Text: Toroidal grid with chords is not spectrally rigid under URF-SG admissible perturbations.
Operator A: Graph Laplacian on toroidal-chord graph.
Perturbation V: Any translation-equivariant local operator.
Gap: Generic consecutive gaps.
ΠVΠ: forced ≠ 0 by abelian symmetry and representation multiplicity.
Status: VERIFIED (non-rigid; representation-theoretic obstruction).