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 $C^2$ 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 $207/100<\sup D_k(s^2)<21/10$, where $g=s^{-2}k$, $s=\operatorname{Scal}_k>0$ and $D_k=\operatorname{div}_k\nabla$. 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.