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.