The arithmetic fingerprint of the Davenport-Heilbronn off-line zeros: a registered test

A registered test of a conjecture about the heights t_k of the 1,268 off-line zero pairs of the Davenport-Heilbronn function f below t = 20,000: that the structure factor S(omega) = |sum_k exp(i omega t_k)|^2 / N peaks exactly at omega = log n for the integers n in the support S of the arithmetic defect D (all prime factors congruent to 2 or 3 mod 5), and nowhere else. Predictions were registered before computing (SHA-256 in the paper) and tested against 4,000 gap-preserving shuffles. The conjecture fails as stated: 30 of the 35 integers n in S below 60 carry lines (z from 12 to 616) sitting at log n to within 0.19 of a line half-width, but the six primes congruent to +-1 mod 5 below 60 also carry lines, one to two orders of magnitude weaker (z = 6.9-17); no line appears at any other n outside S, nor at any of 723 fractions involving such a prime, while strong lines appear at the S-rationals log(a/b). The mechanism is Landau's formula split into on-line and off-line zeros: the full Landau sums of f and of L(s, chi) match -(T/2pi) Lambda_f(n)/sqrt(n) to 0.7%, their difference matches the defect term to 0.7%, and the off-line pairs carry a measured share of it. Inputs: the certified census doi:10.5281/zenodo.23676415. Written jointly with Claude (Anthropic); the human author is responsible for the content.