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_S2 plus t^(-1)h, where the round sphere has curvature 1 and t>0. 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 times the sum of m_nu sqrt(4/9-2nu) over surface eigenvalues nu<2/9, with an O_h(sqrt(t)) remainder. 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.