Eigenvectors, phases, and ranks

If $v_0=b>0$ is real and $\Vert v-{\beta }^{\prime }\Vert \le e<b$, the first exact coefficient is nonzero. Rotating to the frozen positive-real phase gives

The first coefficient lies in the radius-$e$ disk centered at $b$. A later anchor also requires exact zeros for preceding coefficients.

For isometries $V,\hat{V}$, with $\Vert V-\hat{V}\Vert \le v$,

Insert mixed compressions to prove these estimates. Arithmetic errors are additional. Normalize eigenvectors rigorously; FLINT's left normalization $L=R^{-1}$ is not Euclidean orthonormality of $R$.

For a Hermitian $2\times 2$ matrix

Directed interval evaluation yields diagnostic eigenvalue bounds. If $\Vert \tilde{F}-F\Vert \le e_F$, then

Interior occupation rank two follows from a positive validated lower bound for ${\lambda }_{\min }(V^{†}{P}_{m}V)$. Endpoint rank one follows from positive validated trace and structural rank at most one. If rank intervals remain,

Positive floating-point singular values alone do not certify ranks of the exact frozen code.