Hidden higher-order vulnerabilities in simplicial complexes revealed by fixed-index spectral robustness

Robustness of simplicial complexes under triangle deletion is often characterized by the instantaneous smallest positive eigenvalue of the Hodge 1-Laplacian. We show that this observable can change spectral index when the kernel grows, and therefore it need not follow the same nonharmonic edge-space eigenmode along the deletion process. To remove this ambiguity, we define fixed-index spectral robustness by monitoring the eigenvalue whose index is selected by the first nonzero eigenvalue of the intact complex. The corresponding triangle sensitivity follows from first-order perturbation theory and identifies simplices that produce the steepest initial decrease of this monitored eigenvalue. Across synthetic and empirical clique complexes, a small fraction of triangles can drive the monitored eigenvalue to zero while the 1-skeleton and graph-level robustness measures remain unchanged. These results show that higher-order robustness is a genuinely simplicial spectral property that cannot be inferred from graph-level connectivity alone.