Does an operator's spectrum determine it, and what repairs the failure?

Run status: not attempted — no run in lab/runs/ names this case in its settles: field.

Object. A spectrum is not a complete invariant of an operator. Similar matrices A and QAQᵀ share every eigenvalue and differ in every eigenvector; isospectrality is the general phenomenon — "a spectrum forgets the shape" (Milnor's isospectral tori; §8's forgetting hierarchy in docs/rung-vocabulary-v3.54.md). The map operator → spectrum is not injective, so no calculation recovers the operator from its spectrum alone.

The repair. The eigenvector–eigenvalue identity (DPZ) restores what the spectrum forgot, using more spectra: the eigenvalues of the matrix and of its principal minors determine the moduli of the eigenvector entries. Ordinary spectra forget the eigenvectors; minor spectra remember their moduli. Rigidity is recovered — up to a residue.

The residue is a group action. What minor spectra cannot fix is whatever leaves every modulus invariant: a phase/gauge freedom and, in the worked case, a discrete conjugation. For a unitary mixing matrix the residue is exactly one ℤ/2 — the sign of the Jarlskog invariant, CP — at n = 3, and ≥ 2 bits at n = 4 (the "one bit" is a three-generation fact about triangles; see [[N-011]]). A residue of a group action is not spectrally determinable at all — it is resolved only by physically performing the group element (see [[N-012]]).

Why settled, and why a case. The mathematics is standard (isospectrality; DPZ) and the composition was worked this session; there is no open differential, so it settles by argument. It earns a case rather than a dictionary line because two filed artifacts depend on it — [[N-011]] and the rigidity axis in docs/rung-vocabulary-v4-part1.md §8.2 — and it is the sibling of [[CASE-16]] (Mostow rigidity), the other rigidity case here.

Stakes. Whether "a spectrum determines its object" is ever safe to assume. It is not; the failure is named (isospectrality), the repair is named (minor spectra / DPZ), and the residue is named (a group action).