A Sharp Face Theorem for Robust Stability of Quaternionic Polynomial Polytopes

The classical edge theorem reduces robust stability of a real polynomial polytope to its edges. For polynomials with quaternion coefficients, even stability on every two-dimensional face can miss an unstable member. We prove the sharp replacement: a real-affine polytope of globally fixed degree is robustly Hurwitz or Schur if and only if every member on its faces of real dimension at most three is stable. Equivalently, testing all subfamilies spanned by at most four generators suffices. Both bounds are independent of degree and sharp already for quadratics. The proof uses last contact in a four-dimensional value set. For monic polytopes the criterion gives uniform exponential stability of the companion systems. To verify the reduced families, we construct local Bernstein certificates; refinement succeeds for every strictly robust family, with rational certificates for rational data. A fixed PD controller illustrates the five facet tests for a mechanical plant with correlated damping and gyroscopic uncertainty. Local subdivision resolves a false alarm in a cubic family. Author preprint of a manuscript submitted to Automatica. Submitted manuscript; not the version of record. Zenodo preprint version 1.0. This work was funded by the European Union under the project ROBOPROX (reg. no. CZ.02.01.01/00/22_008/0004590). The MATLAB examples use qstab: Quaternion Stability Toolbox, version 0.18.0, archived at https://doi.org/10.5281/zenodo.24086085. The source repository is https://github.com/michaelsebek/qstab-public.