# The design Mobius meter

- 2026-09-07 [Verified] The design Mobius meter oscillates at the zeta ordinates and not at the design's 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`, so the pipeline would have seen a lattice and there is none. Witness: lab/py/design-meter verb spectrum, lab/rs/mobius-designs, A084237.
- 2026-09-07 [Proved] The identity `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 the echo's size splits at `alpha = 1/2`. 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 the constant `sum_n mu(n) A_F(n)/n`, which is nonzero: base 16 `{0,1}` reads `-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}` reads `-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`, so the zeta zeros neither obstruct nor help the square-root conjecture, which is a statement about `R_F` alone. Verified separately at two designs: 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. Witness: lab/py/design-meter verb spectrum, mobius.md.
- 2026-09-07 [Refuted] A family law for the design meter's frequency set: it is not a function of `(base, 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 of mobius.md carries into the spectrum while no `(base, alpha)` law does. Witness: lab/py/design-meter verb family, mobius.md.
- 2026-09-19 [Refuted] The primitive quadratic Dirichlet `L`-zeros of conductor `3`, `4` or `5`, the quadratic conductor each base carries, are no frequency family of the design meter: over eleven designs 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`, so the pipeline would have seen an arc family; the wider prediction over arcs of denominator `base^j` is untestable by this spectrum and is not claimed. Witness: lab/py/design-meter verb spectrum.