The zeta function of a digit design

Fix a base at least 2 and a digit set F inside {0..base-1} with fill = card F >= 2, and let S_F be the positive integers whose digits all lie in F. A design is a set of integers, so it has a Dirichlet series, zeta_F(s) = sum_(n in S_F) n^(-s), and that series is a zeta function with an abscissa, a meromorphic continuation and a lattice of poles. The full digit set gives Riemann's. Every other digit set gives an object carrying the same machinery and none of the same theorems. The critical line demo walks Riemann's own object at s = 1/2 + it, through the origin once per zero, and folds the zeros one at a time into the prime staircase. This page is the zeros of that object, and what the two faces of the Riemann hypothesis become on a design once the Euler product that glues them is taken away: the zeros of zeta_F on one side, the Mobius meter of mobius on the other, and no route running between them.

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

THE SPINE

The abscissa of absolute convergence is alpha = log_base(fill), the design's own mass exponent, and the series continues meromorphically to the whole plane with simple poles confined to the lattice s_(m,j) = alpha - m + 2 pi i j / log base. All of that is built territory: the abscissa is Kohler and Spilker 2009, with position-varying digit rules in Nathanson 2021; the continuation is the automatic-series mechanism of Allouche, Mendes France and Peyriere 2000, carried out for missing digits in Burnol 2026 and unified in Allouche, Shallit and Stipulanti 2025. The object of this page is Burnol's K(s) and is not new. Verified against the sources in REFS.
The continuation is one digit recursion and nothing more. An element of more than one digit is base m + a with m in S_F and a in F, so expanding (base m + a)^(-s) binomially and summing the digit moments gamma_l = sum_(a in F) a^l gives (1 - fill base^(-s)) zeta_F(s) = E_1(s) + sum_(l >= 1) binom(-s,l) base^(-s-l) gamma_l zeta_F(s+l) with E_1(s) = sum_(a in F, a != 0) a^(-s), the shift to s + l moving the argument into faster convergence. That is Burnol's Proposition 4.1, and the peeled form of it that carries the small tail G_P directly instead of as a difference of two large numbers is the engine of lab/py/design-zeta, where every printed value carries a propagated truncation bound. Verified against the source.
The pole lattice is the same object dimensions calls the complex dimensions of the design, one vertical line of period 2 pi / log base per m >= 0, and it is what makes every counting function on a design log-periodic rather than asymptotic to a constant. The off-real poles at m = 0 are genuine and not artefacts of the continuation: at base 3 with F = {0,1} the residue is enclosed in exact interval arithmetic and tied to the Fourier coefficients of the log-periodic profile of A_F(x), certified in dimensions by lab/py/burnol-residue. Proved, computer-assisted.
One factor carries that whole lattice on its own, and dividing it out is what makes the zeros readable. 1/(1 - fill base^(-s)) has poles exactly at s = alpha + 2 pi i m / log base and no zeros, so the cofactor Z(s) = zeta_F(s) (1 - fill base^(-s)) is analytic on Re s > alpha - 1: the m = 0 line is cancelled and no other, the poles of Z are the s_(m,j) with m >= 1 at which zeta_F has a nonvanishing residue, and on a full digit set Z is entire, being zeta(s)(1 - base^(1-s)). One peel level gives it in closed form, Z(s) = E_1(s) + sum_(l >= 1) binom(-s,l) base^(-s-l) gamma_l zeta_F(s+l), checked against brute-force digit summation to 1.6e-14 with and without 0 in F, and Z(s) a_min^s -> 1 to the right, a_min the least nonzero digit. The transfer runs one way without exception and the other way with one: a zero of zeta_F right of alpha - 1 is always a zero of Z, and a zero of Z is a zero of zeta_F except at a pole s_(0,j) whose residue vanishes, where Z vanishes and zeta_F is regular. At s_(0,j) one has fill base^(-s_(0,j)) = 1 exactly for every j, so with u = s - s_(0,j) and lambda = log base the periodic factor is 1 - base^(-u) with no j dependence and zeta_F(s) = Z(s)(1/(lambda u) + 1/2 + lambda u/12 - lambda^3 u^3/720 + ...), which reads the residue, the regular part and its derivative off the Taylor coefficients of Z alone. Proved (lab/py/zeta-locus, lab/py/design-zeta, lab/py/burnol-residue).
What the spine buys is machinery and not a hypothesis. The abscissa, the continuation, the lattice, the positivity at alpha and the residue formula are all cited above and none of them says where zeta_F vanishes. The rest of this page is the zero set, the products that do and do not stand where the Euler product stands on the integers, and what each of them decides about the design's own Mobius meter. The same spine over a memory rule, where the scalar fill base^(-s) becomes a transfer matrix and the one pole lattice becomes one comb per eigenvalue, is beneath, ### The memory zeta.

THE ZEROS

The census runs on the cofactor Z, on one strip, alpha - 0.92 < Re s < alpha + 3.02 and 0.02 < Im s < 60, split at Re s = alpha exactly. Base 3 with F = {0,1}, alpha = log_3 2, carries 3 zeros right of the abscissa and 20 left of it; base 10 with the digit 9 missing, alpha = log_10 9, carries 13 right and 25 left; base 3 with F = {0,2} carries 3 right, in the same three boxes as {0,1}. The largest surviving phase step on any census contour is 0.9896 and the largest propagated bound met at any census evaluation is 9.99e-11, both printed beside every count. A count is resolved and not certified unless a Rouche margin backs it, and eleven of the comb's teeth now are. Verified (lab/py/design-zeta).
Three of those zeros sit inside the design's own half-plane of absolute convergence, where zeta_F is a convergent sum of positive terms and the census uses no continuation at all, the ladder only rearranging it. The integers forbid that: at the base 2 full digit set the same census reads 0 zeros in alpha + 0.02 < Re s < alpha + 3.02, computed and not quoted, and the Euler product is the reason. The claim is this object and not a principle, since absolute convergence of a positive-term series is no zero-free region in general either: 1 + 2^(-s) has abscissa of absolute convergence -infinity and zeros at (2m+1) pi i / log 2. What the census shows is that an infinite design, whose series is the same shape as Riemann's and whose abscissa is a positive number, does the same thing. Verified (lab/py/design-zeta).
Scaled digit columns share a zero set exactly. For a positive integer a with a max F <= base - 1, so that aF stays inside {0..base-1}, the carry-free bijection m -> a m gives zeta_(aF)(s) = a^(-s) zeta_F(s), an exponential factor with no zeros and no poles, so the two designs have the same zeros and residues in the ratio a^(-s_(m,j)), and the proof uses 0 in F nowhere. Base 3 {0,2} against {0,1} agrees to 5.6e-43 at three points, and on what was censused the two agree box for box: winding one in Im [22.01, 24.01], in Im [28.01, 30.01] and in Im [56.00, 58.00] for both, and winding zero in every other box. Left of the abscissa {0,2} is not censused and is inferred from the theorem. On the meter side the same bijection twists by a sign (mobius), so the transfer is exact on both faces and trivial on one of them. Proved (lab/py/design-zeta).
Near the abscissa the zeros are a comb, one tooth per pole, and the residue puts each tooth where it is. A zero near s_(0,j) solves u(R_j + R'_j u + ...) = -r_j with r_j the residue and R_j the regular part, first order u_1 = -r_j/R_j and second order the near root of R'_j u^2 + R_j u + r_j = 0, both built from Laurent data with nothing fitted. Over 20 designs to Im s = 40 one radius 0.45 keeps the pole discs from overlapping, the smallest period in the sweep being 2.2662, and it is not defended by the tooth law, which says nothing past abs(u) = 0.3, so every count below is conditional on it: 164 poles carry one zero of Z, 40 none and 8 two, of which 21 are the residue-null pole centres of the three full-set columns, leaving 143 poles with one zero of zeta_F, 61 with none and 8 with two. Every located tooth is found by a polar grid and not by the prediction, so no tooth is selected by the law it tests. Comparing afterwards, miss2/miss1 has median 0.1637 with miss2 < miss1 at 147 of the 151, and the accuracy is conditional on the tooth being close: the 43 teeth at abs(u) < 0.1 have largest first-order miss 0.01446 and largest second-order miss 0.00164, the 84 at abs(u) < 0.2 have 0.10815 and 0.01526, while the 31 at abs(u) >= 0.3 reach 1.64614 and the prediction says nothing. The densest column is the sharpest: base 10 missing 9 at fill/base = 0.9 locates 15 teeth to a largest first-order miss of 0.013602 and a median of 0.000841. Verified (lab/py/zeta-locus).
The full digit set is the column where that comb is empty and what is left is the critical line. zeta has one pole, s = 1 = alpha, so it is regular at every s_(0,j) with j != 0 and the residue there vanishes as a one-line consequence rather than a measurement, the engine reading 1e-26 to 1e-33 there as its own control. The winding over alpha - 0.92 < Re s < alpha + 3.02, 0.02 < Im s < 40 then splits exactly as six zeros of zeta plus floor(40 log base/2 pi) cofactor-only teeth, those teeth being the zeros of 1 - base^(1-s) on Re s = 1 by exact arithmetic: 10 = 6 + 4 at base 2, 12 = 6 + 6 at base 3 and 14 = 6 + 8 at base 4. The six survivors read Re s = 0.5 at Im s = 14.1347251417, 21.0220396388, 25.0108575801, 30.4248761259, 32.9350615877, 37.5861781588 at all three bases, which share that zero set to 1e-26 because they are one arithmetic object. On a design the same split leaves a second family that is not a line at alpha/2: real parts run -0.273079611 to 0.391038600 over the 7 zeros below Im 40 at base 3 {0,1} against alpha/2 = 0.3154648768, -0.30495894 to 0.28101268 over 6 zeros at base 4 {0,1} against 0.25, and 0.060261843 to 0.97363028 over 5 zeros at base 16 {0,1,2,3} against 0.25. The spread is the witness and no per-design mean is claimed. Verified (lab/py/zeta-locus).
Which comb is stripped does not change what is left, and the next pole line's comb is forced by the first one's residues. For m >= 1 the cofactor Z_m(s) = zeta_F(s) prod_(i <= m)(1 - fill base^(-(s+i))) has exactly the zeros of Z inside alpha - 1 < Re s < alpha + 3.02, since each extra factor vanishes only on Re s = alpha - i with i >= 1, so the survivors are one set under every comb. What Z_m adds is the level-i comb, and the level-one Laurent data is forced: Z is singular at s_(1,j) = alpha - 1 + 2 pi i j/log base through its l = 1 term alone, and with base^(-s_(1,j)-1) = 1/fill and 1 - fill base^(-s_(1,j)) = 1 - base the residue there is r_(1,j) = s_(1,j) gamma_1 r_(0,j)/(fill(base-1)), so the level-one comb is empty wherever the level-zero comb is, and at the full digit set s_(1,0) = alpha - 1 = 0 kills it, which is zeta having no pole at s = 0; the generator prints abs r_(1,0) = 0.0 with its null flag set and abs r_(1,1) = 8.89623e-29 at the base 2 full set. Proved (lab/py/zeta-family, lab/py/zeta-locus).
Stripping both combs at rho = 0.45 over twenty-two designs gives 377 zeros wound by the argument principle, 351 located, 171 teeth of which 9 are level-one teeth, 19 cofactor-only zeros at null-residue poles and 161 second-family zeros, each design censused to its own printed height, 40 except the four base 3 designs at 42.894, base 9 {0,1,2} at 41.464 and base 10 missing two at 25.923. There is no gap at rho on a design: the distance from a second-family zero to the nearest live pole has minimum 0.45510938 at base 4 {2,3}, 0.45909168 at base 3 {0,1}, 0.48696667 at base 4 {0,1,2} and 0.50481072 at base 4 {1,3}, with base 4 {2,3} putting five of its eight inside 0.45 < abs(u) < 0.6, so every count falls as rho rises, N_2 reading 8, 7, 13, 9, 14 at rho = 0.45 against 7, 6, 12, 8, 7 at rho = 0.6. The full digit set is where the gap exists: at base 2 the nearest live pole to a second-family zero is 14.143566 away and no radius below 0.9 moves any count. Where the located count falls short of the winding, base 4 {2,3} at 12 of 18 being the worst, N_2 is a lower bound. Verified (lab/py/zeta-family, verb tests).
The census needs no hand-chosen right edge, because a design zeta has an explicit zero-free half plane. With a_min the least nonzero digit, hence the least element of S_F, any real sigma > alpha with a_min^sigma zeta_F(sigma) < 2 puts no zero in Re s >= sigma: the coefficients are nonnegative, so abs(a_min^s zeta_F(s) - 1) <= a_min^sigma zeta_F(sigma) - 1 < 1 for every Re s >= sigma, and the hypothesis sigma > alpha is load bearing. On the grid alpha + 0.05 n the edge sigma_1 reads 0.5 at base 4 {1} to 1.75 at the three full digit sets over twenty-four designs, a_min^sigma zeta_F(sigma) landing in [1.8635, 1.9995] with largest sigma_1 - alpha equal to 0.95, so the alpha + 3.02 strip of the locus sweep is three times wider than the zeros need. One real evaluation, carrying the ladder's own error bound. Proved (lab/py/transport-census, verb census).
Between the abscissa and that edge every proper design carries zeros and the full digit set carries none. On alpha + 1e-6 < Re s < sigma_1 the argument principle counts 157 zeros below Im s = 40 over twenty-three designs, all 157 located, plus 2 at base 50 missing one digit below Im s = 4; twenty-one of the twenty-four designs carry one, and the three that do not are the base 2, 3 and 4 full digit sets, whose windings read -1.97e-33, 1.73e-33 and 1.53e-33. The count is exact on the box and a lower bound for the half plane, the sliver alpha < Re s <= alpha + 1e-6, the band 0 < Im s < 0.02, everything above the height and the conjugate half plane all uncounted. Each rightmost carries the height it is read below, since the level-zero teeth drift right with the pole index: base 20 missing one digit reads 1.000285484146 at Im s = 2.0988, 1.000549674321 at 4.1971 and 1.002685494779 at 14.6920. Below Im s = 40 the rightmost real parts run 0.441505537191 at base 5 {0,1} to 1.002685494780 at base 20 missing one digit, each certified by a winding 1 box of half width 5e-5 whose sampled contour minimum, 1.2e-4 to 6.1e-3, beats the engine's bound by at least eight orders of magnitude and whose distance to the pole lattice is at least 0.00517845, one hundred box half widths, base 20 missing one digit standing off at 0.0223021 and four hundred of them. Verified (lab/py/transport-census, verb census).
A positive Rouche margin turns a resolved tooth into a proved one, with every input bounded from the digit recursion itself. At a pole s_0 of nonvanishing residue write Z(s_0+u) = P(u) + T(u), P entire with Taylor coefficients the exact finite sums sum_n n^(-s_0)(-log n)^m/m! convolved against those of 1 - e^(-lambda u), and T the l >= 1 part of the ladder numerator, bounded on abs(u) <= R_2 by B_T = sum_(l >= 1) binom(abs(s_0)+R_2+l-1, l) base^(-sigma-l) gamma_l G(sigma+l) at sigma = Re s_0 - R_2. That l sum is closed by a majorant ratio and not an observed one, the term ratio not being monotone: gamma_(l+1)/gamma_l <= a_max and G(sigma+l+1)/G(sigma+l) <= base^(-(P-1)) because every string in the pools is at least base^(P-1), so the term ratio is at most R_l = ((abs(s_0)+R_2+l)/(l+1)) a_max base^(-P), decreasing in l once abs(s_0)+R_2 >= 1 and below a_max base^(-P) otherwise, and stopping at the first l with R_l < 1 and adding term_l R_l/(1-R_l) is a proof. Then on abs(u) = rho, with tau = rho/R_2, abs(Z - (Z_0 + Z_1 u)) <= sum_(m >= 2) abs(P_m) rho^m + B_T tau^2/(1-tau) against abs(Z_0 + Z_1 u) >= abs(Z_1) rho - abs(Z_0); strict inequality gives Z the linear model's zero count, and that count is one because the same inequality forces abs(Z_0/Z_1) < rho. No step uses a differenced quantity, Z_1 being the first Fourier mode of T on a circle of radius R < R_2 with aliasing at most (B_T/R_2)(R/R_2)^N/(1-(R/R_2)^N). Proved (lab/py/zeta-locus, lab/py/design-zeta).
Run with the peel depth raised at each pole until the certificate fires or the string pool caps, that margin certifies exactly one zero at eleven poles of 106 at base 3, base 5, base 9, base 16 and base 10 missing 9 to Im s = 40 inside a fifteen minute budget: 11 certified, 60 failed, 7 residue-null and excluded because there the model's zero is the pole centre, and 28 skipped on budget. The eleven, with depth, margin and radius: base 3 {0,1} j = 2 at P = 7, 0.13418242, rho = 0.205; j = 5 at P = 7, 0.028140545, 0.16; j = 7 at P = 9, 0.00082974181, 0.175; base 5 {0,1} j = 4 at P = 7, 0.15035818, 0.2775; j = 5 at P = 7, 0.12269904, 0.295; base 9 {0,1,2} j = 5 at P = 5, 0.038456894, 0.26; and base 10 missing 9 at j = 1, 2, 3, 4, 7, all at P = 3, margins 0.047105507, 0.030062806, 0.045802462, 0.043508323, 0.046292701 at radii 0.1275, 0.105, 0.1025, 0.09, 0.0725, each on 24 contour samples. Every certified disc agrees with the argument principle's count of one and none disagrees; of the 19 poles carrying zero or two zeros in abs(u) < 0.45 that the budget reached, none certifies, the two double poles reached both failing. The lever is peeling and not a sharper majorant: at base 10 the automatic depth P = 2 gives B_T from 1.08 to 38.1, and P = 3 gives 0.2096 to 1.2010. Proximity is no threshold, the certified abs(Z_0/Z_1) running 0.0282669 to 0.149708 while base 3 {0,1} j = 7 at 0.104443 fails at P = 7 and certifies at P = 9. The margins are evaluated in high precision and not in ball arithmetic, which is the one step short of Proved. Verified (lab/py/zeta-locus, verb rouche).
The locus obeys no law in the design's coarse invariants. There is no curve Re s = f(Im s) shared by designs of equal alpha: base 4 {1,2} and base 16 {0,1,2,3}, both alpha = 1/2, hold zeros 0.015058 apart in Im s near Im s = 4.72 and 0.817047 apart in Re s, and equality of fill/base as well fixes nothing, base 4 {0,1} against {2,3} giving 0.0136014 against 0.719693 near Im s = 17.64. Within one design the worst real-part gap between two zeros of equal frac(Im s log q/2 pi) runs 0.077591803 at base 10 missing 9 to 0.65632474 at base 3 {0,1}, so the fractional part fixes nothing either. The single exception is alpha = 1, where the full digit sets at bases 2, 3 and 4 are one arithmetic object and do share every zero. Refuted (lab/py/zeta-locus).
The pole lattice does not force the zeros either, so the vertical period of the poles is no symmetry of the function. At base 3 {0,1} the two polished zeros right of the abscissa sit at 0.665639628004 + 23.0347504431 i and 0.720787601477 + 28.6056765649 i, an ordinate gap of 5.5709261 against the pole period 2 pi/log 3 = 5.7192017, short by 0.148. The factor 1 - fill base^(-s) is exactly 2 pi i/log base periodic, and so is E_1 at a design whose only nonzero digit is 1, but the shifted terms sum_(l >= 1) binom(-s,l) base^(-s-l) gamma_l zeta_F(s+l) are not, so zeta_F is not periodic and its zeros carry what the lattice cannot. Refuted (lab/py/zeta-locus, verb census).
There is no counting law for the second family in alpha or in the fill, at either radius. The four base 4 two-digit designs share alpha = 1/2 and fill/base = 1/2 exactly and give N_2(40) = 7, 13, 9, 14 at rho = 0.45 and 6, 12, 8, 7 at rho = 0.6, with N_2(80) = 20, 30, 22, 29 and 17, 26, 21, 20: a factor of two at one alpha and one fill/base at both radii, so the refutation is radius-robust even though the integers are not. What spreads is comb occupancy and not the second family, base 4 {0,1} and {2,3} differing by 29 percent in total winding, 14 against 18, and by a factor of two in N_2 because 7 of 8 poles are occupied against 4 of 8. Read as N_2(T) = c_F T log T + d_F T from the two heights, c_F at alpha = 1/2 is 0.10820213, 0.072134752, 0.072134752, 0.018033688, spread 0.09016844, against the base 3 and base 4 full-set controls 0.15486803 and 0.16230319, the classical 1/(2 pi) = 0.15915494 and a control spread of 0.0074351582. Every winding is the nearest integer to a numerically integrated phase whose largest surviving step runs 0.9205 to 0.9998 against a cap of 1, so the counts are measured and not certified. Refuted (lab/py/zeta-family, verbs tests and count).
The second family is not symmetric about any vertical line Re s = c_F either. Reading c_F as the midpoint of the real parts of the two second-family zeros of least Im s and testing the rest, no second-family zero in any design has a reflection partner: the reflection branch needs two zeros within the 0.05 test tolerance in Im s and the smallest ordinate gap inside a design is far above that, so the branch cannot fire. Every pair the sweep records is a self-pair, and self-pairs occur below the chance rate: over ten recensused designs the tally is 8 self-pairs and 0 reflection partners of 47 zeros tested, a rate of 0.170213 against the 0.229904 that drawing each real part uniformly from that design's own observed band predicts, and 22 of 117 over the full sweep. The three full-set controls pair 13 of 13 at c_F = 1/2 to 1e-22, where the functional equation makes every zero its own partner. c_F is not a quantity either: c_F - alpha/2 runs -0.28413232 to +0.47788515 and c_F - 1/2 runs -0.78413232 to +0.28664994, so it is not alpha/2, not theta(F) and not 1/2. Refuted (lab/py/zeta-family, verb symmetry).
Nor do the real parts contract to alpha/2 as a design fills, so the critical line is not the alpha -> 1 limit of this tree. Undivided, max abs(Re s - alpha/2) stays flat along the ladder carrying alpha toward 1, reading 0.2275679549 at base 5 {0,1}, 0.5549589411 at base 4 {0,1}, 0.5885444877 at base 3 {0,1}, 0.4233198337 at base 4 {0,1,2}, 0.5365616661 at base 5 {0,1,2,3} and 0.3151426744 at base 10 missing two, then collapsing to 1.43e-22, 1.10e-21 and 1.76e-22 at the base 2, 3 and 4 full sets. At base 10 missing 9 the second family reads 0.216084781875 to 0.70401657869 about alpha/2 = 0.477121255, a band of width 0.488 against 1 - alpha = 0.0458. Divided by 1 - alpha the statistic runs 0.39971647 to 5.7047812 with no monotone in alpha, falling from 3.8699872 to 3.2519104 on the last two rungs, so the refutation rests on the undivided spread and not on the ratio. Refuted (lab/py/zeta-family, verb limit).
One law does survive the fill, and it is about the heights rather than the real parts. Against the derived null of a quarter of the mean gap between consecutive zeta ordinates in the range, the exact expectation for an equally spaced ordinate set of the same density and conservative for one with gap variance, the mean distance from a second-family ordinate to the nearest zeta ordinate divided by that null falls monotonically in alpha over seven rungs: 2.0495374 at base 5 {0,1} with alpha = 0.430676558, 1.8953371 at base 4 {0,1} with 0.5, 0.75419266 at base 3 {0,1} with 0.630929754, 0.51648744 at base 4 {0,1,2} with 0.792481250, 0.32356636 at base 5 {0,1,2,3} with 0.861353116, 0.090501352 at base 10 missing two with 0.903089987 and 1.0429899e-23 at the base 2 full set. The base and fill confounds are dead: the fall is monotone at fixed base, 2.0495374 to 0.32356636 inside base 5 and 1.8953371 to 0.51648744 inside base 4, and at fixed fill 2 across bases; the nulls move only 1.0425839 to 1.3595166 across the ladder while the raw mean distance falls 2.4032315 to 0.12303809, so the denominator does not drive it. Over the same designs mean abs(Re s - 1/2) reads 0.36482392, 0.39426128, 0.3901396, 0.25540269, 0.31452367, 0.20473972 and 2.4065966e-23 and does not fall monotonically: at alpha = 0.903 the heights are pinned to 2.3 percent of the mean gap while the real parts are still 0.20 off 1/2. A filling design finds zeta's ordinates before its real parts find 1/2. alpha is a trend and not a function, the four base 4 two-digit designs at one alpha = 1/2 spreading 0.79050661 to 2.8404536, and the matching is nearest-ordinate and not injective, 3 distinct ordinates for 4 design zeros at base 10 missing two. Verified (lab/py/zeta-family, verb limit).
That shadow is a first-order perturbation and its constant-free form is a Newton step. The discrete position identity 1_(D_level)(n) = base^(-level) sum_(a mod base^level) G_level(a/base^level) e(-n a/base^level) on 0 <= n < base^level gives zeta_(F,level)(s) = base^(-level) sum_(a mod base^level) G_level(a/base^level) S_level(s, a/base^level), reproduced from the transform to 1.236e-37 at level = 2 over ten designs, and since G_level(0) = fill^level the a = 0 fibre carries the weight (fill/base)^level exactly against the partial sum of zeta to base^level, with no arc and no limit. That splits a polynomial at level level against a TRUNCATED zeta while the object is the continued zeta_F against the full one, and (fill/base)^level falls to 0 with level while both series tend to 1 on the right, so no level is forced and c = fill/base is the level = 1 reading and a definition. For any c the split zeta_F = c zeta + E_F gives E_F(rho_0) = zeta_F(rho_0) at a zero rho_0 of zeta, an identity carrying nothing about c, and a first-order zero of zeta_F at rho_0 - zeta_F(rho_0)/(c zeta'(rho_0)); reading c zeta'(rho_0) as zeta_F'(rho_0) removes the constant and leaves rho_0 - zeta_F(rho_0)/zeta_F'(rho_0), Taylor at a simple zero. The continuous form, the mass of G_level on abs(t) < 1/(2 base^level), is the exact sinc sum 1/base^level + sum_(n in D_level, n > 0) sin(pi n/base^level)/(pi n) and equals kappa_level(F) (fill/base)^level with kappa_level running 0.6015221 to 0.96774464 at level = 1, 2, 3, so it adds no constant the fibre does not give. Proved (lab/py/zeta-shadow verb mass, lab/py/mrly-euler verb position).
The constant-free step predicts the design zero attached to each zeta zero, and it sharpens as the offset shrinks. Over nine designs at twelve zeta zeros to Im s = 56.4462476971, six to Im s = 37.5861781588 at the two densest so the rungs do not share one height, both predictions come from zeta_F(rho_0), zeta_F'(rho_0), zeta'(rho_0) and the digit density alone and the zero is located afterwards by Newton, accepted only at abs(zeta_F) < 1e-16, within 1.5 of rho_0 and 0.02 clear of the pole lattice, largest ladder bound 9.001e-23. The step's median ratio reads 1.3843088, 1.284225, 1.2481449, 1.2060106, 1.2042502, 0.89075541, 1.0195598, 1.005076, 0.99741809 at alpha = 0.430676558 up to 0.994835739, largest abs(ratio - 1) being 0.14041 at base 20 missing one digit and 0.01734 at base 50 missing one digit, bands [0.94875, 1.14041] and [0.98266, 1.01144]; pooled, that largest deviation runs 0.01734, 0.0508884, 0.193158, 0.83912, 3.32327 over the buckets abs off < 0.05, < 0.1, < 0.2, < 0.4 and above, on 7, 4, 14, 18, 44 zeros. The level = 1 reading c = fill/base is the looser column, median ratio 1.4129353, 1.2842149, 1.0955991, 1.1806066, 1.1372453, 1.276577, 1.1347487, 1.0320127, 1.0507079, largest abs(ratio - 1) 0.24964 and 0.0821168 at the two dense rungs, five times looser at base 50, and the coupling does not select it either, median abs(coupling - fill/base) reading 0.24057225 and 0.08291158 against median abs(coupling - 1) 0.27619434 and 0.079335871, a flip between two rungs whose candidates differ by 0.05 and 0.02. Nine zeros at the three sparsest designs have no located zero inside the trust region, predicted offsets 0.95618855 to 3.0967393, so those medians are conditioned on Newton succeeding; the base 2 full set is the exact control, abs(zeta_F(rho_0)) between 1.85e-34 and 1.329e-25 at all twelve zeros. Verified (lab/py/zeta-shadow, verb predict).
The paired offset carries its exponent in the missing-digit density and not in 1 - alpha. The median paired offset over m/base = 1 - fill/base, with m the number of missing digits, reads 1.6463532, 1.2495026, 1.8345578, 1.5731321, 2.2102406, 1.6634381, 2.2424916, 1.8779239, 1.051349 across the nine rungs and over 1 - alpha reads 1.7350628, 1.2495026, 1.6569184, 1.8951686, 3.1883019, 3.432954, 4.9008186, 5.4839111, 4.0716338; a least squares in the logs, a fit and not a theorem, gives (m/base)^1.04544 at R2 0.957842 against (1-alpha)^0.71691 at R2 0.944011, the first column spanning 2.13297 and the second 4.38888, so m/base carries the exponent by a factor of 2.05764 inside the 4.28797 that (1-alpha)/(m/base) itself spans, which is the whole discrimination these two normalisations admit. The two new rungs are base 20 missing its top digit at alpha = 0.9828778777 and base 50 missing its top digit at 0.9948357391, all six zeros located at each, median abs(E_F(rho_0)) 0.11830158 and 0.028066806 and median offset 0.093896196 and 0.021026979. Read in the family row's form the mean distance over a quarter of the mean gap gives 0.81218635, 0.57141859, 0.50488757, 0.37447728, 0.20954319, 0.29197634, 0.14794812, 0.052888241, 0.011098646 and 0 at the full set; the pairing is zeta-zero-first where the family row is design-zero-first, so this is a parallel ladder and not that row recomputed, it bounds no maximum over the second family and touches no jump clause. Verified (lab/py/zeta-shadow, verb rungs).
What it does not do is explain why the ordinates converge before the real parts, because at the zeros it pairs it separates neither. The first-order offset is one complex number, so a paired zero moves isotropically and the ordinate offset and the real-part offset are one quantity with no preferred phase: per zero abs(Im off)/abs(Re off) spans 0.137681 to 6.11895 at base 20 missing its top digit and 0.14167 to 18.7749 at base 50, and rung by rung median abs(Im off) against median abs(Re s - 1/2) reads 0.55734029/0.43095421, 0.49670656/0.28603903, 0.48081533/0.22535435, 0.30633741/0.16748977, 0.12823995/0.36138728, 0.25093621/0.12181102, 0.11574693/0.19332005, 0.058239278/0.049215607, 0.011954894/0.010995712, the ordinate offset larger on seven rungs and smaller on two, at rungs 5 and 7, with both falling broadly and neither monotone. The law binds only the zeros Newton reaches from a zeta zero inside 1.5 of it and enumerates no design zero, so the ordinate shadow of the family row is what a PAIRED zero does and the unpartnered second family is the surplus; the two are consistent with a mixture that approaches in both coordinates on the partnered zeros and not at all on the rest, and that mixture has no witness until the unpartnered count is measured. Refuted (lab/py/zeta-shadow verb rungs, lab/py/zeta-family verb limit).
The one statistic the transport theorem reads obeys no law either: the gain of a design's rightmost zero over its abscissa is not a function of alpha and fill/base. Four equal-key families, one base and one digit count each so both agree exactly and not to a rounding, read unequal gains: at alpha = 1/2, fill/base = 1/2 the four base 4 two-digit designs give 0.0853043873, 0.4400124317, 0.2706238545, 0.3439264581, a spread of 0.35470804; base 5 at alpha = 0.4306766, fill/base = 0.4 spreads 0.37474232; base 3 two-digit at alpha = 0.6309298 spreads 0.17605693; base 4 three-digit at alpha = 0.7924813 spreads 0.060972003, still six hundred box widths. The two columns disagree in direction, the gain being largest at the sparsest designs, 0.5291214025 and 0.4485242462 at alpha = 0, where the rightmost real part itself is smallest. What rises with alpha is the floor: the least rightmost real part per rung reads 0.4485242462, 0.4415055372, 0.5853043873, 0.7207876015, 0.9126562295, 0.9897481059, 1.0015143877, 1.0015892753, 1.0026854948, 1.0000614750 up ten rungs, rising at every step but the first and the last, the last being where the census height drops from 40 to 4; one design per rung above alpha = 0.86 against six at alpha = 0.5, and no fit is taken. Refuted (lab/py/transport-census, verb law).

THE WALL AND THE PRODUCTS

The indicator of S_F is multiplicative exactly at the full digit set, and the wall is constructed rather than described. 1 in F is forced by f(1) = 1; if a digit c >= 2 is missing take the least, and R_c R_(c+1) has no carry because its base^m coefficient is min(m+1, c, 2c-m) <= base-1, so its digit set is exactly {1..c} while gcd(R_c, R_(c+1)) = R_1 = 1; if only 0 is missing then an odd base gives the coprime pair (2, (base^2+1)/2) and an even base the coprime odd pair (base^2-1, base^2+1), whose products leave the set. Over all 8177 sets with 2 <= base <= 12 the constructed witness is asserted at each of the 4083 sets that pass f(1) = 1 and are not full, and an independent search finds a minimal witness for every one, hardest base 12, F = {1}, pair (5, 377). No design outside the full set carries an Euler product over primes. Proved (lab/py/mrly-euler, verb wall).
The polynomial model does not transport Weil's theorem, and the wall's first case is exactly where it breaks. Evaluation P -> P(base) carries the polynomials with coefficients in {0..base-1}, the model F_base[t] at prime base, bijectively onto the nonnegative integers and adds correctly only where no carry occurs: a coefficient of the polynomial product P R reaches (base-1)^2 (min(deg P, deg R) + 1), far above the digit cap base - 1, so the digit string of an integer product is the carry reduction of the polynomial product and not that product, and R_c R_(c+1) is that failure made minimal, the shortest coprime pair whose carry-free product shows the missing digit c. Multiplication in F_base[t] reduces its coefficients mod base and never carries while the integer product does, so the Riemann hypothesis proved over a function field reaches zeta_F along no evaluation bridge, and no other route is spoken to; the density side of the same substitution is coprime. Proved (lab/py/mrly-euler, verb wall).
What stands in its place is a product over digit positions rather than over primes. With G_level(t) = prod_(i<level) sum_(d in F) e(d base^i t), one factor per position, uniqueness of the base expansion gives int_0^1 G_level(t) e(-nt) dt = 1_(D_level)(n) for every integer n, hence sum_(n in D_level, n >= 1) a(n) n^(-s) = int_0^1 G_level(t) A(s,t) dt with A(s,t) = sum_(n >= 1) a(n) e(-nt) n^(-s), for every absolutely convergent Dirichlet series at once. Taking a = 1 returns zeta_F against the periodic zeta of DLMF 25.13; taking a = mu returns the design's Mobius series against the Lerch-Mobius series. The set enters through the digit positions and the arithmetic sits entirely in the kernel; they meet only under the integral. Checked to 1.95e-16 and 2.04e-16 at base 10 missing 9, level = 3, 4, and 2.9e-16 at base 3 {0,1}, level = 3, 4, 5. Proved (lab/py/mrly-euler, verb position).
Read at x = base^level the pairing is exact on a finite grid, M_F(base^level) = base^(-level) sum_(a mod base^level) G_level(a/base^level) S_level(a/base^level), both factors trigonometric polynomials of degree below base^level, and its l^1 mass decomposes level by level as C_level = sum_(j=0)^level fill^(level-j) c_j. The floor C_j >= base C_(j-1) forces the top-level share c_level/C_level >= 1 - fill/base at every base and digit set, measured 0.485846, 0.602606, 0.687994, 0.510055 against floors 0.333333, 0.500000, 0.600000, 0.100000, with the levels j >= level/2 carrying 0.995116, 0.996061, 0.997043, 0.942350 of it. So weighting the Mobius input per denominator saves exactly log_base(C_level/c_level)/level, a constant factor capped by base/m, the numerator reading 0.657068 at base 3 {0,1} identically at every level = 6..14: the mass is where the method cannot spend it. Proved (lab/py/mrly-pairing, verb split).
The large sieve does not rescue that split on the major arcs of any design below the full set, and the exact second moment prices it. Writing a = base^v a' with base not dividing a' and j = level - v, the l^2 mass of the grid pairing over the levels j <= J is exactly fill^(2 level - J) base^J, since G_level(a/base^level) = fill^(level-j) G_j(a'/base^j) and Parseval on Z/base^J gives sum_(a mod base^J) abs(G_J(a/base^J))^2 = base^J fill^J. Those base^J points are base^(-J) spaced, so the spacing form of Montgomery and Vaughan 1973 applies at every base, including the composite ones where an a' not divisible by base need not be coprime to it and the coprime-residue form misses the point, and it gives sum_(j <= J) sum_(a') abs(S_level)^2 << (base^level + base^J) base^level. With x = base^level the levels below J = u level then cost x^(alpha + u(1-alpha)/2), which under the hypothesis alpha < 1 is strictly above alpha at every u > 0, equals alpha only at u = 0, and returns the whole-grid value (alpha + 1)/2 at u = 1; at alpha = 1 it is alpha at every u, so the full digit sets witness nothing here. Proved (lab/py/mrly-pairing, verb split).
The principal fibre of that grid is the classical Mertens function and it is asymptotically too small. The a = 0 term is base^(-level) fill^level M(base^level), of exponent alpha - 1/2 under RH, and alpha - 1/2 < alpha/2 for every alpha < 1, so in the limit M cannot carry the conjectured size of the design meter. At finite depth it carries a great deal: shares -0.028961, 0.018555, -0.006583, 0.408208 at base 3 {0,1} at level = 14, base 4 {0,1} at level = 11, base 5 {0,1} at level = 9 and base 10 missing 9 at level = 6, and exactly all of the meter on the two full-set controls. The exponent gap at base 10 missing 9 is 0.454243 against 0.477121, and a factor of 10 between them needs x = 10^44. Proved (lab/py/mrly-pairing, verb split).
The design does carry an exact Euler product, and it is the one with nothing in it. On the free monoid over F with norm N(w) = base^(abs(w)), sum_w N(w)^(-s) = 1/(1 - fill base^(-s)) = prod_(level>=1) (1 - base^(-level s))^(-c_fill(level)) with c_fill(level) the Lyndon count, by Chen-Fox-Lyndon: every word factors uniquely as a non-increasing product of Lyndon words, so the free monoid on F is equinumerous by norm with the free abelian monoid on Lyndon words and is not equal to it. Its primes are the Lyndon words, it is zero free, its Mobius is supported on the empty word and the letters so its Mertens is 1 - fill beyond norm 1, and its poles are exactly the design pole lattice. All RH content of zeta_F therefore sits in the cofactor Z, which is what the census reads. Expansion verified through u^16 at fill = 2, 3, 4, 9, 10, c_2(level) being A001037. Proved (lab/py/mrly-euler, verb word).
Inside the design there is a third product, genuinely over primes, and it lives on a different set: the free semigroup N_F on the primes that lie in S_F, a Beurling system with its own Mobius and its own RH-shaped question. It is blind to whole columns. If gcd(F) = a > 1 then every element of S_F is a multiple of a, so when a is prime the design's primes are {a}, N_F is the powers of a and M_B(x) = 0 for x >= a, and when a is composite the design holds no prime at all, N_F = {1} and M_B is identically 1, witness base 10, F = {0,4,8}. The eight scaled census families of mobius are exactly the columns the Beurling route cannot see and the scaling transfer reads exactly. Proved (lab/py/mrly-euler, verb beurling).
The Beurling census to x = 10^6 says the same thing in numbers, and it reads a level rather than a trend. Base 3 {0,2} has the single prime 2 and M_B identically zero past it; base 3 {0,1} has 525 primes, N_F(920483) = 2198, running max abs(M_B) = 98 and exponent 0.3339 against alpha/2 = 0.3155; base 10 missing 9 has 35139 primes, N_F(10^6) = 488864 against x^alpha = 531441, M_B(10^6) = 1860, running max 1866, and log(running max)/log x reading 0.4203, 0.4882, 0.5452 at 10^4, 10^5, 10^6 against alpha/2 = 0.4771, where full base 10 as control reads 0.4084, 0.4241, 0.4276 against its own alpha/2 = 0.5. That is +0.068 over alpha/2 for the design against -0.072 for the control, with a running maximum climbing in both: there is cancellation, 0.545 against the trivial alpha = 0.954, and it sits above alpha/2, so the census supports cancellation on N_F and does not support the design's own square-root shape. N_F is not S_F. Full base 10 reproduces -23, -48, 212, A084237. Verified (lab/py/mrly-euler, verb beurling).

THE IDENTITY

One identity does survive the wall, and it is not zeta M = 1. For every base and F with 1 in F the indicator of S_F has a Dirichlet inverse nu_F, given by nu_F(1) = 1 and nu_F(n) = -sum_(d divides n, d > 1, d in S_F) nu_F(n/d), so zeta_F(s) N_F(s) = 1 with N_F(s) = sum nu_F(n) n^(-s), and nu_F is mu exactly at the full digit set, where the classical identity is the special case. The support of nu_F lies inside the multiplicative semigroup generated by S_F and strictly inside it: 9, 27 and 36 lie in the semigroup with nu_F = 0 while 16, 48 and 52 lie in the semigroup and outside S_F. So a design carries four sets, not two: S_F, the Beurling integers on the primes of the design, the semigroup, and the support of nu_F. Checked against mu term for term to n = 131072 at base 2 and n = 177147 at base 3. Proved (lab/py/mrly-pairing verb inverse, lab/py/design-zeta).
That identity is a one-way bridge from the zeros to the design's own Mertens function, and it runs the wrong way for the hypothesis. If rho is a zero of zeta_F with Re rho > alpha then sigma_c(N_F) >= Re rho, by the identity theorem on the connected pole-free half plane Re s > max(sigma_c(N_F), alpha), so sum_(n <= x) nu_F(n) is not O(x^(Re rho - eps)) for any eps > 0; the converse bound sigma_c(N_F) <= sup Re rho is not claimed. The partial sums of N_F(sigma) meet 1/zeta_F(sigma) to 1.60e-3 at sigma = 0.8008 and 1.96e-4 at sigma = 0.9208 at base 3 {0,1}, and to 1.72e-2 at sigma = 1.0816 and 2.39e-3 at sigma = 1.2016 at base 10 missing 9, both offsets sitting above Re rho. Proved (lab/py/mrly-pairing verb inverse).
Two of the censused zeros are certified by winding, and they turn that bridge into anti-cancellation. The argument principle gives winding 1 on Re in [0.72074, 0.72084], Im in [28.60563, 28.60573] at base 3 {0,1}, contour minimum abs(zeta_F) = 8.298e-4 against the engine bound 6.284e-30, and winding 1 on Re in [1.00150, 1.00168], Im in [2.73915, 2.73925] at base 10 missing 9, contour minimum 6.865e-4 against 2.798e-23, while the control rectangle Re in [0.99900, 1.00050], Im in [2.73810, 2.74030] there returns winding 0. Both boxes lie strictly right of alpha = 0.6309297536 and 0.9542425094, so sum_(n <= x) nu_F(n) is not O(x^(0.72074 - eps)) and not O(x^(1.00150 - eps)) respectively: the limsup of the design's own Mertens function exceeds the design's own mass, and at base 10 missing 9 exceeds x itself, the box lying right of Re s = 1. Proved (lab/py/mrly-pairing verbs box and inverse, lab/py/design-zeta).
The square-root shape is therefore false for nu_F, and the census is far below both limsups at every depth reached, which is what a limsup statement allows: max/A_F = 0.0738 at base 3, level = 16 and max/x = 0.0847 at base 10, level = 7, the running maximum of sum nu_F(n) growing by 9.4474, 11.5000, 10.2220, 10.0354 per level at base 10 missing 9, level = 4..7, against base^(Re rho) = 10.036661 and the trivial fill 9, only the last of the four landing on the predicted rate, with max/A_F(base^level) rising 0.1043, 0.1094, 0.1398, 0.1588, 0.1771; at base 3 {0,1} the geometric mean of the four steps level = 12..16 is 2.059 against 2.207512 and 2 while the arithmetic mean of the five printed level ratios is 1.9972, a census too short to separate them. Proved (lab/py/mrly-pairing verbs box and inverse).
One certified zero right of the abscissa refutes every square-root-shaped bound for the design's own Mobius, and the digit 1 is the hypothesis that bites. Let 1 in F, let rho be a zero of zeta_F certified by a winding 1 box with left edge x_0 > alpha containing no pole. Transport gives sigma_c(N_F) >= Re rho >= x_0 > alpha, so sum_(n <= x) nu_F(n) is not O(x^(x_0 - eps)) for any eps > 0; since A_F(x) has exponent alpha the conjectured exponent is alpha/2 <= alpha < x_0, so nu_F misses even the trivial O(x^(alpha - eps)), the first inequality failing to be strict only at the two designs with alpha = 0, where A_F(x) grows like log x. Nineteen of the twenty-four designs censused meet all three hypotheses and get a bound, seventeen of the twenty-two the locus and family sweeps carry, running theta(nu_F) >= 0.4414555 at base 5 {0,1} to theta(nu_F) >= 1.0026354 at base 20 missing one digit. Four have Re rho > 1, so their own Mobius outruns the count of ALL integers below x: base 10 missing two digits, base 10 missing 9, base 20 missing one digit and base 50 missing one digit, at fill/base = 0.8, 0.9, 0.95, 0.98, three censused to Im s = 40 and base 50 to Im s = 4, and only the SIGN of Re rho - 1 is read and never its size. It is not a k/q effect: base 5 {0,1,2,3} at the same fill/base = 0.8 has rightmost 0.989748105861. Two designs carry a zero right of alpha and no bound, base 4 {2,3} and base 4 {0,2,3}, which omit the digit 1, so the indicator vanishes at 1 and nu_F does not exist. Proved (lab/py/transport-census, verb law).
So the design's own Mobius is not the object the hypothesis is about. The square-root shape on a design can only be carried by mu restricted to S_F, the meter of mobius, and the decoupling is what protects it: zeta M = 1 on the full set, S_F is not multiplicatively closed for any proper F, and zeta_F M_F is not 1, so no known route runs from a zero of zeta_F to the meter's exponent and the zero census carries no bound on it. What survives is not the zeros but the position product, which pairs a = 1 and a = mu alike. Coons 2010 Theorem 2.3, that mu is not k-automatic for any k, rules out the automatic-continuation route to the Mobius series and nothing wider. Proved (mobius, lab/py/design-zeta, lab/py/mrly-euler).
The obvious repair, gluing the two series anyway and calling the remainder small, gains nothing: writing zeta_F M_F = 1 + D_F, the abscissa of D_F is exactly alpha. Absolute convergence of zeta_F^2 puts sigma_a(D_F) <= alpha, and if sigma_c(D_F) were below alpha then M_F(sigma) would tend to 0 as sigma -> alpha+, since zeta_F has nonnegative coefficients and is singular at its abscissa by Landau; that half rests on a measurement, unconditional in shape since sigma_c(D_F) < alpha would force P(x) = o(x^alpha), and P(x)/x^alpha reads 0.493767, 0.699235, 0.758519, 0.587055 at four sampling phases at base 3 {0,1}, the four phases being needed because the ratio is log-periodic and sampling only at x = base^level aliases every Fourier mode onto one number. Since M_F = (1 + D_F) N_F and sigma_c(N_F) > alpha, the glue is not neutral but lossy. Verified (lab/py/mrly-pairing, verb glue).

THE METER

The design's Mobius meter does oscillate at the zeta ordinates, and not at the design's own pole lattice. Read M_F(x)/x^(alpha/2) uniformly in log x, Hann-windowed, against a local-median floor and a null of rigid shifts of each candidate list: at base 10 with the digit 9 missing all six strongest peaks sit within one bin of a nontrivial zeta zero, offsets 0.068 to 0.216, with the full-set control at the same depth reading ten of ten, offsets 0.018 to 0.196. Thirteen zeta ordinates are reachable in the band 4 < gamma < 60, so a peak lands within one bin of one by chance with probability 0.159 and six of six is P = 1.6e-5. The pole lattice 2 pi j / log base scores -0.592, -0.640, -0.734 at base 3 {0,1}, base 3 {0,2} and base 5 {0,1}, below its own null, while the counting function over the identical elements scores 3.602, 3.764 and 3.973: the pipeline would have seen a lattice and there is none. Verified (lab/py/design-meter, verb spectrum, and the echo demo).
The echo is a mean field and not a signature, and its size is a theorem. M_F(x) = sum_(n <= x) mu(n) A_F(n)/n + R_F(x) defines R_F at every base and digit set, and partial summation gives sum_(n <= x) mu(n) A_F(n)/n = A_F(x) H(x) - sum_(m in S_F, m <= x) H(m-1) with H(y) = sum_(n <= y) mu(n)/n, which is O(y^(-1/2 + eps)) under RH; so the echo is O(x^(alpha - 1/2 + eps)) when alpha > 1/2, while for alpha < 1/2 the second sum converges absolutely and the echo tends to a nonzero constant, base 16 {0,1} reading -0.0937, -0.1330, -0.1242, -0.1051, -0.1099 at 10^3 to 10^7 against x^(alpha - 1/2) falling 0.1778 to 0.0178, and base 10 {0,1} reading -0.0500 at 10^7 against 0.0405. Against the square-root bar x^(alpha/2) the echo dies at x^(-min(alpha, 1 - alpha)/2), equal to 1 only at alpha = 1. Proved (lab/py/design-meter, mobius).
The split separates the two by hand and the frequencies follow the echo. At base 10 missing 9 the echo carries six of six top peaks at zeta zeros and the residual none of the two it has, the echo being 0.1342 of the meter in root mean square against 0.6476, and the echo's share of the meter falls 0.356028, 0.242495, 0.207229 there and 0.208549, 0.099001, 0.047902 at base 3 {0,1}, share over prediction reading 1.0000, 0.7387, 0.6846 and 1.0000, 0.9219, 0.8663, each design decaying at least as fast as its own rate. So the zeta zeros neither obstruct nor help the design's square-root shape, which is a statement about R_F alone. Proved (lab/py/design-meter, verb spectrum).
There is no family law behind the frequencies, and the obvious one is dead. The frequency set is not a function of (q, alpha): base 3 {0,1} and base 9 {0,1,2,3} share alpha = 0.630930, element count 1048575 and log range to within 1.4%, and their meters split ten peaks against none, where support-matched random-sign meters reach 0 to 4 peaks on the first support and 0 to 2 on the second over eight draws each, so the ten sit above their own null and the none does not; base 9 {0,1,2,3} and base 9 {0,1,3,4} share base and alpha and split the same way, the second being base 3 {0,1} element for element since its digits are the base-3 pairs 00, 01, 10, 11. The scaled pair base 3 {0,1} and {0,2} shares eight of ten peaks, so the scaling transfer carries into the spectrum where no (q, alpha) law does. Refuted (lab/py/design-meter, verb family, mobius).
The arcs are not a frequency family the meter carries. The primitive quadratic Dirichlet L-zeros of conductor 3, 4 or 5, the quadratic conductor each base carries, separate from their shift null at no design over the eleven censused: the largest score is 0.372 against a null of 0.341 at base 4 {0,1,2} and the widest gap over a null is 0.371 against 0.290 at base 3 {1,2}, while the zeta ordinates on the same meters reach 1.130 against 0.392 at base 10 missing 9 and 0.742 against 0.291 at base 3 {0,1}, so the pipeline would have seen an arc family and there is none. The wider prediction, over arcs of denominator base^j, is untestable by this spectrum rather than false, the generator carrying L-zeros at the three quadratic conductors alone, and it is not claimed here. Refuted (lab/py/design-meter, verb spectrum).

WHERE THE REST LIVES

The other face of the Riemann hypothesis on a design is the Mobius meter M_F, and it has its own page: the exact transfer between scaled columns, the 47-column cancellation census, the power saving under GRH at large base, the pair route with its l^1 threshold, and the open exponent are mobius.
The bar that route has to clear is arithmetic about the digit set rather than about mu, and the certified bases that clear it, the least base below the quarter threshold at one and at two missing digits and where each family closes, are coprime.
The poles themselves, the certified residue at the off-real lattice points and what they say about Minkowski measurability are dimensions; the design's counting function, its log-periodic ripple and the checkpoint identities are mobius.
The verbs of lab/py/mrly-euler this page does not print: the disjunction that replaces zeta_F M_F = 1 and the least n where the product leaves 1, the Holder reading of the position identity that recovers the pair route's exponent, the Lerch-Mobius fibres written as inverse Dirichlet L-functions of modulus dividing the denominator, and Hurwitz's formula rebuilt so that the reflection moves the kernel and not the design.
The per-design tables behind the census and the shadow, which this page reads only in aggregate, are lab/py/transport-census for sigma_1, the strip, the winding and the certified box of every design, and lab/py/zeta-shadow for the remainder, the coupling and both predictions at every zeta zero of every rung.
The per-design cells of the family sweep live in lab/py/zeta-family: the counts, real parts and rightmost Mertens reading of every design at both assignment radii, the level-one residue table with its null flags, and the residue of each zero's ordinate against 2 pi/log base.
The verb of lab/py/mrly-pairing this page does not print: the one-step constant of the digit transform against its triangle-split bound, strict at every family measured beyond level = 1.
Every finding on a tagged line: DISCOVERIES. Every source resolved: REFS.

GENERATORS

lab/py/design-zeta is the ladder, the contour engine and the census: uv run python research/lab/py/design-zeta/design_zeta.py, and --full runs the census to Im s = 60 and adds the base-10 columns. Every printed value carries a propagated truncation bound and the ladder raises rather than returns when the tolerance is not met.
lab/py/zeta-locus imports that engine and adds the cofactor, the Laurent data, the comb law and the falsification sweep: uv run python research/lab/py/zeta-locus/zeta_locus.py shadow 40 all for the law at every design and census 40 all for the strip census, N_F(T) and one row per zero, and rouche 40 rou for the certificate at every pole of the nine designs the census counts, with its margin, radii, B_T and depth.
lab/py/transport-census prints the zero-free edge, the census right of the abscissa and the bound each design proves: uv run python research/lab/py/transport-census/transport_census.py census 40 locus, and the verb law; the third argument selects locus, rungs, b20, b50, ctl or all, and all is the twenty-two of the two sweeps, base 20 and base 50 being their own runs.
lab/py/zeta-shadow prints the position identity, the fibre weight and the constant-free step against its fill/base rival: uv run python research/lab/py/zeta-shadow/zeta_shadow.py mass ladder, and the verbs predict and rungs; the second argument selects ladder, old or new.
lab/py/zeta-family splits the second family from the combs and runs the three derived tests: uv run python research/lab/py/zeta-family/zeta_family.py family 40 all, and the verbs symmetry, count, limit and tests, the second argument the height and the third a design family.
lab/py/mrly-euler prints the wall, the position identity and the three products: uv run python research/lab/py/mrly-euler/euler.py wall, and the verbs pair, position, dual, fibre, word and beurling.
lab/py/mrly-pairing prints the identity, the winding boxes, the glue and the grid split: uv run python research/lab/py/mrly-pairing/pairing.py split, and the verbs glue, inverse and box.
lab/py/design-meter prints the meter's spectrum, its echo and the equal-alpha comparison: uv run python research/lab/py/design-meter/design_meter.py spectrum, and the verbs sieve and family.
lab/py/burnol-residue certifies the residues at the off-real poles in interval arithmetic: uv run python research/lab/py/burnol-residue/burnol_residue.py.