The math is sound. The method is not. We recast the azimuthal mode-coupled surface equations as a block-Toeplitz convolution in the mode index and diagonalize this coupling via a discrete Fourier transform in \(n\), then briefly review the basic definition and core properties of the DFT used in this step.

The math is sound. The method is not. claim: \[ \begin{aligned} & {\widehat{\left(\partial_x f\right)_n}}(k)=\frac{k}{2}\left(\hat{f}_{n-1}(k)-\hat{f}_{n+1}(k)\right) \\ & {\widehat{\left(\partial_y f\right)_n}}(k)=\frac{k}{2 i}\left(\hat{f}_{n-1}(k)+\hat{f}_{n+1}(k)\right) \end{aligned}\tag{1} \]