Parseval Scattering Stem with Certified Depth

Integrate the stem before the learned CNN or ViT patch embedding. For an input batch x with d=2, choose K complex or real convolution filters psi_lambda and one low-pass phi. First construct an unconstrained bank in the Fourier domain, then normalize every frequency bin by q(xi)=sqrt(|phi_hat(xi)|^2+sum_lambda|psi_hat_lambda(xi)|^2+epsilon), replacing each multiplier by its value divided by q; separately ensure phi_hat(0) is nonzero. Implement paths breadth-first: initialize paths with U_empty=x; for depth m=0,...,N-1, compute y_lambda=abs(conv_fft(U_path,psi_lambda)), append conv_fft(y_lambda,phi) to the feature tensor, and retain y_lambda only if its measured energy justifies expanding the path. Use the residual estimator r_N^2=max(0, ||x||_2^2-sum_{|p|<N}||S[p]x||_2^2), evaluated on a minibatch; stop expanding when r_N^2/||x||_2^2 is below a target such as 1%. The Parseval partition and modulus inequality come from the mathematics; the finite-grid FFT normalization, boundary handling, and the constant C_f are empirical engineering choices. Start with CIFAR-10, a 3-depth 8-channel-per-layer scattering stem, and compare against a same-FLOP learned ConvNet stem and a standard fixed wavelet scattering baseline. Measure energy drift, gradient norm variance, accuracy at equal parameter count, and accuracy versus depth. Success means near-zero energy-accounting error, no exploding gradients, and equal-or-better accuracy with fewer learned front-end parameters; a secondary signal is that depth chosen by the residual estimator avoids accuracy loss while reducing convolutional path count.