Gradient-Side Error-Feedback SignMuon

Integrate this into the Muon optimizer at the gradient-to-update boundary for every 2-D weight matrix; leave biases, layer-normalization parameters, and 1-D tensors on AdamW or ordinary SGD. On each optimizer step, first form the usual Muon momentum matrix G from the current minibatch gradient. Maintain a full-precision residual R of identical shape locally on each communicating worker or centrally on a parameter server. Use: H = G + R; scale = mean(abs(H)); bits = sign(H); transmit(bits, scale); Hq = scale * bits; R = H - Hq; U,S,Vh = svd(Hq); D = U @ Vh; W -= lr * D. For distributed training, test both local error-feedback followed by sign aggregation and signing the averaged compensated matrix, because these operations do not commute. The mathematical components are the residual recursion, entrywise compressor, and polar LMO; estimate the scale from the current tensor and compute the polar factor with stable SVD or Newton-Schulz iteration. Keep Muon's momentum decay and learning-rate schedule unchanged initially. First run a ResNet-18 on CIFAR-10 and a small Transformer on a language-modeling subset. Compare full Muon, SignMuon without residuals, post-LMO sign compression, and this method at equal communicated bits and equal FLOPs. Log loss and gradient norm versus optimizer steps, final validation accuracy, residual norm ||R||_F/||G||_F, and communication volume. Success means slower residual growth and faster loss descent than sign-only Muon, with accuracy approaching full Muon at one bit per matrix entry.