Sharp Growth Exponents for Positive Robin Eigenvalues at Fixed Volume

Fix a positive Robin parameter alpha and a volume M. We prove that the kth Robin eigenvalue of every bounded convex domain in R^n with volume M is at least a k^(2/(2n-1)), where a>0 depends only on n,alpha,M. The proof combines a one-dimensional trace estimate, a John ellipsoid and convex-domain Poincare cube counting. For general bounded Lipschitz sets, the bound b k^(1/n) is recorded as a direct consequence of van den Berg and Bucur's established 2014 heat-trace estimate, with explicit constants and attribution. Thin boxes and unions of equal balls show both exponents are optimal over their respective classes; dimension one is also covered. This answers the two-part Euclidean Robin conjecture on p.2094 of Oberwolfach Report36/2023. The separately bundled manifold question in UnsolvedMath record OWR-14297736-026 remains unaddressed. This English preprint includes an AI-assisted analytic proof, source comparison, self-audit and supplementary exact/numerical checks. No optimal-constant, extremizer, historical-priority, independent-review or formal-verification claim is made.