When Is Muon Optimal? Radial Fragility and Minimax Geometry

Ao Xu

Muon replaces a matrix gradient by an approximate polar factor and has shown strong empirical performance in neural-network training. The Isotropic Curvature Model (ICM) provides a curvature-based explanation for this update by asking when gradient orthogonalization minimizes a one-step local-loss surrogate. We investigate the stability of this explanation when the fixed input radius of the ICM is replaced by a general radial law. We introduce the Random-Radius Isotropic Curvature Model (RR-ICM) and the Polar-Factor Admissibility Set, which encodes exact scaled-polar optimality as a geometric condition on the positive polar factor of the gradient. We derive an exact characterization of this set for arbitrary radial laws and show that all nonscalar admissible responses arise from radial atoms aligned with curvature kinks. Consequently, every atomless radial law exhibits zero-capacity rigidity: exact scaled-polar optimality is possible only when the gradient is already a scaled column isometry. To connect the curvature-based framework with the established spectral geometry of Muon, we introduce the Minimax RR-ICM by replacing directional averaging with worst-case optimization over…