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.