Einstein link spectra and the Morse index of Ricci-flat cones

We study the low transverse-traceless spectrum of positive Einstein manifolds and its effect on the Morse index of Ricci-flat cones. Free quaternionic quotients of four equal round spheres have exactly determined first scalar and tensor eigenvalues. For four round two-spheres, the quotient cone is nine-dimensional, is stable and has restricted holonomy SO(9), whereas its eightfold cover has infinite negative index. For a general Einstein link, its invariant eigenspaces below the Hardy threshold determine the exact index on cone annuli and the leading logarithmic accumulation of negative eigenvalues on complete Ricci-flat manifolds with weighted C² convergence to the cone. No convergence rate is required for the leading term. At the threshold, an error of order o((log r)⁻²) makes cone stability equivalent to finite index. A pointwise comparison for canonically conformally Kähler Einstein four-manifolds gives, for the Chen–LeBrun–Weber metric at Einstein constant 3, the inequality 207/100 < sup Dₖ(s²) < 21/10, where g = s⁻²k, s = Scalₖ > 0 and Dₖ = divₖ ∇. This proves the Hall–Haslhofer–Siepmann derivative inequality and improves the low tensor eigenvalue bound. Applications include an explicit leading coefficient for the complete Ricci-flat Böhm metrics, an index formula on long Einstein necks, and Euclidean rigidity for finite-index five-dimensional fillings under a uniform half-Weyl eigenvalue inequality. First published on Zenodo: 3 September 2026 (v1). DOI: 10.5281/zenodo.22914003.