The Second-Order Threshold for Standardized Laplacian Eigenvalues

Chebotarev conjectured that at and above a chord-weight threshold, each pair of conjugate Fourier modes contributes nonpositively to the second-order outward displacement of a circulant standardized Laplacian eigenvalue. We prove the inequality for every admissible order and show that it is strict above the threshold. The proof reduces the sum of coefficients to a weighted average of two real rational terms. A trigonometric bound on the product of their arguments determines the sign. The strict inequality also gives a local bound under perturbations of all arc weights, with equality only for the nearby circulants with uniform chord weights.