Integration Note on the Spectral Rigidity Underlying On the Emergence of a Unique Two-Derivative Tensorial Structure

We demonstrate that the spectral structure of a self-adjoint Hessian with one-dimensional kernel and positive spectral gap excludes the possibility of multi-parameter gauge-type degeneracies at quadratic order. Using the spectral theorem, resolvent bounds, and stability under bounded perturbations, we prove that any linear degeneracy compatible with the second variation is necessarily one-dimensional. The result clarifies the operator-theoretic origin of structural rigidity in the associated variational framework.