Connected Hankel positivity: two counterexamples and a comparator obstruction

This eight-page AI-assisted expository technical note presents an explicit finite-order Polya-frequency counterexample, a rational conjugate-pair example, and a persistent covariance obstruction to a specified cumulative-source Gram comparator. It distinguishes positive source moments from the nonlinear connected moments of a logarithmic derivative. The constructions combine classical criteria with explicit calculations and are offered as reusable controls for invalid positivity inferences. A short application covers a retained theta-layer class. Section 6 asks five prioritized questions about precise antecedents, correctness, and expository usefulness. The note does not claim a proof of the Riemann hypothesis or establish novelty priority, and it has not undergone human peer review. Drafting, literature comparison, and exact checks were assisted by OpenAI Codex, with additional AI contrast. The accompanying archive includes the unchanged version-3 PDF and LaTeX source, standalone selected rational/symbolic checks, a recorded verification result, and SHA-256 hashes. Computational PASS applies only to those checks; universal statements rest on the proofs and cited results. Correspondence: [email protected].