Complete polynomial inequality and bounded nonnormal-matrix evidence

Family 325. Edition: 9 October 2026. Source metadata, pinned revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb.

The direct main manuscript's 623 lines, selected 24-line scope, 44-line Comparator, 15-line config, 41-line import entry, 119-line bound and 95-line sharpness texts were read. Full imports/kernel/source program/independent proof and companion structural manuscript remain open. The finite statement uses arbitrary complex matrices with positive dimensions, Euclidean operator 2-norm and sum A^k tensor B_k. It claims sharp constant two on the entire numerical range.

Potential software is a polynomial-error/model review module, a repeated-filter enclosure framework or a matrix-coefficient SDK. Existing numerical linear algebra/direct norms and normal-specific analyses are strong baselines. Tiny exact cases already admit better independent direct evidence, so no source speedup or paying use is demonstrated.

The new reference supplies exact finite threshold evidence independently of the source theorem; twenty rational support halfspaces and covered-cell Lipschitz/Frobenius bounds give a separate entire-domain upper. 857 exact finite controls and 195 existing SciPy diagnostics passed. The sharp nilpotent example's conditional bound fits 2.2 while the rounded prior bound does not, but direct exact norm 2 already suffices. This is a bounded model/evidence experiment, with low commercial confidence. Revised dossier.

Next: test reuse across an actual polynomial error workflow with uncertain-input/physical-model bridges, baseline cost and independent source acceptance. Do not infer nonlinear stability, rational-pole safety, general runtime or profitability from the inequality.