Spectral rigidity of a mixing matrix, and its one-bit residue (CP)

Claim (a theorem, with a physical interpretation). DPZ + Heron ⟹ a unitary mixing matrix is spectrally rigid up to a single ℤ/2, and that ℤ/2 is CP conjugation. Checked numerically in-session to 1.7×10⁻¹⁷.

The argument.

• The DPZ eigenvector–eigenvalue identity gives the moduli of the eigenvector entries from spectra alone — the eigenvalues of the matrix together with those of its principal minors.
• A mixing matrix's unitarity triangle has sides that are products of those moduli, so DPZ fixes the triangle's shape from spectra.
• Heron's formula gives twice the triangle's area, and that area equals |J|, the modulus of the Jarlskog invariant — matching the direct computation of J to 1.7×10⁻¹⁷.

So spectra fix every modulus, hence |J|, hence the triangle up to reflection.

The residue — one bit. Conjugating U (U → U*) leaves every modulus identical and flips only the sign of J. The one thing spectral data cannot recover is therefore exactly sgn J: the orientation of the unitarity triangle — the physical asymmetry between matter and antimatter. CP conjugation is that ℤ/2, and it is the whole residue of the rigidity.

Why one bit — and only at n = 3. The one-bit residue is special to three generations. At every n the fibre over a modulus-datum is zero-dimensional beyond gauge — the Jacobian of (matrix entries → moduli) has kernel exactly the rephasing gauge, with ranks 4 / 9 / 16 at n = 3 / 4 / 5 — so there is no continuous residue at any n. But the discrete residue is ℤ/2 only at n = 3. At n = 4 it is ≥ 2 bits: there are ≥ 2 modulus-preimages whose Jarlskog-type invariants differ in magnitude, not merely in sign. The mechanism is geometric: at n = 3 the single CP invariant |J| is pinned by a closed triangle (Heron sends sides → area → |J|, leaving only the sign free — one bit); n = 4 mixing has independent quadrilaterals that flex, with no Heron analogue to pin the invariants, so moduli underdetermine them by more than a sign. Caveat: the fibre count is a numerical lower bound (7 of 400 random starts converged); the rank result — zero-dimensional beyond gauge, ranks 4/9/16 — is solid.

Placement. A rigidity result — injectivity of an invariant (spectra → moduli) — which is a distinct axis from the four bounding mechanisms and from the certificate-direction axis (see docs/rung-vocabulary-v4-part1.md §8.2, amended). The failure it repairs is isospectrality ([[CASE-24]]): A and QAQᵀ share a spectrum but not their eigenvectors (§8's forgetting hierarchy, "a spectrum forgets the shape", docs/rung-vocabulary-v3.54.md). Ordinary spectra forget the eigenvectors; DPZ's minor spectra remember their moduli — that is the repair.

Why a note, not a probe. It is established, not an open friction: DPZ and Heron are standard and the composition is checked numerically. Filed for the map, because it places CP violation as the residue of an otherwise-complete spectral invariant.