The zeros of the design zeta

• 2026-09-07 [Verified] This specific infinite design zeta has zeros in its own half-plane of absolute convergence, which the integers forbid, and the claim is the object and not the principle, since the positive-term Dirichlet series 1 + 2^(-s) has abscissa of absolute convergence -infinity and zeros at (2m+1) pi i/log 2. The census counts zeros of the Lyndon cofactor Z(s) = zeta_F(s)(1 - fill base^(-s)), analytic on Re s > alpha - 1, so it needs no pole-free strip and leaves no sliver against the pole line, on the single strip alpha - 0.92 < Re s < alpha + 3.02, 0.02 < Im s < 60, split at Re s = alpha exactly. Base 3 F = {0,1} at alpha = log_3 2 carries 3 zeros right of the abscissa and 20 left of it inside that strip; base 10 missing the digit 9 at alpha = log_10 9 carries 13 right and 25 left; base 3 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. The base 2 full digit set is the control on both sides and each side names its object: zeros of zeta in alpha + 0.02 < Re s < alpha + 3.02 count 0, which is what the Euler product forbids, computed and not quoted; zeros of zeta in alpha - 0.98 < Re s < alpha - 0.02 count 13, the first thirteen below Im s = 60; and the teeth of the cofactor 1 - 2 base^(-s), which sit exactly ON Re s = alpha and are not zeros of zeta, bring the one-strip count to 19 = 13 + 6 with 6 = floor(60 log 2/2 pi). Right of the abscissa no continuation is used, since the positive series converges absolutely there and the ladder only rearranges it. The count is resolved and not certified: the largest surviving phase step is printed and nothing bounds zeta_F'/zeta_F on the contour, so a zero pair closer than the surviving spacing would stay invisible. Witness: lab/py/design-zeta.
• 2026-09-07 [Proved] 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 zeta_(aF) and zeta_F 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 strip alpha < Re s < alpha + 3.02, 0.02 < Im s < 60, the two censuses coincide 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 zero in every other box. Left of the abscissa {0,2} is not censused and is inferred from the theorem. On the Mobius side the same bijection twists the meter by a sign, so the transfer is exact on both faces and trivial on one of them. Witness: lab/py/design-zeta, mobius.md.
• 2026-09-07 [Proved] The Euler-product bridge between the two faces of RH is absent on a design: 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 theta(F) and the zero census carries no bound on the square-root conjecture. What survives is not the zeros but the position product: G_level pairs against a = 1 and a = mu alike and the arithmetic sits entirely in the kernel. Coons 2010 Theorem 2.3 rules out the automatic-continuation route to M_F and nothing wider. Witness: mobius.md, lab/py/design-zeta, lab/py/mrly-euler, REFS.md.
• 2026-09-07 [Proved] The zeros of the design zeta are read off one analytic function and their positions near the pole lattice are forced by the residues. The Lyndon cofactor Z(s) = zeta_F(s)(1 - fill base^(-s)) is analytic on Re s > alpha - 1, since 1 - fill base^(-s) cancels exactly the m = 0 line of the digit recursion's poles and no other; the poles of Z are those s_(m,j) = alpha - m + 2 pi i j/log base with m >= 1 at which zeta_F has a nonvanishing residue, the nearest line to that half-plane being Re s = alpha - 1 with residue -s_(1,j) gamma_1 r_j/fill, and on a full digit set Z has no pole at all, being zeta(s)(1 - base^(1-s)), entire. One peel level gives Z(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) and gamma_l = sum_(a in F) a^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 zero of zeta_F right of alpha - 1 is always a zero of Z; conversely a zero of Z is a zero of zeta_F except at a pole s_(0,j) with r_j = 0, 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) the periodic factor is 1 - base^(-u) with no j dependence and zeta_F(s) = Z(s)(1/(L u) + 1/2 + L u/12 - L^3 u^3/720 + ...), L = log base, giving residue Z_0/L, regular part Z_1/L + Z_0/2 and its derivative Z_2/L + Z_1/2 + Z_0 L/12 from the Taylor coefficients of Z alone, on a disc of radius at least 1 and exactly 1 when r_j does not vanish. Witness: lab/py/zeta-locus, lab/py/design-zeta, lab/py/burnol-residue.
• 2026-09-07 [Verified] The zeros of the design zeta near the abscissa are a residue comb whose tooth position the residue and the regular part predict. A zero near the pole s_(0,j) solves u(R_j + R'_j u + ...) = -r_j, 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 the Laurent data with nothing fitted. Over 20 designs to Im s = 40 (every scaling class at base = 3 and base = 4, two at base = 5, base 9 {0,1,2}, base 16 {0,1,2,3}, base 10 missing 9, base 2 full set) one assignment radius 0.45, fixed by the discs not overlapping and not by the tooth law so that every count is conditional on it, serves both the count and the tooth, discs never overlapping since the smallest period in the sweep is 2.2662: the argument principle on that circle gives 164 poles carrying one zero of Z, 40 none and 8 two, of which 21 are the residue-null pole centres of the three full-set columns and are zeros of Z that are not zeros of zeta_F, leaving 143 poles with one zero of zeta_F, 61 with none and 8 with two. All 151 poles carrying a zero have their zeros located by a polar grid inside that same disc and not by the prediction, so no tooth is selected by the law it tests and no pole carrying a zero is left without one. Comparing prediction to tooth 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. Witness: lab/py/zeta-locus.
• 2026-09-07 [Verified] The critical line is the second family of the full digit set. On a full digit set zeta_F is zeta, whose only pole is 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 machinery reads those residues as 1e-26 to 1e-33, which is a control of the engine, and the comb is empty. The winding of the cofactor 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, and the three columns 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: its 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 being the witness and no per-design mean claimed. Witness: lab/py/zeta-locus.
• 2026-09-07 [Proved] The second family of the design zeta does not depend on which comb is stripped, and the next pole line's comb is computed from 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(s) = zeta_F(s)(1 - fill base^(-s)) inside alpha - 1 < Re s < alpha + 3.02, since each extra factor vanishes only on Re s = alpha - i for i >= 1, so the survivors of the assignment are one set under every comb. What Z_m adds is the level-i comb, and its Laurent data is forced by the level-zero data: Z(s) = E_1(s) + sum_(l >= 1) binom(-s,l) base^(-s-l) gamma_l zeta_F(s+l) 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) = base^(-s_(0,j)) = 1/fill and 1 - fill base^(-s_(1,j)) = 1 - base the residue of zeta_F 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 makes it vanish, which is zeta having no pole at s = 0; the lab prints abs r_(1,0) = 0.0 with the null flag set and abs r_(1,1) = 8.89623e-29 at the base 2 full set. Witness: lab/py/zeta-family, lab/py/zeta-locus, lab/py/design-zeta.
• 2026-09-07 [Verified] The second family of the design zeta, split out and censused over twenty-two designs at a stated assignment radius, with no gap at that radius on any design. Stripping the level-zero and level-one combs at rho = 0.45, a constant fixed only by the pole discs not overlapping and not by the tooth law, which is accurate only inside abs(u) < 0.2, the twenty designs of the locus sweep plus base 5 {0,1,2,3} and base 10 missing two digits give 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 is conditional on rho and 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 on base 3 {0,1} and the four base 4 two-digit designs. 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. Witness: lab/py/zeta-family verb tests.
• 2026-09-07 [Verified] Seventeen designs carry a lower bound on the Mertens exponent of their own Mobius, and the strongest bound is radius-robust. Nineteen of the twenty-two designs have a censused zero of zeta_F strictly right of alpha, twelve of them in the second family, and at the seventeen of them whose digit set contains 1, so that nu_F exists, the transport theorem gives that sum_(n <= x) nu_F(n) is not O(x^(Re rho - eps)); base 4 {2,3} and {0,2,3} omit the digit 1 and carry a zero but no nu_F. Base 10 missing two digits has a zero at 1.00151438765 + 2.77402670058 i against alpha = 0.903089987, a second base-10 column where the design's own Mobius has a Mertens exponent above 1 and so above x itself; that zero is a level-zero tooth at abs(u) = 0.1083 of the j = 1 pole, deep inside every assignment radius tested, so the bound does not depend on where the comb is cut. Base 4 {1,2} has a second-family zero at 0.940012431696 + 13.0678968771 i against alpha = 1/2, an exponent of 0.94 against a design mass exponent of 0.5, and base 3 {0,1} reads 0.720787601477 at Im 28.6056765649 against alpha = 0.630929754. Witness: lab/py/zeta-family verb tests, lab/py/mrly-pairing verb inverse.
• 2026-09-07 [Verified] What converges as a design fills is the ordinate set and not the real part. 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: 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, 2.0495374, 1.8953371, 0.75419266, 1.0429899e-23 at alpha = 0.430676558, 0.5, 0.630929754, 1; 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, so 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. alpha is a trend and not a function: the four base 4 two-digit designs at one alpha = 1/2 spread 0.79050661 to 2.8404536. The matching is nearest-ordinate and not injective, 3 distinct ordinates for 4 design zeros at base 10 missing two. Witness: lab/py/zeta-family verb limit.
• 2026-09-07 [Proved] The ordinate shadow is a first-order perturbation and its constant-free form is a Newton step from the zeta zero. 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) with S_level(s,x) = sum_(1 <= n < base^level) e(-nx) n^(-s), reproduced from the transform to 8.326e-40 at level = 2 on 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 identity splits the polynomial at level level against a TRUNCATED zeta while the object is the continued zeta_F against the full zeta, 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 constant 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 no information 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 gives rho_0 - zeta_F(rho_0)/zeta_F'(rho_0), Taylor at a simple zero of zeta_F. The offset is one complex number, so at the zeros this law pairs the ordinate offset and the real-part offset are one quantity. 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 over the ladder at level = 1, 2, 3, so it adds no constant the fibre does not give. Witness: lab/py/zeta-shadow verb mass, lab/py/mrly-euler verb position.
• 2026-09-07 [Verified] The constant-free first-order 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 are computed 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 from rho_0, 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, 0.5, 0.630929754, 0.792481250, 0.861353116, 0.903089987, 0.954242509, 0.982877878, 0.994835739, with largest abs(ratio - 1) 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 over the ladder 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 with largest abs(ratio - 1) 0.24964 and 0.0821168 at the two dense rungs, five times looser than the step at base 50, and the coupling zeta_F'(rho_0)/zeta'(rho_0) does not select it either, median abs(coupling - fill/base) reading 0.24057225 and 0.08291158 there against median abs(coupling - 1) 0.27619434 and 0.079335871, a flip between the two rungs while the candidates differ only 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 rungs' 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, so E_F = 0 and both offsets are 0. Witness: lab/py/zeta-shadow verb predict.
• 2026-09-07 [Verified] The paired shadow offset carries its exponent in the missing-digit density rather than in 1 - alpha, and two new rungs sample the interval between alpha = 0.954 and 1. The median paired offset divided by m/base = 1 - fill/base reads 1.6463532, 1.2495026, 1.8345578, 1.5731321, 2.2102406, 1.6634381, 2.2424916, 1.8779239, 1.051349 across the nine rungs and divided by 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 over this ladder, which is the whole discrimination the two normalisations admit here. The new rungs are base 20 missing its top digit at alpha = 0.9828778777 and base 50 missing its top digit at alpha = 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, so the PAIRED offset falls fast across that interval; this bounds no maximum over the whole second family and touches no jump clause, since the pairing selects zeros for closeness to a zeta zero and censuses nothing. Read in the form of the family row, the mean distance from a located design ordinate to the nearest zeta ordinate over a quarter of the mean gap between consecutive zeta ordinates in the range 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's is design-zero-first, so this is a parallel ladder and not that row recomputed. Witness: lab/py/zeta-shadow verb rungs.
• 2026-09-07 [Proved] A positive Rouche margin proves exactly one zero of the design zeta in a disc about a pole, with every input bounded from the digit recursion itself. Write Z(s_0+u) = P(u) + T(u) at a pole s_0 = s_(0,j) with nonvanishing residue, where P(u) = (1 - base^(-u)) D_(P-1)(s_0+u) + E_P(s_0+u) is entire with Taylor coefficients the exact finite sums sum_n n^(-s_0)(-log n)^m/m! convolved against those of 1 - e^(-L u), and T is 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 with G the peeled majorant. That l sum is closed by a majorant ratio and not by an observed one, the term ratio itself not being monotone: since 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), the term ratio is at most R_l = ((abs(s_0)+R_2+l)/(l+1)) a_max base^(-P), which decreases in l once abs(s_0)+R_2 >= 1 and is below a_max base^(-P) otherwise, so stopping at the first l with R_l < 1 and adding term_l R_l/(1-R_l) is a proof. Then abs(Z_n) <= B_T/R_2^n for n >= 2 beyond the explicit part, so on abs(u) = rho one has abs(Z - (Z_0 + Z_1 u)) <= sum_(m >= 2) abs(P_m) rho^m + B_T tau^2/(1-tau) with tau = rho/R_2, while abs(Z_0 + Z_1 u) >= abs(Z_1) rho - abs(Z_0); when the first is strictly less than the second the linear model and Z have the same zero count in abs(u) < rho by Rouche, and that count is one because abs(Z_0/Z_1) < rho follows from the same inequality. Since the residue does not vanish, Z(s_0) != 0 and the zero is a zero of zeta_F. No step uses a differenced quantity: Z_0 is the ladder value with its propagated bound and Z_1 is the first Fourier mode of T on a circle of radius R < R_2 with N samples, whose aliasing is at most (B_T/R_2)(R/R_2)^N/(1-(R/R_2)^N), plus an exact p_1. The peel depth P and the radii rho and R_2 are free parameters of the proof. Witness: lab/py/zeta-locus, lab/py/design-zeta.
• 2026-09-07 [Verified] The residue comb carries exactly one zero of the design zeta at eleven certified poles, the peel depth is the lever that decides which, and the certificate fails at every pole carrying none or two. Running the Rouche margin with the peel depth raised at each pole until the certificate fires or the string pool caps, over 106 poles at base 3, base 5, base 9, base 16 and base 10 missing 9 to Im s = 40 inside a fifteen minute budget, gives 11 certified, 60 failed, 7 residue-null and excluded because there the model's zero is the pole centre, a zero of the cofactor that is not a zero of zeta_F, and 28 skipped when a design spent its budget. The certified eleven, with depth, margin and the radius the proof used: base 3 {0,1} j = 2 at P = 7, 0.13418242, rho = 0.205; j = 5 at P = 7, 0.028140545, rho = 0.16; j = 7 at P = 9, 0.00082974181, rho = 0.175; base 5 {0,1} j = 4 at P = 7, 0.15035818, rho = 0.2775; j = 5 at P = 7, 0.12269904, rho = 0.295; base 9 {0,1,2} j = 5 at P = 5, 0.038456894, rho = 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 and 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 count of one and none disagrees; of the 19 poles carrying zero or two zeros in abs(u) < 0.45 that the budget evaluated none is certified, the two double poles reached, base 3 {1,2} j = 3 and j = 5, both failing, while base 5 {1,2} j = 7 and base 10 j = 15 were skipped for budget. Base 10 is not closed by any sharper majorant but by peeling: at the automatic depth P = 2 its B_T runs 1.08 at j = 1 to 38.1 at j = 15, and at P = 3 it runs 0.2096 to 1.2010 over the eight poles reached, five of which certify. Proximity of the tooth is no threshold, the certified abs(Z_0/Z_1) running 0.0282669 to 0.149708 and base 3 {0,1} j = 7 at 0.104443 failing at P = 7 and certifying at P = 9. The margins are evaluated in high precision and not in ball arithmetic, which is the one step between this row and Proved. Witness: lab/py/zeta-locus.
• 2026-09-07 [Refuted] The locus of the zeros of the design zeta is no curve Re s = f(Im s) shared by designs of equal alpha, no comb in the pole-period residue, and no law in alpha and fill/base. The witness against a shared curve is a pair of zeros of nearly equal imaginary part and very different real part on two designs of equal alpha, which a single curve cannot carry: 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; base 4 {0,1} against {2,3}, equal in alpha and in fill/base, gives 0.0136014 against 0.719693 near Im s = 17.64; base 4 {0,1} against {1,2} gives 0.0063091 against 0.280397 near Im s = 22.87 and base 4 {1,2} against {2,3} gives 0.00638631 against 0.198012 near Im s = 31.79, each pair drawn from censuses of the same box and the same height. Equality of both alpha and fill/base therefore fixes nothing. Within one design the worst real-part gap between two zeros of equal frac(Im s log base/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, and the zeros per period at alpha = 1/2 reads 1.1897445 at base 16, 1.2868204 at base 9 and 1.586326, 2.0395621, 2.0395621, 2.2661801 at base 4, so no counting law in alpha alone survives. The single exception is alpha = 1, where the full digit sets at base = 2, 3, 4 are one arithmetic object and do share every zero. Witness: lab/py/zeta-locus.
• 2026-09-07 [Refuted] The second family of the design zeta is not symmetric about any vertical line Re s = c_F. 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 second-family zeros within the 0.05 test tolerance in Im s, and the smallest ordinate gap inside a design is far above that on every design tested, so the branch cannot fire at all. Every pair the sweep records is a self-pair, a real part landing within 0.05 of c_F, and self-pairs occur below the chance rate: over the ten designs recensused 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 over the full sweep 22 of 117. 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. Witness: lab/py/zeta-family verb symmetry.
• 2026-09-07 [Refuted] There is no counting law for the second family in alpha or in fill, at either assignment 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. The subject of the spread 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 Verified and not Proved. Witness: lab/py/zeta-family verbs tests and count.
• 2026-09-07 [Refuted] The real parts of the second family do not contract to alpha/2 as a design fills, so the critical line is not the alpha -> 1 limit of MrlyMath. The refuted law is that max abs(Re s - alpha/2) -> 0 as alpha -> 1. Undivided, that statistic stays flat along the ladder carrying alpha toward 1, reading 0.2275679549 at base 5 {0,1} with alpha = 0.430676558, 0.5549589411 at base 4 {0,1} with 0.5, 0.5885444877 at base 3 {0,1} with 0.630929754, 0.4233198337 at base 4 {0,1,2} with 0.792481250, 0.5365616661 at base 5 {0,1,2,3} with 0.861353116 and 0.3151426744 at base 10 missing two with 0.903089987, 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, alpha = 0.954242509, 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. Witness: lab/py/zeta-family verb limit.
• 2026-09-07 [Refuted] The ordinate shadow does not explain why a design's ordinates converge before its real parts, because at the zeros it pairs it separates neither. The first-order offset is one complex number, so for a paired zero 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 missing its top digit, and rung by rung the medians median abs(Im off) against median abs(Re s - 1/2) read 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 at alpha = 0.430676558 to 0.994835739, the ordinate offset larger on six rungs and smaller on three with both falling along the ladder. The law binds only the zeros Newton reaches from a zeta zero inside 1.5 of it and this lab enumerates no design zero, so it neither explains nor forbids what a design-zero-first census reports; the census contrast and this row are both consistent with a mixture in which the partnered zeros approach in both coordinates while the rest of the second family does not approach at all, and that mixture has no witness until the unpartnered count is measured. Witness: lab/py/zeta-shadow verb rungs, lab/py/zeta-family verb limit.
• 2026-09-14 [Verified] The peeled continuation of a digit-design Dirichlet series runs in double precision inside the public crate. mrlynum::ladder carries Design, zeta, cofactor and residue on mrlynum::design::elements and mrlynum::zeta::Complex, and returns every value beside a bound. Against the arbitrary-precision lab at the base 2 full set the gaps are 7.4e-11 at s = 2, 4.6e-14 at s = 0.3 + 40i and 7.5e-12 at s = -1 + 2i against reported bounds 3.8e-10, 1.3e-11 and 6.8e-10; the base 3 residues meet the certified enclosures to 3.8e-15 against bounds near 8e-14. Five adversary breaks are repaired and no pinned number moved. Witness: mrlynum::ladder, lab/py/design-zeta.
• 2026-09-14 [Proved] The carried scale of a double-precision ladder majorises every intermediate magnitude of the value recursion, term by term. Beside the propagated truncation bound the module carries scale_j = (sum_(n in E_P) n^(-Re w) + cut + sum_l abs(binom(-w,l) base^(-w-l) gamma_l) scale_(j+l)) / abs(1 - fill base^(-w)). The induction is immediate: the base entries start at tail >= 0, poly_scale(E_P, Re w) >= abs(poly(E_P, w)) term by term, and every step applies the same nonnegative weights and the same divisor modulus, so scale_j >= abs(value_j) at every level. Witness: mrlynum::ladder.
• 2026-09-14 [Verified] The rounding charge of the double-precision ladder is measured and not counted, and it holds with a factor of fifteen to spare. The reported bound is truncation + ROUNDING * scale with ROUNDING = 1e-13; a count of about 75 roundings over up to 115 levels gives 9.5e-13, an order above the constant, so the charge is not a standalone bound. Measured over four designs at 28 points each plus the real axis the worst ratio of true error to returned bound is 0.0667, at base = 3, F = {0,1}, s = 0.5 + 40i, and the error never exceeded the bound anywhere probed; on a rejected rung the truncation half carries the bound and that ratio reaches 451. Witness: mrlynum::ladder.
• 2026-09-14 [Verified] The wall of the double-precision port is cancellation and not truncation, and it is visible at Re s = -1. At s = -1 + 2i on the base 2 full set the truncation bound falls to 1.1e-20 at shift 12 and to 7.8e-206 at shift 114 while the carried bound sticks at 5.6e-10, because the peeled tail G_P(-1+2i) has modulus 5.6e3 against zeta_F(-1+2i) of modulus 0.183, a loss of four and a half digits; the module raises rather than returns at any tolerance under that. The lab reaches 2.8e-32 there only by lifting the working precision with the height. Witness: mrlynum::ladder, lab/py/design-zeta.
• 2026-09-14 [Proved] The residue column below a pole of a digit-design zeta is a finite recursion in the peeled variables, Burnol's Proposition 7.1 in peeled form. With R_m the residue at s_(m,j) = alpha - m + 2 pi i j / log base, taking residues in the peel identity at w = s_(m,j) kills E_P and leaves only the terms whose shifted argument is a pole: (1 - base^m) R_m = sum_(l = 1)^m binom(-s_(m,j), l) base^(-s_(m,j)-l) gamma_l R_(m-l) with R_0 the numerator over log base, since fill base^(-s_(m,j)) = base^m. At base 3 on F = {0,1} the m = 1, j = 1 value 0.6950303416383606 + 0.37779086109705695i meets the eight-node contour average on a circle of radius 0.05. Witness: mrlynum::ladder, REFS.md Burnol 2026.
• 2026-09-19 [Refuted] The pole lattice does not force the zeros of a design zeta: at base 3 {0,1} the two polished zeros right of the abscissa are 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, because the shifted terms sum_(l >= 1) binom(-s,l) base^(-s-l) gamma_l zeta_F(s+l) of the digit recursion are not 2 pi i/log base periodic although 1 - fill base^(-s) is. Witness: lab/py/zeta-locus verb census.