Adjoint spectral opening for stable pooling

Use this as a pooling or skip-connection operator in a frequency-aware U-Net or CNN. At a chosen resolution, compute X=fft2(x), magnitude f=abs(X), and phase p=X/(f+eps). Parameterize a positive spectral structuring element psi=softplus(theta)+psi_min; initialize theta so psi is large outside the desired passband and near one inside it. Compute the erosion e=f/(psi+eps), then the adjoint dilation d=e*(psi+eps). In exact arithmetic d=f, so a literal same-grid quotient/multiplication pair is only a validity test, not a useful pooling layer. To make the opening nontrivial, insert a low-resolution projection P between them: downsample e to a coarse Fourier grid, optionally apply a nonnegative max/average aggregation, upsample it with U, and set o=U(P(e))*(psi+eps), clipping o to f to retain anti-extensivity. Alternatively define delta as a learned supremum over neighboring frequency bins, then verify the adjunction numerically. Reconstruct y=real(ifft2(o*p)), and use y as the pooled branch or as a filtered skip feature. The code should test ||gamma(gamma(f))-gamma(f)||, gamma(f)<=f, and monotonicity on random nonnegative f. First cheap experiment: compare a standard stride-2 convolution, average-pool-plus-convolution, and this opening block in a small U-Net on CIFAR-10 or 128x128 BSD denoising. Track aliasing under frequencies above the post-stride Nyquist limit, validation accuracy or PSNR, and repeated-application drift. Success means lower high-frequency alias energy and materially smaller representation drift when the opening is applied twice, with no worse accuracy at equal memory.