Paneitz Resonances and Index Growth on a Sphere Times a Hyperbolic Surface

For a fixed closed connected hyperbolic surface \((\Sigma,h)\) of curvature \(-1\), we determine the full Paneitz spectrum for \(g_t=g_{S^2}\oplus t^{-1}h\), where the round sphere has curvature \(1\) and \(t\gt0\). The exceptional scales with nonconstant kernel form a locally finite set, with explicit resonances of unbounded nullity. An exact finite weighted count gives the negative index. As \(t\) tends to infinity, this index equals \(2t\sum_{\nu\lt2/9}m_\nu\sqrt{4/9-2\nu}+O_h(\sqrt t)\). Eigenvalues at the threshold contribute only through explicit narrow scale intervals. We also obtain square-root nullity bounds and infinitely many conformal classes distinguished by their negative indices. This is a complete result for the specified product family, not a classification on arbitrary manifolds. The Singer–Eastwood equal-scale example and established conformal-index theory are credited. The preprint is AI-assisted, self-audited and unrefereed; historical priority is not certified.