A harmonic law for detection thresholds of Gaussian-windowed Hankel ladders at off-critical zeros

Companion archive for the paper A harmonic law for detection thresholds of Gaussian-windowed Hankel ladders at off-critical zeros. Contains the paper (LaTeX source and PDF), the full constant chain for the main theorem, the four experiment scripts, the raw threshold data with SHA-256 manifests, the two sealed predictions with their frozen text and outcomes, and the record of every adversarial audit. Subject. Given a multiset of points on the real line contaminated by one conjugate pair off the line at height b, how expensive is it to detect the intruder using only polynomial moments seen through a Gaussian window? Writing M_k(A,u) for the Hankel matrices of the windowed moments and u_m for the detection thresholds, the main results are: (1) a harmonic law with a universal constant, namely that the normalized threshold m*u_m*g^2 exceeds 2 always and converges to 2, independently of the intruder height and uniformly in the admissible window center; (2) a two-sided detection criterion valid on arbitrary backgrounds, in which a folded-measure Christoffel function at the negative extrapolation point -b^2 gives sufficiency and an exact Schur-complement reduction gives necessity; and (3) a counting bound stating that the number of negative eigenvalues never exceeds the number of distinct intruder pairs, uniformly in all window parameters and truncation orders. The mechanism throughout is a reduction of detection to a classical polynomial separation problem. Scope. This work proves nothing about the Riemann Hypothesis itself. It is a quantitative theory of the detection side of the positivity criteria to which RH is equivalent; the theorems concern synthetic lattice families and general model backgrounds, and the extrapolation to genuine zeta zeros is explicitly exploratory. Authorship and verification. The mathematics, code and text were produced by Claude Fable, an artificial intelligence system developed by Anthropic, working under human direction. No human mathematician has yet verified these results. Three mechanisms were used in place of that verification and their records are included: adversarial audits of every theorem by an independent AI auditor (which found real errors in every first draft), two sealed predictions committed to a version-controlled history before the verifying computations were run (both fired), and an honest-failure log of retracted claims. Readers are invited to check the work; the proofs are short, the constants explicit, and every number traces to an archived file. Code is licensed MIT; the paper, proofs, data and documentation are licensed CC BY 4.0.