design-meter

• Reads the oscillation spectrum of the digit-design Mobius meter M_F(x) = sum of mu(n) over n in S_F, n <= x, with F the kept digits, normalised by x^(alpha/2) with alpha = log_base fill, and asks which frequencies it carries.
• The full set is the control: M(x)/sqrt(x) to 10^8 reproduces the nontrivial zeta zeros, so any frequency a design does or does not carry is measured against a pipeline that provably sees zeros.
• Three candidate frequency families are tested per design: the zeta ordinates gamma, the design's pole lattice 2 pi j / log base, the ordinates of the primitive quadratic Dirichlet L-function of conductor base, plus the alpha-rescaled zeta ordinates and two rigid displacements of the zeta list as null controls.
• Every series is resampled uniformly in log x on 32768 points by exact step evaluation, Hann-windowed, mean-removed, and read as gamma = 2 pi f from the real FFT power spectrum; the noise floor is a 101-bin running median, so a score is power over local floor.
• A family is scored by the mean of log10 of its targets' scores against a null of 4000 rigid circular shifts of the same target list inside the band 4 < gamma < 60; the null therefore keeps the family's own spacing statistics and only moves it.
• Two controls run on every design at the same depth: the counting function A_F(x)/x^alpha on the same elements, and a sign vector supported exactly where mu is nonzero. The first says whether a lattice would be seen; the second says what a meter with no arithmetic looks like.
• The split test cuts the log range in half and correlates the two disjoint halves' spectra, calibrated against 24 support-matched random-sign draws; the family verb reports, for each design it compares, the peak count reached by eight support-matched random-sign meters, which is the null the peak counts are read against.
• The mean-field split defines R_F by M_F(x) = sum of mu(n) A_F(n)/n + R_F(x) and reads the spectrum of all three terms, which separates a design's own frequencies from the classical Mertens function seen through the design's density; it also prints the echo's share of the meter at three depths against the rate x^(-min(alpha, 1 - alpha)/2), and prints two designs below alpha = 1/2 where the echo converges instead of decaying.

RUN

• uv run python research/lab/py/design-meter/design_meter.py sieve
• uv run python research/lab/py/design-meter/design_meter.py spectrum
• uv run python research/lab/py/design-meter/design_meter.py family
• About fifty seconds, three minutes and two minutes; prints only, writes nothing. Peak memory is the 10^8 linear Mobius sieve.
• sieve enumerates each design and factors every element exactly (trial division to the cube root with a shrinking survivor set, then a perfect-square test and deterministic Miller-Rabin on the twelve witnesses 2..37), and asserts the meters against an independent generator. spectrum is the full read. family is the equal-alpha comparison across bases.

READS

• Control, base 10 full set to 10^8: M(10^8) = 1928, which is A084237; bin width 0.3544; all ten of the strongest peaks fall within one bin of a nontrivial zeta zero, offsets 0.018 to 0.196.
• Base 10 with the digit 9 missing, A_F = 43046721 elements, alpha = 0.954243: all six of the strongest peaks fall within one bin of a zeta zero, offsets 0.068 to 0.216.
• The same design split by the density echo: the echo carries six of six top peaks at zeta zeros, the residual R_F carries none of the two it has, and the echo is 0.1342 of the meter in root mean square against 0.6476.
• The echo's share decays with depth: 0.356028, 0.242495, 0.207229 at base 10 missing 9 and 0.208549, 0.099001, 0.047902 at base 3 {0,1}, against the rate x^(-min(alpha, 1 - alpha)/2); the printed share over prediction reads 1.0000, 0.7387, 0.6846 and 1.0000, 0.9219, 0.8663, so each design decays at least as fast as the rate. The generator prints ratios and never an exponent, and the share is measured against the meter, so reading it as a rate against x^(alpha/2) assumes the square-root conjecture.
• Base 16 {0,1} to 10^7, alpha = 0.25, 63 elements: the echo sum mu(n) A_F(n)/n settles near -0.11, reading -0.0937, -0.1330, -0.1242, -0.1051, -0.1099 at 10^3 to 10^7 against x^(alpha - 1/2) reading 0.1778, 0.1000, 0.0562, 0.0316, 0.0178, the ratio climbing 0.53, 1.33, 2.21, 3.33, 6.18. Base 10 {0,1}, alpha = 0.301030, 128 elements: -0.0500 at 10^7 against 0.0405, ratio 1.24.
• Below alpha = 1/2 the tail sum_(m in S_F) H(m-1) converges absolutely, so the echo tends to a nonzero constant and O(x^(alpha - 1/2 + eps)) is false there; the surviving rate against x^(alpha/2) is x^(-alpha/2).
• Base 3 F = {0,1} at level 14, 16, 18, 20, 22: the zeta family scores 0.528, 0.621, 0.693, 0.742, 0.759 against nulls 0.167 to 0.302, every rung at p <= 0.008; the same list displaced by +2 scores 0.187 and reflected scores 0.445.
• The pole lattice on the meter of base 3 {0,1}, base 3 {0,2} and base 5 {0,1} scores -0.592, -0.640, -0.734 against nulls 0.30, 0.27, 0.23, below its own null; on the counting function over the identical elements it scores 3.602, 3.764, 3.973 against nulls 0.72, 0.95, 1.15.
• Base 3 {0,1} against base 9 {0,1,2,3}, equal alpha = 0.630930 and equal element count 1048575: ten peaks against none, where support-matched random-sign meters on the same two supports reach 0 to 4 and 0 to 2 peaks over eight draws each. The spectra correlation printed beside it, -0.047, runs against no null here and carries nothing; the peak counts are the read.
• Base 9 {0,1,3,4} is base 3 {0,1}, its digits being the base-3 pairs 00, 01, 10, 11, so the generator asserts the elements and the mu values equal and the printed correlation 1.0000 is an assertion of that identity and not a measurement.

CHECKS

• Meters asserted against the exact census of mobius-designs: base 3 {0,1} reads (11, 105), (149, 173), (-30, 312), (496, 539), (1009, 1089) at level 14, 16, 18, 20, 22, base 3 {0,2} reads (-382, 485) at level 20, base 3 {1,2} reads (-1461, 1582) at level 18, base 10 missing 9 reads (2181, 5234) at 10^8, each as (M_F, max abs M_F).
• The two mu paths agree: exact factorisation of a design's elements against the linear sieve.
• The L-function zero finder is checked by its own output, the first ordinates reading 8.0397 at conductor 3, 6.0209 at conductor 4 and 6.6485 at conductor 5, each with real part 0.5 to the printed precision.

NOTE

• A design has far fewer samples than the full set at the same depth, so resolution comes from the log range and not the element count; the bin widths here are 0.2579 to 0.3871 against the control's 0.3544.
• The family score against its shift null is deterministic to the printed digits; the p beside it is Monte Carlo over 4000 shifts and moves by a factor of two in the tail between runs, so a claim rests on the score and its null, never on the p alone.
• A score of 3 over the local median is not a detection: for exponential noise it fires on one bin in eight, and the printed base rate per design says so. The threshold here is 8, and no claim rests on a single peak.
• The lattice band starts at gamma = 4, below the first lattice line of every base read, so a lattice is inside the band it is tested in.