Instability of Einstein warped products and a one-sided Weyl curvature gap

Let Ric_g = Λg, where Λ > 0. For Einstein warped products of closed manifolds, we prove that the least transverse-traceless Einstein eigenvalue is strictly less than −(n−2)Λ/(n−1), without an upper bound on the dimension. The same estimate holds for the smooth spherical-fibre completions specified here, unless the completed metric is round. A trace-free Hessian correction expresses the shifted quadratic form as a negative sum of squared base-curvature tensors and admits an H¹ extension across the collapsing fibres. In dimension four, there is a universal positive gap for ‖(λ_max(W⁺)−2Λ/3)_+‖_{L²} on closed simply connected oriented positive Einstein manifolds. Below this gap the excess vanishes, and the metric is anti-self-dual or Kähler–Einstein in the given orientation. The proof uses Einstein orbifold compactness, extension of parallel real line bundles through anti-self-dual ALE limits, and LeBrun–Ozuch’s Kähler–Einstein desingularization theorem. For circle-warped four-metrics, the corrected quadratic form equals a negative multiple of either chiral Weyl energy. The small-excess TT-semistable metrics are the round sphere and the Fubini–Study plane. Zenodo publication date: 1 September 2026. DOI: https://doi.org/10.5281/zenodo.22851382 Author page: https://anassnifa.com/papers/einstein-instability-weyl-gap/