Muon-C: Operator-Aligned Muon for Convolutional Kernels

Muon replaces matrix momentum with an approximately orthogonal polar direction, but its geometry depends on the matrix representation. For convolution, standard unfolding describes a local patch map rather than the convolution operator. We introduce Muon-C, an operator-aligned optimizer that represents kernel momentum as frequency-wise channel-transfer matrices, polarizes these blocks independently, and uses a critical Fourier grid to return updates exactly to the original finite kernel support. We show that the new geometry arises from combining the block partition and Fourier coordinates. The exact-polar direction is a linear minimization oracle under the critically sampled convolution norm. Its worst-case guarantee relative to the continuous convolution-operator norm is never weaker than unfolding and is strictly stronger for 3 times3 kernels. On CIFAR-10 flow matching with matched applied-update RMS, Muon-C reaches 9.87 FID at 40k iterations, compared with 22.26 for unfolded Muon and 51.31 for Adam. It reaches their final quality using 0.62 times and 0.64 times their model FLOPs, respectively. Under equal tuning budgets, Muon-C achieves 3.42 FID. Gains persist across data scales and transfer to classification across convolutional architectures.