Exact inputs and routine guarantees

Use the exact integer matrices and denominator in (eq:integerH), not a decimal reconstruction.

NumPy's returns ascending approximate eigenvalues and normalized eigenvector columns, uses a selected triangle, and treats imaginary diagonal parts as zero NumPyEigh. It does not validate selection or audit full Hermiticity.

The supplied FLINT documentation distinguishes FLINTMat:

1. approximate QR eigenvalues, with heuristic convergence only;

2. global spectrum coverage, without a one-to-one ball assignment;

3. successful all-simple isolation;

4. successful exhaustive counted clusters;

5. local eigenpair or invariant-subspace enclosures.

A counted cluster does not prove exact degeneracy. A local “at least $k$” statement is not an exhaustive selection certificate. Outputs are not automatically energy-sorted.

The documented eigenvalue routines are experimental and may fail to converge as precision increases. Archive the actual installed version or commit, precision, return status, and guarantees used. No FLINT invocation is reported in the supplied P8-new record.

Approximate routines ignore radii; zero-radius numerical outputs are not exact-solution certificates. Rigorous routines must establish certificate premises. The documented infinity-norm bound is a row-sum bound, whereas

or $\Vert X\Vert \le \sqrt{{\Vert X\Vert }_1^{\mathrm{ind}}{\Vert X\Vert }_{\infty }^{\mathrm{ind}}}$ suffices for the operator norm here.