Muon as a Kronecker-Factored Curvature Approximation without Accumulation

Kronecker-factored methods such as K-FAC, Shampoo, and SOAP approximate layer curvature with input and output factors that they accumulate across steps, whereas Muon orthogonalizes the current momentum and stores no curvature matrix. The algebraic link between Muon and Shampoo without accumulation does not show whether Muon's momentum statistics follow the curvature of trained networks. We study Muon as an accumulation-free Kronecker-factored curvature approximation with two substitutions: the input factor is read from the Gram of the current momentum, and the output factor is replaced by the identity. A signal and noise decomposition explains when the momentum Gram follows the leading curvature subspace and separates subspace recovery from spectral accuracy. Measured against a sequence-level generalized Gauss-Newton reference in GPT-2, the momentum Gram captures most of the leading input curvature energy, whereas the output factor is nearly diagonal pairwise but has unequal diagonal scales and collective structure. Controlled quadratic and nonlinear experiments show that estimation noise, unequal output scales, and output coupling cause distinct failures. These results support a conditional curvature interpretation of Muon and specify where it falls short.