Nonlinear Fourier amplitude budget regularizer

Attach the regularizer to the terminal transmission coefficients of the SU(1,1) scan described above, or to any differentiable discretized scattering layer. For each batch example, run the scan over positions k and frequencies j, obtain a_j, and compute H_j=sqrt(clamp(log(abs(a_j)^2+epsilon), min=0)). Choose p=1.5 initially, so p'=3, and use uniformly spaced frequencies with quadrature weight Delta_xi. Compute input_budget=(h*sum_k(abs(f_k)^p))^(1/p) and spectral_budget=(Delta_xi*sum_j(H_j^pprime))^(1/pprime). Add lambda*relu(spectral_budget-input_budget)^2 to the task loss; detach neither quantity, so the network learns to reduce excess spectral concentration or increase the appropriate input representation norm. Use the identity residual invariant_error=abs(abs(a)^2-abs(b)^2-1) as a separate diagnostic, not as the main loss. The exact mathematics supplies the one-sided budget and the logarithmic amplitude; finite frequency truncation, quadrature error, epsilon, and lambda are empirical approximations. Start with a small character-level language model or sequential-MNIST classifier using 8-16 frequencies and a 64-dimensional potential generator. Compare no regularizer, ordinary L2 on f, and this hinge penalty at matched training loss. Look for reduced validation loss, less sensitivity to sequence length and learning rate, and lower out-of-band spectral spikes at equal accuracy. A negative result is clear if the constraint activates frequently, harms training loss, or gives no generalization or stability improvement over L2.