We study the eigenvalue structure of the degree-2 Lasserre moment matrix M for the Traveling Salesperson Problem (TSP), evaluated at the fractional LP optimum of a family of hard metric instances known as comb graphs. For the comb instance Gh with n=2h nodes, we prove a closed-form characterization of the LP optimum, showing that handle and tip-tip edges receive weight 1/ (h-1) while tooth edges receive weight 1. We prove unconditionally that the moment matrix block-decouples into an integer subspace and a fractional subspace VF via a complete case analysis on metric lifting constraints. This decoupling ensures that eigenvectors in the observed "flat" eigenvalue band are supported entirely within the fractional subspace. We establish that the flat band has dimension exactly h (h-1) -1, growing as Theta (n²) with the instance size. These results are verified numerically for h = 4, 5, and 6, and used to analytically predict the h=7 (N=14) case. We close with open problems regarding eigenvalue scaling laws and the connection between spectral localization and hierarchy complexity. Institutional Note: This research was conducted as part of Project SPEC-Comb at Precision Data Strategies LLC, led by Ronald Parent Jr. https: //precisiondatastrategies. com
Ronald Parent Jr (Fri,) studied this question.