a spectrum forgets the shape (isospectrality); minor spectra remember the eigenvector moduli (the eigenvalue–eigenvector identity) — rigidity recovered up to a residue

The shape: an invariant that looks complete forgets something (a spectrum forgets the eigenvectors — isospectral drums, A vs QAQᵀ), and the repair is not a different tool but more of the same invariant (minor spectra recover the moduli), restoring rigidity up to a residue that is a group action ([[EX-05]]). Calibration question: when an invariant fails to determine its object, is the fix a richer version of the same invariant — and what residue is left?