Fourier moment-capped cyclic layers

Integrate this into a cyclic 1D convolution, block-circulant linear layer, or attention module whose sequence dimension has length ell. The layer should expose its channel-mixing kernels as tensors with shape [out_channels, in_channels, ell], or as a shorter kernel embedded into an ell-point circular convolution. First compute the unitary FFT of the kernel along the cyclic dimension, producing W_j for every frequency j. Form M_j=W_j W_j^*+epsilon I without materializing the ell-by-ell convolution matrix. For the soft-regularization variant, compute bar_mu=(ell r_1)^(-1) sum_j tr(M_j) using stop-gradient, compute R_4=(ell r_1)^(-1) sum_j tr((M_j/bar_mu)^4) by matrix multiplications, and add lambda R_4 to the task loss. For a hard stability variant, estimate the largest eigenvalue of each M_j with two or three power iterations and apply the supplied rescaling s_j to the Fourier block, followed by inverse FFT before the ordinary forward pass. The exact mathematics supplies the block decomposition and trace moment; lambda, tau, and epsilon are empirical hyperparameters. Start with a small circular CNN on CIFAR-10 or a tiny structured Transformer on WikiText-2, comparing ordinary weight decay and spectral normalization at equal FLOPs. Measure gradient spikes, per-frequency gain, validation loss, and FFT overhead. Success means fewer instability events and lower validation loss or faster loss descent with less than 10% overhead.