] The exceptional-set threshold obeys the same Parseval floor as the `l^1` exponent, and the pair route reaches only `fill >= base^(3/4)`. The normalised transform is at most `1` pointwise, so the moment exponent `m_t` is non-increasing in `t`; and `m_2 = 1 - alpha` exactly, since two length-`level` digit strings congruent modulo `base^level` are equal. Hence `m_t >= 1 - alpha` for every `t <= 2`, and since `2 - t <= 1` for `t >= 1` the threshold `beta = inf over t in [1,2) of m_t/(2 - t)` is at least `1 - alpha` at every base and every digit set, the same floor `alpha + alpha_1 >= 1` puts on the `l^1` exponent. The route's window condition `beta <= 1/4` alone then forces `alpha >= 3/4`, that is `fill >= base^(3/4)`, with equality only when the `l^1` floor is also an equality. That window condition is a convenience rather than a necessity, and dropping it does not widen the route: on the single-window branch, which carries the greedy step under `alpha_1 <= 1 - (13/4) beta`, the same two floors give `alpha >= 13/17 = 0.764705..706`, so that branch reaches only `fill >= base^(13/17)` and the gate rises. The weaker reading `fill > sqrt(base)`, which follows from `alpha_1 < 1/2` alone, stays true and is simply not sharp, so no earlier row is contradicted. Witness: lab/py/mobius-region verb check.
- 2026-09-07 [Verified] The census of the pair criterion over 49 digit designs, and a second machine at base 21. 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 and the infimum matrix loses a row. A pass is decided at the pessimistic corner and a failure at the optimistic one, every cap being monotone in each parameter. A second implementation of the window method returns `alpha_1 in [0.2499715, 0.2499822]` for base 21 missing `0` at five window digits and sub-scan `8`, against the five-digit `[0.2499715, 0.2499821]` already certified, agreeing on the lower bound to all seven printed digits and differing by one unit in the last on the upper; both run the same method at the same depth, so the agreement witnesses transcription and the upper-bound gap is the only independent information. The same machine reproduces base 10 missing `5` at `alpha_1 in [0.3505101, 0.3506471]`, `m_(235/154) <= 0.1362891` and `beta <= 0.2875140` against the three published values `27/77`, `59/433` and `23/80`. Witness: lab/py/mobius-region verbs criterion and params.
- 2026-09-07 [Proved] The line half of the digit-restricted bilinear estimate and its two bookkeeping steps, at every base. With `x = base^level`, a threshold `beta` admissible and at most `2/5`, which with the Parseval floor `beta >= 1 - alpha` forces `alpha >= 3/5` on 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 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`, 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 write-outs complete it. For coefficients bounded by the `j`-fold divisor function, orthogonality on the grid with `tau_j^2 <= tau_(j^2)` gives `#{a mod x : the exponential sum is at least x/C} <<_j C^2 (log x)^(j^2 - 1)`, so a Heath-Brown decomposition costs a log power where a 1-bounded sequence costs none. And Cauchy-Schwarz in the long variable turns the bilinear sum into `x/N` times the pair sum of the transform against the sum over `l_1, l_2 <= N` of `min(x/N, the inverse distance from (a_1 l_1 - a_2 l_2)/x to the nearest integer)`, 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. Witness: mobius.md The pair route.
- 2026-09-07 [Conjecture] A digit set satisfying condition (E) in one dimension whose sup-over-shift `l^1` exponent obeys `alpha_1 < 1/4` has `Sum_{level in S_F, level <= x, gcd(level,base) = 1} mu(level) = O(A_F(x) (log x)^(-B))` for every `B`. The program is named: two Proved steps for `mu`, the rest set-only or coefficient-free, and the lattice branch of the source's Section 14 in general parameters owed. Base 10 fails on two independent numbers, `27/77` against `1/3` and `23/80` against `1/4`. Witness: mobius.md The pair route. Superseded by the criterion row that names five lattice conditions and the threshold `beta <= 1/4`, under the same subsection in OPEN.
- 2026-09-07 [Conjecture] , whose owed list and whose base-10 diagnosis are both superseded. A digit set satisfying condition (E) in one dimension whose sup-over-shift `l^1` exponent obeys `alpha_1 < 1/4` has `Sum_{level in S_F, level <= x, gcd(level,base) = 1} mu(level) = O(A_F(x) (log x)^(-B))` for every `B`. The program is named: the major-arc lemma and the level of distribution on an initial segment are Proved for `mu`, and the lattice branch is Proved in general parameters. Three things are owed and none is a new idea: the line branch at general base, whose two lemmas are set-free and coefficient-free but whose own conditions `m_t < (2 - t) beta` and `N >= x^max((5/4) beta, (5 beta - 1/2)/3)` are gathered into no statement yet; and the write-out at general base of two bookkeeping steps, the Parseval count of large frequencies for the Heath-Brown pieces and the dyadic reduction of the bilinear sum to the pair sum, both stated at source for arbitrary 1-bounded sequences. Base 10 now fails on one number only, the sharp threshold `beta = inf_t m_t/(2 - t)`, at `23/80` against `1/4`. Witness: mobius.md The pair route.
- 2026-09-07 [Refuted] The pair criterion cannot be met at base 10 at any excluded digit. An upper bound on a moment exponent bounds the threshold above and can never show the criterion fails, so the published miss of `3/80` prices a gap and refutes nothing. Two monotonicities close it: on a cell `[t_0, t_1]` every `t` has `m_t/(2 - t) >= m_(t_1)/(2 - t_0)`, and above a cut the Parseval value `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 and the route is dead at base 10 at every digit rather than merely unreached. The refuting certificates do not order the columns, their brackets `[0.2510933, 0.2625620]` at the digit `9` and `[0.2515026, 0.2875159]` at the digit `4` overlapping; 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. The miss is at most `0.0125620` at the cheapest column and at least `0.0139557` at the digit `4`; the factor `2.99` between the two printed upper bounds is a ratio of upper bounds and not of misses. Witness: lab/py/mobius-region verbs threshold and threshold 0.2626.
- 2026-09-14 [Proved] The one-step constant of the `l^1` recursion is exact and cheap at one excluded digit. Let `F = {0..base-1}` less `{e_0}`, `phi_r = (t+r)/base`, `A_r = (-1)^r sin(pi t)/sin(pi phi_r)` and `c = e_0 - (base-1)/2`. Then `abs(g_F(phi_r)) = abs(A_r - e(c phi_r)) = sqrt(A_r^2 + 1 - 2 A_r cos(2 pi c phi_r))` for every `t` not in `Z`, which is where `A_r` is defined, since `D_base(phi_r) = e((base-1)phi_r/2)(-1)^r sin(pi t)/sin(pi phi_r)` and the unimodular factor divides out, so `B_base(F) = sup_t sum_(r mod base) abs(g_F((t+r)/base))` is a sup of `base` real square roots and costs `O(base)` per `t`; the reduction reproduces the direct sum over `F` to `12` digits and reproduces the grid sups `4.0000000000` at base 3 `{0,1}` and `19.8885438199` at base 10 missing `9`, the floored readings of `split`'s `4.000000000` and `19.888543820`. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Proved] Two exact symmetries of that constant: `B_base(F)` is unchanged by `e_0 -> base-1-e_0`, because the digit reflection multiplies `g_F` by a unimodular factor, and the shifted-grid sum is symmetric in `t` about `1/2`, because `r -> base-1-r` carries `t` to `1-t` with `sign(A_r) cos(2 pi c phi_r)` fixed; so the scan for the sup runs on `t` in `[0, 1/2]` and on `e_0 <= (base-1)/2`. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Proved] The phase identity behind the one-step constant: for every `base`, every `e_0` and every `t` in `(0,1)`, `sum_(r mod base) (1 + sign(A_r) cos(2 pi c phi_r)) = base + cos(2 pi c (t - 1/2)/base)/cos(pi c/base)`, by summing the geometric series `sum_r (-1)^r e(c r/base) = e(-c/(2q))/cos(pi c/base)`, which is where `e(c) = (-1)^(base-1)` collapses the numerator to `2`; the right side is at least `base + 1` at every `t`, since `abs(2 pi c (t-1/2)/base) <= abs(pi c/base) < pi/2`, and it reaches `base + 1/sin(pi/(2q))` at `t = 1/2` and `e_0 in {0, base-1}`. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Proved] The triangle split `abs(g_F) <= abs(D_base) + abs(g_E)` of the shifted-grid step can be sharpened by a fixed share of `base` at one excluded digit, with no new input. For `base >= 17` and `m = 1`, `B_base(F) <= (4/pi) base + Psi_base + base/2 - sec(pi (e_0 - (base-1)/2)/base)/2`, where `Psi_base = (2 base/pi) H(ceil((base-2)/2)) + (1 - 2/pi) base` with `H(n) = ln n + gamma + 1/(2n)`, the desk convention of mobius.md, is the step 3 kernel constant less its two-point part, against the step 3 bound `base PB_base(1) = base + Phi_base = (4/pi) base + Psi_base + base + 0.00023954`, the constant being `0.727 - 2(1 - 2/pi)` exactly. The proof is `abs(a - e(psi))^2 = (a+1)^2 - 2a(1 + cos psi)` with `sqrt(1-X) <= 1 - X/2`, then `abs(A_r) >= sin(pi t) = s` and `s/(1+s) >= s/2`, then the phase identity, then the `t`-dependent kernel bound `sum_r abs(D_base(phi_r)) <= (4/pi) base + s Psi_base` that step 3's own two-point and pairing estimates give, and finally `h(tau) = cos(pi tau)(Psi_base - base/2) - cos(pi tau) cos(2 beta tau)/(2 cos beta)` with `beta = pi (e_0 - (base-1)/2)/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 x <= x` and `sec beta <= base`. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Proved] That sharpening lowers the base of the conditional power saving with no new idea and no change to any other step: the least `base` with `(base-1) base^(-b(a)) > B_base(F)/base` falls from `3690` to `2446` at every excluded digit and to `1812` at `e_0 in {0, base-1}` at the GRH rung `b = 3/4`, from `8578` to `5700` and `4242` at `b = 1417/1850`, and from `33547` to `22416` and `16816` at `b = 913/1160`, each an exhaustive scan from `base = 17` in the generator, whose `held` column prints `3997555 = 4000000 - 2446 + 1` and the five like counts, so every wall is an up-set over its whole scan and not a first crossing, while the three step 3 baselines are quoted from mobius.md and not rescanned. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Proved] The `l^1` floor is higher than `base` at one excluded digit: letting `t -> 0` in the shifted-grid sum gives `abs(g_F(0)) = base-1` and `abs(g_F(r/base)) = 1` at every `r != 0`, so `B_base(F) >= 2(base-1)` and `c_base >= log(2 - 2/base)/log(base) > 0` for every `base >= 3`, and that endpoint is the seat at `base = 3` by hand, the three terms collapsing to `4 cos u` on `[0, pi/6)` and `4 cos(u - pi/3)` on `[pi/6, pi/3)`, both at most `4`. Hence no exact constant can push the method below the base where `(base-1) base^(-3/4) > 2 - 2/base`, which is `base^(1/4) > 2` and so `base >= 17`, and the sup-times-`l^1` method needs `fill > (2 - 2/base) base^(3/4)` and not `fill > base^(3/4)`; the floor is too weak to give `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`. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Verified] The exact one-step constant falls with the excluded digit, while the step 3 bound is one number for all of them: at `base = 3690`, `B_base(F)/base >= 5.750052` at `e_0 = 0` and `>= 6.392410` at `e_0 = 1844`, against the exact kernel sup `K_base/base >= 6.191324`, the proved kernel bound `Phi_base/base <= 6.791445` and `1 + Phi_base/base = 7.791445`; the split defect `(K_base + base) - B_base(F)` reads `1.441272 base` at `e_0 = 0`, flat to `1.3e-5` across `base = 100, 1000, 2234, 3690` and agreeing to six digits with `1 + (2/pi) ln 2 = 1.4412712`, which nothing here proves is its limit, and `0.798914 base` at the middle digit, where those same four `base` read `0.808644`, `0.799643`, `0.799091` and `0.798914`, a spread of `9.8e-3`, so that branch is stable only to `1e-2`; `Phi_base - K_base` reads at most `0.600121 base` there, so the two losses are the same order and the excluded digit's position is worth `0.64 base`. Each number is a grid scan on `t` in `[0, 1/2]` at cut `1/4000` with the sup seated at `t = 1/2`. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Verified] The ceiling of the exact-constant lever, and what it is not: the measured `B_base(F)` itself would put the GRH base at `927` at every excluded digit, the last failure being `base = 926` at `e_0 = 462` on a downward scan of `17..2000` over every digit, and at `304` at `e_0 in {0, base-1}`, last failure `303` on `17..8000`, against the proved `2446` and `1812`, so a further `0.9 base` of slack is left in the sharpened bound, of which `0.6 base` is the gap between `Phi_base` and the exact kernel sup. The middle digit is not the maximiser at odd `base` and reading it alone reports the crossing `232` bases early: at `base = 695` the worst digit is `e_0 = 463` with `B_base(F)/base = 5.327344` against `(base-1) base^(-3/4) = 5.127089`, a failure, while the middle digit passes by `2.1e-5`. Both bases are readings of `Sigma(1/2)`, which is the sup on the cut at every `(base, e_0)` checked in the range but is not proved to be the sup, and neither crossing is proved monotone in `base`, so they bound nothing and never enter a statement. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Verified] Nothing measured contradicts the sharpened bound: over every excluded digit at `base = 17..60` the worst ratio of `B_base(F)` to the bound is `0.807189` at `base = 60`, `e_0 = 29`, at the seats `base = 100, 1000, 2234, 3690` it is `0.876716` at `base = 3690`, `e_0 = 1844`, and `4000` draws at seed `1009` over `base in {17, 23, 60, 101, 333, 1000, 3690}` with `e_0` and `t` uniform give worst ratio `0.872146` at `base = 3690`, `e_0 = 1701`, `t = 0.499866` and no violation. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Proved] The pair route's gate is `fill >= base^(3/4)`, and dropping its window condition narrows the route rather than widening it. `F_x <= 1` pointwise makes `m_t` non-increasing and `m_2 = 1 - alpha` is exact by Parseval on the grid, so `beta >= 1 - alpha` at every base and every digit set, and the window condition `beta <= 1/4` forces `alpha >= 3/4`. The branch that drops it asks `alpha_1 <= 1 - (13/4) beta`, and with `beta <= m_1 <= alpha_1`, the `t = 1` term of the infimum against the grid sum being one shift of the supremum, that reads `beta <= 4/17 = 0.235294` and `alpha >= 13/17 = 0.764705`, so the gate rises to `fill >= base^(13/17)`. The step `beta <= alpha_1` is load-bearing (lab/py/mobius-region, verb boundary prints the branch): without it `alpha = 0.9`, `alpha_1 = 0.155`, `beta = 0.26` clears both floors, `2 alpha_1 < alpha`, the greedy condition and all five lattice conditions with `beta > 1/4`. The weaker reading `fill > sqrt(base)` stays true and unsharp. Witness: mobius.md The pair route, lab/py/mobius-region verb check, which prints the two floors at four designs and `beta <= alpha_1` at `t = 1` at the base-21 recompute.
- 2026-09-14 [Proved] The region the pair route asks for is exactly two inequalities. Of the eight, 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 at the wall `alpha_1 = alpha/2`. There `(2 - alpha) 2 beta < 1 - alpha_1` reads `1/4`, `2 beta < (1 - alpha_1)(1 - alpha_1 + alpha/2)` reads `(2 - alpha)/4` and `5 beta < 1 + alpha/2` reads `(1 + alpha/2)/5`, three identities in `alpha` with the last two strictly above `1/4` on `(1/2, 1)`; the `u`-condition has wall coefficient `(3 alpha/2 - 1) + u (1 - alpha/2)`, so it reads exactly `1/4` for `alpha >= 2/3` and is vacuous for `alpha` in `(1/2, 2/3)`, where that coefficient is negative at small admissible `u`. Vacuous or `1/4`, none of the four cuts, so for `alpha` in `(1/2, 1)` the region is `alpha_1 < alpha/2` with `beta <= min(1/4, (2/5)(1 - alpha_1))`, the greedy cap cutting below `1/4` exactly from `alpha_1 = 3/8`, a threshold free of `alpha`. Witness: mobius.md The pair route, lab/py/mobius-region verbs region and boundary.
- 2026-09-14 [Verified] The pair route and the Mobius census cannot break each other, and no proved conditional exponent sits below a measured one. 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, `0.4465` to `0.5358`. The chain that does print an exponent is the GRH one, `theta(F) <= 1 - (1/4 - alpha_1)/alpha`, and it reads above `1` at every base-10 one-missing-digit 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 neither chain is falsified by the census at any design of it, and a design whose measured exponent rose above its own proved ceiling would refute one of them. Witness: lab/py/mobius-region verb criterion, mobius.md The pair route.
- 2026-09-14 [Proved] The chord `1/sin x <= 1/x + (2/pi)(1 - 2/pi) x` holds on `(0, pi/2]`, where `csc x - 1/x` has an all-positive Taylor series and so lies under its own chord; pairing `r` with `base-1-r` puts every shifted-grid argument inside `(0, pi/2]` at `t in (0, 1/2]`, and `K(t) = K(1-t)` carries the rest, so `K(t) = sin(pi t) sum_(r mod base) 1/sin(pi (t+r)/base) <= (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` and `(base/pi)(2 H(P-1) - 1 + 2/P) + (1 - 2/pi)(base/2 + 1/(2 base))` at odd `base`, `P = floor(base/2)`, `H(n) = ln n + gamma + 1/(2n)` the desk convention of mobius.md; the paired argument sum is `base^2/4` at even `base` and `P(P+1) + t` at odd `base`, the source of the odd `1/(2 base)`, and at even `base` the constant is `Psi_base - base/2 + 2/pi`. Witness: lab/py/mrly-pairing, verb `onestep`, the chord kernel block.
- 2026-09-14 [Verified] The chord kernel bound cuts the gap between the proved kernel constant and the exact kernel sup `K_base` by a factor `5.98`: at `base = 3690` the up-rounded gap columns give `Psi_base - (K_base - (4/pi) base) <= 0.600121 base` and `Psi'_base - (K_base - (4/pi) base) <= 0.100293 base`, the same quantity the lemma-slack column floors to `0.100292 base`; the chord column reads `0.100290, 0.100293, 0.100293` at `base = 100, 1000, 2234`, so the slack is flat to `1e-5`, and it is attained at the seat `t = 1/2`. Witness: lab/py/mrly-pairing, verb `onestep`, the chord kernel block.
- 2026-09-14 [Proved] At one excluded digit and `base >= 36` the one-step constant of the shifted-grid `l^1` recursion obeys `B_base(F) <= (4/pi) base + Psi'_base + base/2 - sec(pi (e_0 - (base-1)/2)/base)/2`, the phase identity and the `sqrt(1-X) <= 1 - X/2` chain of the earlier sharpening run against the chord kernel bound; the monotone step needs `Psi'_base >= (1 + pi) base/2`, which first holds at `base = 36` with `H(n) = ln n + gamma + 1/(2n)`, the desk convention of mobius.md, and at `base = 37` with the harmonic number itself, the over-estimate safe elsewhere since it only raises an upper bound; the hypothesis is sufficient and not necessary, the max of `h(tau)` sitting at `tau = 0` at every `e_0` from `base = 8` up on the exhaustive scan `4..79`. Witness: lab/py/mrly-pairing, verb `onestep`, the chord wall block; lab/rs/mertens-numerology, `the_chord_floor_carries_its_harmonic_convention`.
- 2026-09-14 [Verified] The chord bound moves the GRH wall `base_0(1/2)` from `3690` at step 3 and `2446` at the phase sharpening to `1499` at every excluded digit, and from `1812` to `1032` at `e_0 in {0, base-1}`, each an up-set over the exhaustive scan `36..4000000`; the rungs `b = 1417/1850` and `b = 913/1160` move from `5700` and `22416` to `3525` and `14078`, and from `4242` and `16816` to `2459` and `10013`, up-sets over `36..8000000` and `36..40000000`. Witness: lab/py/mrly-pairing, verb `onestep`, the chord wall block.
- 2026-09-14 [Verified] Every chord wall costs out against the level-`x^(alpha/2)` defect within five steps of itself: the saving `delta_base` first exceeds `1/(2(base-1) ln base)` at `base = 1502` for the wall `1499` and at `base = 1036` for the wall `1032`, against `2450` for `2446`, `1815` for `1812` and `3692` for `3690`, every crossing an up-set to `100000`, each row scanned from its own floor, `base >= 3` at step 3, `17` at the phase sharpening and `36` at the chord, and no wall inherited. Witness: lab/rs/mertens-numerology, sharpened cost-out block, `sharpened_cost_out_is_pinned`.
- 2026-09-14 [Verified] Nothing measured contradicts the chord bound: the worst ratio of the exact `B_base(F)` to it is `0.902124` over every `e_0` at `base = 36..60`, `0.941239` at the larger seats and `0.936333` over `4000` seeded draws of `(base, e_0, t)`, against `0.807189`, `0.876716` and `0.872146` for the phase sharpening alone. Witness: lab/py/mrly-pairing, verb `onestep`, the chord falsification block.
- 2026-09-14 [Verified] The weight the chord bound leaves behind is not free: at the seat `t = 1/2` the kept weight `w_r = abs(A_r)/(abs(A_r) + 1) >= s/(1 + s) >= s/2` is worth `base/2`, while dropping the singular terms by `(1 + sign(A_r) cos) <= 2` costs `2 sum_(r mod base) 1/(abs(A_r) + 1) = (2 - 4/pi) base`, measured `0.726761 base` at `base = 1000, 3690, 20000` against its exact limit `2(1 - 2/pi) = 0.726761`, a net `-0.226761 base`, so the route loses more than it wins. Witness: lab/py/mrly-pairing, verb `onestep`, the weight route block.
- 2026-09-14 [Proved] Above each rung's step 3 wall the sharpened `m = 1` certificate needs no scan: `base PB_base(1) - base PB_base(1, e_0) = base/2 + sec(pi (e_0 - (base-1)/2)/base)/2 + 0.727 - 2(1 - 2/pi)` with `0.727 - 2(1 - 2/pi) = +2.3954 * 10^(-4)` and `abs(pi (e_0 - (base-1)/2)/base) < pi/2`, so `PB_base(1) - PB_base(1, e_0) > 1/2` at every `base >= 17` and every excluded digit, and the step 3 certificate `(base-1) base^(-b(a)) - PB_base(1) > 0`, proved positive from `q_0(a) = 3690`, `8578`, `33547` on by the monotone floor at `Q(b) = 723`, `1486`, `4754`, carries the sharpened certificate over `[q_0(a), infinity)` unscanned. Witness: mobius.md step 3 sharpened and step 5, on lab/py/mrly-pairing verb `onestep`.
- 2026-09-14 [Verified] Below each rung's step 3 wall the sharpened `m = 1` certificate `(base-1) base^(-b(a)) - PB_base(1, e_0) > 0` is exhaustive and not a first crossing, so with the proved half above `q_0(a)` each sharpened wall is a half line and not a window: the scans `17..4 * 10^6`, `17..8 * 10^6` and `17..4 * 10^7` each run past their own `q_0(a)` and hold at every `base` from `2446` and `1812` at `b = 3/4`, `5700` and `4242` at `b = 1417/1850`, `22416` and `16816` at `b = 913/1160`, the `held` counts printing `3997555`, `3998189`, `7994301`, `7995759`, `39977585`, `39983185`, each equal to `hi - w + 1`. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Proved] The per-denominator Mobius input states, at the exact frequencies, and buys no exponent. Group the grid `a/base^level` of the orthogonality step by `base`-power level `j`, so `a = a' base^(level-j)` with `base` not dividing `a'` and `hat F_level(a'/base^j) = fill^(level-j) hat F_j(a'/base^j)` because `g_F` at an integer is `fill`; with `c_j` the primitive level-`j` sum `sum abs(hat F_j(a'/base^j))`, `c_0 = 1` and `C_level = sum_j fill^(level-j) c_j`, Baker-Harman's PROPOSITION p.194 eq. 6 under its hypothesis (4), that `L(s, chi)` is zero-free in `sigma > a` for EVERY Dirichlet character, taken at `(r,Q)` the frequency itself, where its second factor is `1`, gives `abs(M_F(base^level)) <<_(base,eps) x^eps base^(-level) sum_(j <= level) fill^(level-j) c_j min(x^(b(a)), x^a base^(j/2))` at `x = base^level`, the reduced denominator of `a'/base^j` dividing `base^j`. That sum lies in `[m/base, 1]` times the uniform `base^(-level) C_level x^(b(a))`: Parseval on the shifted grid is exact, `sum_(s mod base) abs(g_F((t+s)/base))^2 = qk`, and `abs(g_F) <= fill`, so `sum_s abs(g_F((t+s)/base)) >= base`, hence `C_j >= base C_(j-1)` at every `j` and the top level carries `c_level = C_level - fill C_(level-1) >= (m/base) C_level`, while `b(a) <= a + 1/2` at every rung keeps the uniform constant `x^(b(a))` on that level. So the exponent stays `b(a) + c_base`, the saving is at most `-log(c_level/C_level)/(level log base) <= log(base/m)/(level log base)` and vanishes with `level`, and the whole lever is worth one bounded factor `base/m`. The bracket is proved for this corollary and for it alone, the level charge being an upper bound and not the pointwise truth: at composite `base` a top-level `a' = 5^level u` has true denominator `2^level = x^0.301`. Witness: lab/py/mrly-pairing verb perden, the exponent block, `level(den - unif)` reading `-0.657068` at `base = 3` one digit off and `-0.292383` at `base = 10` missing `9`, constant in `level`, the largest term sitting at `argmax j = level` at every printed row by measurement and not by proof.
- 2026-09-14 [Verified] The same tool at full strength buys no exponent either. Letting any reduced `r/Q` serve any frequency, `Q(1 + x abs(a/base^level - r/Q)) = Q + abs(aQ - r base^level)`, so the per-frequency constant is `min(x^(b(a)), x^a nu(a)^(1/2))` with `nu(a) = min_Q (Q + norm(aQ)_(base^level))`, and the honest ratio to the uniform input rises with `level` while its exponent gain decays faster than `1/level`. Witness: lab/py/mrly-pairing verb perden, the full minor-arc block, `nu` checked against a full search over every reduced `r/Q` at `base^level = 81` with `0` mismatches, ratio `0.817368`, `0.835986`, `0.851049`, `0.861910` at `base = 3` `level = 6, 8, 10, 12` and `0.684136`, `0.705013`, `0.737968` at `base = 10` `level = 4, 5, 6`, with `level` times the gain falling from `-0.183564` to `-0.135265` and from `-0.164858` to `-0.131963`; that the ratio is bounded below in `level` is measured over these seven rows and not proved.
- 2026-09-14 [Proved] The `l^1` mass of a digit transform decays geometrically downward from the top denominator. From `C_j >= base C_(j-1)` the top level's share is `c_j/C_j >= m/base` at every `j`, and the levels below `J` carry `sum_(j <= J) fill^(level-j) c_j = fill^(level-J) C_J <= (fill/base)^(level-J) C_level`, so the mass on the levels of reduced denominator at most `x^(1/2)`, the tie at `j = level/2` included, is at most `(1 - m/base)^(ceil(level/2))` of the whole and falls geometrically in `level`. Witness: lab/py/mrly-pairing verb perden, the level-profile block, which asserts the decomposition identity, the floor and the cap at every printed row.
- 2026-09-14 [Verified] The measured level profile at one excluded digit. Top-level shares `0.485846`, `0.510055`, `0.573574` and `0.676152` at `base = 3` level `12`, `base = 10` level `6`, `base = 101` level `3` and `base = 1499` level `2`, against the proved floor `m/base = 0.333333`, `0.100000`, `0.009901` and `0.000667`, the last two short rows where `C_j/C_(j-1)` is still moving, `244.658399` then `234.507307` at `base = 101`, and so not converged constants; the levels of reduced denominator at most `x^(1/2)` carry `0.018474`, `0.117603`, `0.174294` and `0.323848` against the proved cap `0.087791`, `0.729000`, `0.980296` and `0.999333`; the one-step ratios `C_j/C_(j-1)` read `3.889889`, `18.369403`, `234.507307` and `4625.632148`, each above the proved floor `base`. Witness: lab/py/mrly-pairing verb perden, level-profile block, reproducing verb split's `18.369402635` and its top share `0.510055`, and matched by brute force over the digit strings at `C_j = 234.856179`, `913.566768` and `331.978584` with top shares `0.485833`, `0.485848` and `0.512017` for `base = 3` `j = 4, 5` and `base = 10` `j = 2`.
- 2026-09-14 [Proved] Baker-Harman's PROPOSITION is a `d`-form minor-arc bound, and that is where it pays, inside eq. 6's printed range on `Q`, which the desk has not read. On a Dirichlet arc `abs(theta - l/d) <= 1/d^2` with `(l,d) = 1` it reads `S_mu(theta) << x^(a+eps) d^(1/2) (1 + x/d^2)^(1/2) = x^(a+eps) (d + x/d)^(1/2) <= x^(a+eps) (d^(1/2) + x^(1/2) d^(-1/2))`, which at `a = 1/2` is `x^eps ((x d)^(1/2) + x d^(-1/2))` under the generalized Riemann hypothesis. Unlike the rows above, where the `min` caps the PROPOSITION by the uniform THEOREM at the level that decides, this one applies eq. 6 at an arbitrary arc denominator `d` and so carries that unread range. Witness: lab/py/mrly-pairing verb perden, the arc block, derived from the statement quoted at source in REFS.md.
- 2026-09-14 [Proved] The chord kernel constant beats the step 3 one at every base with no scan: `Psi'_base < Psi_base` at every `base >= 5`, by `Psi'_base = Psi_base - base/2 + 2/pi` at even `base` and by `Psi_base - Psi'_base = (2 base/pi)(H(P) - H(P-1)) + (base/pi)(1 - 2/P) + (1 - 2/pi)(base/2 - 1/(2q))` at odd `base`, `P = floor(base/2)`, positive since `H(P) - H(P-1) = ln(P/(P-1)) - 1/(2P(P-1))` and `ln(P/(P-1)) > 1/(P - 1/2) > 1/(2P(P-1))` at `P >= 2`. Witness: `cargo test -p mertens-numerology the_chord_constant_saves_half_a_base`, `psi_chord(base) < psi(base)` at every `base` in `36..3999`, `34 passed; 0 failed`.
- 2026-09-14 [Proved] Hence each chord wall is a half line and not a window: `base PB_base(1) - base PB'_base(1, e_0) = (Psi_base - Psi'_base) + base/2 + sec(pi (e_0 - (base-1)/2)/base)/2 + (0.727 - 2(1 - 2/pi)) > 0` at every `base >= 36`, so the step 3 certificate, positive from `base_0(a)` on by the monotone floor, carries the chord certificate over `[base_0(a), infinity)` unscanned. Witness: mobius.md step 5, on lab/py/mrly-pairing verb `onestep`, chord walls `1499` and `1032` holding `3998502` and `3998969` of the scan `36..4 * 10^6`.
- 2026-09-14 [Conjecture] The criterion, with the owed list it now carries. A digit set satisfying (E1) whose `l^1` exponent obeys `alpha_1 < 1/4` has `sum_(level in S_F, level <= x, (level,base) = 1) mu(level) = O_B(A_F(x) (log x)^(-B))` for every `B`. The two steps that read the sequence rather than the set are proved here, the major arcs for `mu` at base-smooth moduli and the level of distribution on an initial segment; the lattice half, the line half and both bookkeeping steps are now written out at general base, so the three write-outs the program listed as owed are written and this supersedes the owed list carried by the earlier criterion rows in OPEN. One arithmetic item is left, the level of distribution at base-divisible moduli, needed only to drop `(level, base) = 1`: the level carries by the divisor split but the main term does not, being a digit-string count times `1/d_2` and not `1/d`. Of the source reading, the line branch's close is read once and owes a second reading. Witness: mobius.md The pair route.
- 2026-09-14 [Refuted] The seat of `B_base(F)` is not `t = 1/2` at every `base >= 10`: at `base = 11`, `e_0 = 0` the sup on the cut is `22.5094271855` at `t = 0.47275` against `Sigma(1/2) = 22.4703926508`, and at `base = 13`, `e_0 = 0` it is `27.9876970872` at `t = 0.47875` against `27.9570308138`, so the seat is interior at both. `t = 1/2` is the seat at every family printed from `base = 100` up, and is where the constant is read and not where it is proved to sit; the floor `2(base-1)` and the sup scan are untouched, while any wall read off `Sigma(1/2)` alone is a reading and not a bound. Witness: lab/py/mrly-pairing, verb `onestep`.
- 2026-09-14 [Refuted] Base 10 is refuted at every excluded digit, on both branches, rather than merely unreached. A published or certified moment exponent is an upper bound on `beta` and can never show the criterion fails, so the published `beta <= 23/80` prices a gap of `3/80` and refutes nothing. From below, `F_x <= 1` makes `m_t` non-increasing and `m_2 = 1 - alpha` is exact, so 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`. One such certificate kills both branches: the merged window asks `beta <= 1/4`, and the single-window branch asks `alpha_1 <= 1 - (13/4) beta`, which with `beta <= alpha_1` reads `beta <= 4/17 < 1/4`. At the target `0.2626` the same chain gives `beta >= 0.2632014` at each of the eight non-extreme digits, up to `0.2645208` at the digit `7`, with the digits `0` and `9` undecided, so the two extreme digits are strictly the cheapest columns of the base. Witness: mobius.md The pair route, lab/py/mobius-region verbs threshold and threshold 0.2626.
- 2026-09-14 [Refuted] The per-denominator weighting lowers neither the exponent cost `c_base` nor the base `q_0` of the conditional power saving on the dense digit columns. At the top level `j = level` the charge is `x^(a + 1/2)` and the uniform constant `x^(b(a))`, and `a + 1/2 - b(a)` is `1/4`, `47/185`, `61/232`, `19/70`, `3/10` and `1/3` as exact rationals at `a = 1/2, 13/25, 11/20, 4/7, 3/5, 2/3`, so the top level is strictly worse at EVERY rung of both exponent tables and the crossing `2(b(a) - a)` never exceeds `1/2`; the certificate `(base-1) base^(-b(a)) - PB_base(1, e_0) > 0` takes no per-denominator quantity at all, so the GRH walls of the chord certificate stand untouched. Beating `x^(3/4)` on the top level means bounding the Mobius exponential sum at denominator `base^level = x`, which is the conjectural `x^(1/2 + eps)` at essentially every frequency and lies past the method's ceiling. Witness: lab/py/mrly-pairing verb perden, the rung block, every rung printing `no` in its `beats uniform at j = level` column.
- 2026-09-19 [Proved] Above the supremum the `L^p` norms of the digit transform buy nothing: `max(fill^p, base fill^(p/2)) <= Lambda(p) <= base fill^(p-1)` forces `Lambda(p)^(1/p)/fill` to `1`, reading `1.224744, 1.029883, 1.004288, 1.000653, 1.000107` at `p = 2, 4, 6, 8, 10` for `{0,1}` at `base 3`. Witness: lab/rs/rho-decoupling, riesz higher moments.
- 2026-09-19 [Proved] The unbalanced kernel carries no Type II estimate uniform over bounded coefficients at any digit set containing `0`: `a_m = b_l = 1` makes the sum the box representation count and some admissible box carries `R >= x^(alpha - o(1))`, while on the balanced sum the bound-to-trivial ratio rises through `1`, `1.0134` at `level 12` and `1.0730` at `level 14` at `{0,1}` base 3, the margin of `alpha` over the achieved exponent falling `0.035675` to `0.027009`, and inside the arc regime the digit column is worse for the method than a random column of the same density at every cell. Witness: lab/rs/rho-decoupling, menergy type II.
- 2026-09-19 [Verified] Two box witnesses floor every coefficient-free route at `x^alpha` over the nine dense cells: `a_m = b_l = 1` reads `0.19` to `0.41` of `fill^level` and `a_m = 1_(base divides m)`, `b_l = 1` reads `0.0024` to `0.104`, the second carried by the frequencies `a'/base^j` at bounded `j`. Witness: lab/rs/rho-decoupling.
- 2026-09-19 [Conjecture] That the same floor holds at every digit set, which rests on `R >= fill^level (log x)^(-C)` and is void at `alpha = 0.15` where the box is empty. Witness: lab/rs/rho-decoupling.
- 2026-09-19 [Verified] The cost-out of the GRH saving against the level-`x^(alpha/2)` defect runs at the rung `b = 3/4` and at no rung above it, so no rung past `a = 1/2` is set against the defect anywhere. Witness: lab/rs/mertens-numerology, the cost-out block.