# 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.