Tight Worst-Case Bounds for the Smallest Eigenvalue of ReLU NTK Gram Matrices

For n unit vectors x_1,\ldots,x_n \in \mathbb{R}^d, we study the continuous ReLU derivative Gram matrix H, whose entries are obtained by averaging pairwise gated inner products over a standard Gaussian direction. Writing Δ_\pm := \min_{i \neq j} \min{ |x_i-x_j|2, |x_i+x_j|2 } for their projective separation, we prove the universal dimension-free lower bound λ{\min}(H) = Ω( Δ\pm/\sqrt{\log n} ) . Conversely, we construct worst-case families satisfying the matching upper bound λ_{\min}(H) = O( Δ_\pm/\sqrt{\log n} ) , showing that this rate is tight up to universal constants.