Quantitative Edge Eigenvector Universality for Random Regular Graphs: Berry-Esseen Bounds with Explicit Constants

We establish the first quantitative Berry-Esseen bounds for edge eigenvector statistics in random regular graphs. For any d-regular graph on N vertices with fixed d ≥ 3 and deterministic unit vector q ⊥ e, we prove that the normalized overlap √N q, u₂ satisfies \[ x ∈ R |P(√N q, u_2 ≤ x) - Φ(x)| ≤ C_d N-1/6+ε \] where u₂ is the second eigenvector and Cd ≤ C̃d³ε⁻¹⁰ for an absolute constant C̃. This provides the first explicit convergence rate for the recent edge eigenvector universality results of He, Huang, and Yau {HHY25}. Our proof introduces a single-scale comparison method using constrained Dyson Brownian motion that preserves the degree constraint H̃ₜe = 0 throughout the evolution. The key technical innovation is a sharp edge isotropic local law with explicit constant C(d,ε) ≤ C̃dε⁻⁵, enabling precise control of eigenvector overlap dynamics. At the critical time t_* = N-1/3+ε, we perform a fourth-order cumulant comparison with constrained GOE, achieving optimal error bounds through a single comparison rather than the traditional multi-scale approach. We extend our results to joint universality for the top K edge eigenvectors with K ≤ N1/10-δ, showing they converge to independent Gaussians. Through analysis of eigenvalue spacing barriers, critical time scales, and comparison across multiple proof methods, we provide evidence that the N-1/6 rate is optimal for sparse regular graphs. All constants are tracked explicitly throughout, enabling finite-size applications in spectral algorithms and network analysis.